an initial decoded codeword is reprocessed in a particular structured strategy based on the reordering. Such strategy confines the search to a usually small class ...
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[5] F. C. Chow and J. L. Hennessy, “The priority-based coloring approach to register allocation,” ACM Trans. Program. Lang. Syst., vol. 12, pp. 501–536, 1990. [6] T. Etzion, “New lower bounds for asymmetric and unidirectional codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 1696–1704, 1991; vol. 38, pp. 1183–1184, 1992. [7] T. Etzion and S. Bitan, “On the chromatic number, colorings, and codes of the Johnson graph,” Discr. Appl. Math., vol. 70, pp. 163–175, 1996. ¨ [8] T. Etzion and P. R. J. Osterg˚ ard, “Greedy and heuristic algorithms for codes and colorings,” Tech. Rep. CS 909, Comput. Sci. Dept., Technion, Haifa, Israel, 1997. [9] G. Fang and H. C. A. van Tilborg, “Bounds and constructions of asymmetric or unidirectional error-correcting codes,” Appl. Algebra Eng. Commun. Comput., vol. 3, pp. 269–300, 1992. [10] T. A. Feo, M. G. C. Resende, and S. H. Smith, “A greedy randomized adaptive search procedure for maximum independent set,” Oper. Res., vol. 42, pp. 860–878, 1994. [11] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: Freeman, 1979. [12] M. R. Garey, D. S. Johnson, and H. C. So, “An application of graph coloring to printed circuit testing,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 591–599, 1976. [13] F. Glover, “Tabu search—Part I,” ORSA J. Comput., vol. 1, pp. 190–206, 1989. [14] A. Hertz and D. de Werra, “Using tabu search techniques for graph coloring,” Computing, vol. 39, pp. 345–351, 1987. ¨ [15] I. S. Honkala and P. R. J. Osterg˚ ard, “Code design,” in Local Search in Combinatorial Optimization, E. Aarts and J. K. Lenstra, Eds. New York: Wiley, 1997, pp. 441–456. [16] D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, “Optimization by simulated annealing: An experimental evaluation; Part II, graph coloring and number partitioning,” Oper. Res., vol. 39, pp. 378–406, 1991. [17] R. J. McEliece and E. R. Rodemich, “The Constantin–Rao construction for binary asymmetric error-correcting codes,” Inform. Contr., vol. 44, pp. 187–196, 1980. ¨ [18] K. J. Nurmela, M. K. Kaikkonen, and P. R. J. Osterg˚ ard, “New constant weight codes from linear permutation groups,” IEEE Trans. Inform. Theory, vol. 43, pp. 1623–1630, Sept. 1997. [19] C. L. M. van Pul and T. Etzion, “New lower bounds for constant weight codes,” IEEE Trans. Inform. Theory, vol. 35, pp. 1324–1329, 1989.
Reliability-Based Syndrome Decoding of Linear Block Codes Marc P. C. Fossorier, Member, IEEE, Shu Lin, Fellow, IEEE, and Jakov Snyders, Member, IEEE
Abstract—In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRB-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among ( i ) candidates, where is the dimension of the code, for an order of reprocessing. At practical bit-error rates, for codes of length 128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to +1 errors with an order of reprocessing of at most 2, where is the error-correcting capability of the code considered.
oK
K
t
t
N
i
Index Terms—Block codes, generalized Hamming weights, soft-decision decoding, syndrome decoding.
I. INTRODUCTION Recently, several reliability-based maximum-likelihood decoding (MLD) algorithms for binary linear block codes have been proposed [1]–[11]. These algorithms first reorder the symbols, within each received block, according to their reliability measures. Thereafter, an initial decoded codeword is reprocessed in a particular structured strategy based on the reordering. Such strategy confines the search to a usually small class of candidates of high likelihood, whereby the decoding complexity is significantly reduced. Basically, two general different approaches exist, namely, the vector space associated with either the code considered C or with its dual space C ? are processed. These two spaces are, respectively, referred to as the G-space and the H -space of the code. In [7] and [11], an algebraic decoder generates the successive candidates processed by the algorithm. The vectors entered into the decoder are obtained by systematically adding to the (bit-by-bit hard detected version of the) received sequence error patterns whose supports are confined to positions where the detection is of relatively low reliability. These reprocessing methods yield codewords to be scored, hence the algorithms are of the G-space type. In a different kind of G-space algorithms [6], [8]–[10] an equivalent code is first determined by identifying the K most reliable independent positions Manuscript received February 1, 1996; revised June 24, 1997. This work was supported by the National Science Foundation under Grant NCR-9115400 and Grant NCR-94-15374, and in part by the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities. The material in this correspondence was presented in part at the IEEE Symposium on Information Theory, Whistler, BC, September 1995. M. P. C. Fossorier and S. Lin are with the Department of Electrical Engineering, University of Hawaii, Honolulu, HI 96822 USA. J. Snyders is with the Department of Electrical Engineering-Systems, TelAviv University, Tel-Aviv 69978, Israel. Publisher Item Identifier S 0018-9448(98)00072-8.
0018–9448/98$10.00 1998 IEEE
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associated with the received sequence, where K is the dimension of the code. The codeword which concurs with bit-by-bit hard detection at these positions is then reprocessed based on a systematic search guided by a heuristic cost function. The methods presented in [1]–[5] are in a sense dual to the previous methods since they reprocess an H -space corresponding to the N 0 K least reliable independent positions, where N is the length of the code. However, while the algorithms in the G-space allow efficient decoding of a large variety of block codes of various rates and quite large lengths, all previously published reprocessing methods of the H -space require the codimension N 0 K to be relatively small, as otherwise the number of the column patterns to be scored is extremely large. In this correspondence, we first exhibit the duality between the G-space and H -space approaches in an explicit form. Based on this duality, we then describe a structured search of the H -space, in a way that the two approaches can be unified within the same framework. We then further address maximum-likelihood (ML) soft-decision syndrome decoding in the H -space defined by the least reliable independent positions. Let N 0K be the basis for the column space of a check matrix of the code specified by these positions. Consider the bit-by-bit hard-detected version of the received sequence and the syndrome associated with it. We determine sufficient conditions, based on the support of the syndrome, for this vector to be either identified as the optimum codeword or modified to the optimum codeword by a restricted search. Contrarily to stopping criteria employed in the MLD algorithms of [6]–[13], our new criteria do not require any soft informations and can be evaluated after a partial ordering of the received sequence only. In particular, many real value operations are saved in comparison with algorithms based on the dual approach of [8] and [10]. Since the syndrome-based tests are code-dedicated, strategies for their effective application to different classes of codes are then devised. This results in decoders with low computational complexity. For example, MLD of the (24; 12; 8) Golay code is accomplished with an average number of 13 real value operations per received block at the bit-error rate (BER) of 1006 . Finally, a new reprocessing algorithm is analyzed. After the determination of the least reliable basis N 0K this algorithm no longer requires real-value operations as it simply computes syndromes and tests their Hamming weight. For all simulated codes, a performance within 1.5 dB of the optimum BER has been achieved, even for long codes. As a result, at the BER of 1006 , a degradation of less than 1 dB with respect to the maximum-likelihood performance is achieved for the (128; 64; 22) extended BCH code with an order of o(K 3 ) syndrome computations. On the other hand, a coding gain exceeding 2 dB with respect to algebraic decoding is achieved. We emphasize that after ordering, our algorithm processes mainly binary operations. Therefore, in terms of decoding complexity, it is closer to algebraic decoding than to conventional MLD, such as the Viterbi soft-decision decoding. This fact is particularly consequent for practical applications where an l-bit quantizer is used, since the ordering is directly obtained from the contents and the cardinalities of the 2l sets of quantized received symbols [14]. The paper is organized as follows. A duality between the Gspace and H -space approaches as well as the resulting maximumlikelihood soft-decision syndrome decoding algorithm are presented in Section II. Sufficient conditions for the hard-decision vector to be the optimum codeword are derived in Section III and, for particular classes of codes, corresponding reprocessing strategies are proposed in Section IV. These conditions are then generalized to other candidate codewords expressed in terms of N 0K in Section V. Finally, the new hard-decision decoding algorithm based on N 0K is analyzed in Section VI and some concluding remarks are provided in Section VII.
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II. MAXIMUM-LIKELIHOOD SYNDROME DECODING BASED ON THE NOISE STATISTICS AFTER ORDERING A. Definitions
Let C be an (N; K; dH ) binary linear code with length N , ~ and dimension K , and minimum Hamming distance dH . Let G ~ H be generator and parity-check matrices of C , respectively. Assume that C is used for error control of information transmitted through a binary-input memoryless channel which, for simplicity of exposition and without essential loss of generality, is assumed to be an additive white Gaussian noise (AWGN) channel. Let c = (c1 ; c2 ; 1 1 1 ; cN ) be a codeword in C . For binary phase-shift keying (BPSK) transmission, the codeword c is mapped into the bipolar sequence x = (x1 ; x2 ; 1 1 1 ; xN ) with xi = (01)c 2 f61g. The demodulator output, hereafter called the received sequence, is r = (r1 ; r2 ; 1 1 1 ; rN ) with ri = xi + wi , where fwi : i 2 f1; 1 1 1 ; N gg are statistically independent Gaussian random variables with zero mean and variance N0 =2. If a hard decision (HD) is performed on each ri independently then the natural choice of reliability measure is jri j since, for bipolar signaling, this value is proportional to the magnitude of the log-likelihood ratio associated with the hard decision applied to the ith symbol. We first rearrange the components of the received sequence r in nondecreasing order of reliability values. The resultant sequence is denoted y = (y1 ; y2 ; 1 1 1 ; yN ), with jy1j jy2j 1 1 1 jyN j. This reordering defines a permutation function 1 for which y = 1 (r ). A second permutation 2 is then applied to the ordered received 0 ) = 2 (y ), the last sequence y such that, for z 0 = (z10 ; z20 ; 1 1 1 ; zN ~ K columns of 2 (1 (G)) form a basis BK for the column space ~ ) = GF (2)K of G~ and, for each i 2 f1; 1 1 1 ; N 0 K g, both cs (G conditions
0i =
0j
j2S (i) and for all j 2 S 0 (i) are fulfilled, where S 0 (i) fN 0 K +1; 1 1 1 ; N g and 0l represents the lth column of 2 (1(G~)).
jzi0 j
jzj0 j
~)) are arranged such that Also, the columns of 2 (1 (G
jzN0 0K j jzN0 0K j 1 1 1 jzN0 j: +1
+2
Consequently, the rightmost K positions are the location set L(BK ) ~ )) is of the ordered basis BK . Then by row operations 2 (1 (G brought into the systematic form G = [Q0 IK ], hence the columns of IK constitute the basis BK . The last K components of z 0 are called the most reliable independent (MRI) symbols of z 0 and BK is referred to as the (ordered) most reliable basis (MRB) (for ~ ) = cs (G) = GF (2)K ). The permutation (2 1 ), which cs (G establishes a one-to-one correspondence between the equivalent codes corresponding to the noisy sequences z 0 and r , is described in the foregoing as a composition of two permutations to suit the viewpoint of a convenient implementation. Also, this description emphasizes the distinction between the ordering of positions 1 utilized in [7] and [11] and the one employed here and in [6] and [8]–[10]. Another ordered basis, N 0K , is utilized by the syndromedecoding algorithm of [2]. It is the least reliable basis (LRB) for ~ ) = GF (2)N 0K of H~ . The LRB can be the column space cs (H constructed by applying a permutation 2 to y , to obtain the sequence
z = (z1 ; z2 ; 1 1 1 ; zN ) = 2 (y ) ~ )) forming a basis of cs (H~ ) with the first N 0K columns of 2 (1 (H and such that, for each i 2 fN 0 K + 1; 1 1 1 ; N g, both conditions i = j and jzi j
j2S(i)
jzj j for all j 2 S (i) are satisfied, where S (i) f1; 1 1 1 ; N 0 K g and l represents the lth column of ( (H~ )). 2
1
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~ )) are arranged such that z1 Also, the columns of 2 (1 (H jz2 j 1 1 1 jz 0 j. Then the leftmost N 0 K positions are the location set L( 0 ) of the ordered basis 0 . It follows that j
N
H N
= [I
N
0K
j
K
0 Q] is the systematic form of 2 (1 (H~ )) and the first N
K
N
G = [QT IK ]
K
K
components of z are called the least reliable independent (LRI) symbols of z . In the following, we first establish a duality between the MRB BK for the column space of G, associated with a received sequence, and the corresponding LRB N 0K for the column space of H . Based on this duality, we then present an algorithm equivalent to that of [8] but expressed in terms of the LRB associated with the received sequence. This algorithm may be viewed as a method to implement the maximum-likelihood decoding via the approach of [1]–[5], as it provides an efficient and structured search on the columns of the systematic parity-check matrix H .
B. Duality Between the MRB and LRB Assume for a moment that during the formation of BK no linear dependence is encountered, i.e., the K most reliable symbols happen to be associated with a linearly independent set of columns of G. Recalling that the complement of an information set is an information set of the dual code, we deduce that the location sets L BK and L N 0K are complementary. The following Theorem 2.1 states that this duality holds for any received sequence. However, 2 1 r and 2 1 r are not necessarily equal even under the previous assumption. If during the formation of 2 1 G and 2 1 H the additional conditions
~
(
1 2 111; i
than the reliability associated with any of the columns ; ; of IN 0K . This implies that in the generator matrix
( ) ( ( )) ( ( ~ ))
)
( ( ))
( ( ~))
jz10 j jz20 j 1 1 1 jz0 0 j N
and
jz 0 N
K
K
( ( )) = ( ( )) =
respectively, are also maintained then 2 1 r 2 1 r , and the generator matrix G and the parity-check matrix H constructed in QT , where Section II-A determine the same code, or simply Q0 the superscript T stands for transposition. Theorem 2.1: The complement of the location set of most reliable basis BK is the location set of the least reliable basis N 0K , i.e.,
L(B ) = L( 0 ): K
c
N
(1)
K
Proof: Assume, without loss of generality, that C is specified by the following parity-check matrix:
H
= [I
N
Q]
0
K
(2)
where the columns of IN 0K constitute the LRB N 0K , with nondecreasing reliability from left to right. It can also be assumed without loss of generality that the rightmost K positions are permuted such that
Q = [Qi
Qi
111
Qi
]
(3)
= [q
i
1 qi 2
1 1 1 q ];
` = 1; 2; 1 1 1 ; v
i j
3( ) =
satisfy the following conditions: all the members of each i` ; q ; 1 1 1 ; q g have their last -entry at position i from the ` i 1 i 2 i j top and
1
fq
( )
3( ) = 1 2
( )
C. Decoding Algorithm Assume hereafter that the parity-check matrix of C is given by (2), where the columns of both IN 0K and Q, constituting the LRB and the MRB, respectively, are arranged in nondecreasing order of reliability from left to right. From the sequence b that represents the vector obtained from the HD decoding of z , we compute the syndrome
s = b HT
= b I Q0 N
K
(5)
T
defined here as a row vector. As stated in [1], the MLD rule becomes: “If s 6 0, find the set L of columns of H , such that the sum of these columns is s and the sum of the decoding costs associated with these columns is minimized among the sums of the decoding costs of any e1 ; e2 ; 1 1 1 ; eN with collection of columns summing to s .” Set e ei if i 2 L and ei otherwise. Then
=
=(
=0
N
1(e) = j
=1
1 i1 < i2 < 1 1 1 < i = N 0 K (the equality i = N 0 K holds provided there is no location where all the codewords have a zero entry). Then for each ` = 1; 2; 1 1 1 ; v the reliability at the position of any element of 3(i ) is not smaller v
v
`
)
ej jzj j
(6)
is the decoding cost associated with L and b 8 e is the most likely (ML) codeword. For l K , the order-l reprocessing of b , expressed in terms IN 0K Q , is defined as follows: of H
0 =[ ] For 0 i l, sum all possible collections L of i rows of Q . Let s 2 = (s2 1 ; s2 2 ; 1 1 1 ; s2 0 ) be such sum. Construct the vector s1 = s 8 s2 , and then the codeword b 8 [e 1 e 2 ], where e1 = s1 and e2 =(e2 1 ; e2 2 ; 1 1 1 ; e2 ) with e2 =1 if j 2 L and e2 = 0 otherwise. For each reconstructed codeword, determine its corresponding BPSK sequence x . Compute the squared Euclidean distance d2 (z ; x ) between the T
i
;
;
;N
;
i
K
;
;K
;j
;j
ordered received sequence z and the signal sequence x, and record the codeword a 3 for which x 3 is closest to z . When all the li=0 Ki possible codewords have been tested, order-l reprocessing of b is completed and the recorded codeword a 3 is the final decoded codeword.
0
where the submatrices
Qi
(4)
that corresponds to H of (2), the collection of columns of IK forms the MRB BK . Remark: The basis of cs G obtained in the proof of Theorem 2.1, namely, the collection of the columns of IK , is identical to BK in the sense of equality of sets; to construct the ordered basis BK it remains to sort the reliabilities and permute the positions associated i` ; ` ; ; 1 1 1 ; v. with each of Based on Theorem 2.1, we proceed to formulate the algorithm of [8] in terms of the LRB of the cs H .
=1
+1 j jzN 0K +2 j 1 1 1 jzN j
`
For i l, the procedure of summing all the possible collections of i rows of QT , reconstructing the corresponding codewords, and computing their squared Euclidean distances from z is referred to by the name phase-i of the order-l reprocessing. Note that by Theorem 2.1, phase-i reprocessing considers the sum of any collection of i rows of QT . Then it adds the obtained N 0K -tuple to the syndrome s to form the vector s1 . Upon completion of order-l reprocessing, the recorded codeword differs from the ML codeword if and only if more than l HD errors are present in the K MRI positions of the ML solution. The order-l reprocessing algorithm of [8] based on G is therefore equivalent to the foregoing order-l reprocessing of H , hence both have the same error performance.
(
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As presented above, order-l reprocessing can be viewed as a structured version of the algorithm of [14]. In [14], the error patterns e are tested with monotonically increasing values K
12 (e) = j
=1
e2; j jzN 0K+j j
so that the order in which each e is reprocessed depends on each received sequence r . Based on the results of [8], our structured search in successive stages processes the error patterns e in an order which remains close to that corresponding to the monotonicity of 2 e . In addition, a tight upper bound on the error performance associated with each reprocessing phase is now available [8]. A different structured search based on similar principles is also proposed in [10]. The effectiveness of the decoding by algorithm of [8] relies on a powerful stopping criterion, called resource test, whose efficiency increases as the algorithm progresses. There is a dual version of the resource test of [8], derived in Appendix I. Therefore, the foregoing algorithm, described in the dual space, is equivalent to the algorithm of [8], both from the performance and the computation points of view. Its only advantage is that whenever N 0 K < K , a smaller matrix is stored and processed. Also, the derivation of the resource test is simpler with this approach since b is the starting point of the reprocessing algorithm while in [8], a cost difference with respect to b is carried out. This new version of the reprocessing procedure determines a fast search method valid for any binary linear code for MLD, performed in the dual space of the code. However, in comparison with the method of [2], we observe the following inefficiencies. The proposed algorithm involves real-number computations, whereas in [2] integervalued computations are used to a large extent. Also, a decoder devised on the basis of the approach of [2] necessarily takes into account the structure of the code. Whereas decoders with a wide range of applicability are desirable for some applications, they are in general considerably less efficient than decoders that exploit the structure of the individual code. In the following sections, we propose ways to overcome the previously mentioned inefficiencies. First orderreprocessing is considered.
1()
0
0
III. ORDER- GENERALIZED SYNDROME DECODING If s is the syndrome associated with the bit-by-bit hard detected version b of the reordered received sequence z then, based merely 0 is the only condition which guarantees that MLD on s, s has been accomplished (i.e., b is the most likely codeword). In this section we show that there are other cases where examination of the support of the syndrome s, in relation to the ordered L N 0K , either guarantees optimality of order- i 0 reprocessing, or implies a restricted search for phase-i reprocessing.
=
(
( 1)
A. Covering Properties
1
)
Let hi 2 N 0K ; i 2 f ; 1 1 1 ; N 0K g be indexed in nondecreasing order of reliability. Assuming s 6 0, expand s in terms of the LRB N 0K , i.e.,
s=
w
j
=1
hp ;
supp( ) =
=
where
p1 > p2 > 1 1 1 > pw :
1
(7)
Then ws and s fpj ; j 2 f ; 1 1 1 ; ws gg are the Hamming weight and the support of s, respectively. The results of this section are based on the notion of “covered positions.” s is said to be covered for Definition 3.1: A position pj 2 phase-i reprocessing if for every codeword c b 8 e constructed e such by phase-i reprocessing, there exists a position p 2
supp( )
=
supp( )
391
that jzp j jzp j. The syndrome s is called a covered syndrome for phase-i reprocessing if, for every c constructed by the reprocessing, s are covered by distinct positions of all the positions of e ; e c 8 b. For example, for phase-i reprocessing, the last nonzero position p1 of the syndrome is always covered. Indeed, phase-i reprocessing either includes a position p such that jzp j jzp j or no such position sp , hence p1 is self-covered. exists, in which case ep Theorem 3.1: If the syndrome s is covered for phase-i reprocessing, then phase-i reprocessing of b does not improve orderdecoding. Proof: The proof follows by the definition of a covered synb 8 e0 be the codeword recorded by orderdrome. Let a0 reprocessing, with decoding cost e0 associated with e0 s 0 . e0 at positions j By (6), the codeword c 3 b 8 e 3 improves where ej3 and sj and increases e0 at positions where ej3 and sj ; while if ej3 sj , the cost contribution from position j is unchanged. Since the syndrome s is covered for phase-i reprocessing, it is guaranteed that for each position j such that ej3 and sj , there exists at least one distinct position p 2 f ; 1 1 1 ; N g for which ep 3 and e0; p with jzp j jzj j. Therefore, e 3 e0 , which completes the proof. Example 3.1: Based on the ideas conveyed by the previous proof, it is readily seen that for phase- reprocessing, the syndrome s is covered if dH . Remark: In the sense of the N 0K -based partial ordering imposed on the collection of the error patterns, introduced in [2]–[4], the condition of Theorem 3.1 means that e 0 is the only minimal element among the error patterns corresponding to phase-i reprocessing, whereby a 0 is at least as likely as any other codeword obtained by the reprocessing. Let us address the extent of the covering of the positions in terms of constructing the LRB. If the first N 0 K columns of H examined during this construction are linearly independent, then the first N 0 K reliabilities jzj j are smaller than any of the last K reliabilities jzj j, hence all p1 ; p2 ; 1 1 1 ; pi are covered for order-i reprocessing. However, the conclusion does not apply if among the columns encountered there are some which are linearly dependent on the preceding columns, i.e., if positions of low reliability are excluded from the support of the LRB. We first present a necessary condition for a position pj , with j i, to be uncovered for phase-i reprocessing. Theorem 3.2: For phase-i reprocessing, the position pj , j i; is uncovered only if prior to identification of the pj th member of the occasions of linear least reliable basis N 0K at least i 0 j 0 dependence of the columns of H are observed. Proof: Assume that < i 0 j 0 -dependent columns have been permuted before constructing the pj th LRP. Then, for phasei reprocessing, at least i 0 positions p satisfy jzp j jzp j with i 0 j . Hence pj is covered. Theorem 3.2 implies that, in general, the positions pj that are covered depend on the received block. Nonetheless, it is possible to derive some conditions that guarantee the covering of positions. These conditions are based on the generalized Hamming weights dr of the code (see [15]). Theorem 3.3: If for j 2 f ; 1 1 1 ; i 0 g
supp( )
supp( ) =
=
=1
0
^
^ =
=0
=1
1(^ )
= =1
=0
=1 =1 1( ) 1(^ )
=
^
1(^ ) ^ 1( )
1
=0 1
(1010010000)
0 ^ =[ ]
=
6
=0
^
^
2
(
1
pj +1 < di0j
( 1) 1)
1 0i+j+1
(8)
then the position pj +1 is covered for phase-i reprocessing. Proof: If pj +1 is uncovered for phase-i reprocessing then, by Theorem 3.2, at least i 0 j occasions of linear dependence have occurred before the pj +1 th dimension of the LRB is processed. Assume i 0 j such occurrences for . In [10] it is shown
(
(
+ )
)
0
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that the rth occasion of dependence cannot occur before processing the dr th position of the ordering defined by the permutation 1 . It follows that
(pj +1 0 1) + (i 0 j + ) di0j+ :
and either
ws
i or, in case that ws i + 1, max fpl + 2(l 0 1) + 1g dH l2fi+1;111;w g
(10)
(11)
then order-(i 0 1) reprocessing is optimum. Since the first condition of Theorem 3.4 guarantees that pj +1 for all j 2 f1; 1 1 1 ; i 0 1g are covered in case that dependent columns are permuted when constructing the LRB, it can be dropped whenever no such columns are encountered. Also, for i = 1 no such condition exists. If the generalized Hamming weights dr of the code considered are unknown, lower bounds on dr can be substituted for them in Theorem 3.4. Theorem 3.4 can be viewed as a hard-decision version of the first part of the resource test (38) derived in Appendix I. Therefore, Theorem 3.4 is not as efficient as the resource test of Section II-C. On the other hand, it has the advantage of requiring no additional computations. Note finally that Theorem 3.4 can replace [16, eq. (3)], where the author assumed that any K positions form an information set. This fallacy was first mentioned in [17]. From the preceding theorem we deduce two corollaries. Corollary 3.1: If for all j 2 f1; 1 1 1 ; i 0 1g
pj +1 < di0j 0 i + j + 1
FOR
jS
0 (1)j = a(N 0 K ) 0 b
(9)
The proof is completed by taking the contrapositions of (9) and by utilizing the fact that dr+ dr + for 0. For phase-i reprocessing, the set S0 (i) of covered syndromes is determined by the following theorem, whose proof is deferred to Appendix II. Theorem 3.4. Covering Test: If for all j 2 f1; 1 1 1 ; i 0 1g
pj +1 < di0j 0 i + j + 1
TABLE I COEFFICIENTS (a; b)
(12)
and either ws i or pi+1 dH 0 ws 0 i, then order-(i 0 1) reprocessing is optimum. Corollary 3.2: If ws > dH =2, then there exists no i < dH =2 + 1 for which order-(i 0 1) reprocessing satisfies the covering test of Theorem 3.4. Both corollaries follow from Theorem 3.4: consider x ws in (39) of the proof of Appendix II for Corollary 3.1 and l = ws in (11) for Corollary 3.2. Corollary 3.1 is a weaker version of Theorem 3.4, but it allows to check rapidly whether the covering test is satisfied for the most likely syndromes. In general, practically optimum error performance is achieved with an order of reprocessing much lower than dH =2 [8]. Corollary 3.2 implies that whenever ws > dH =2, the covering test becomes useless to guarantee optimum MLD. Example 3-2: For the (8; 4; 4) Reed–Muller (RM) code, S0 (1) contains the all-zero vector, the four 4-tuples of weight 1 and the three 4-tuples of weight 2 starting by 1. Consequently, 8 out of 16 syndromes satisfy the covering test for i = 1. For these 8 syndromes, Theorem 3.1 implies that order-0 reprocessing achieves MLD. In fact, for any code with dH = 4, Theorem 3.4 is satisfied only by syndromes with ws 2 for i = 1. Based on this example, it follows that for such codes, jS0 (1)j = 2(N 0 K ). Extended Hamming codes fall into this category of codes. We observe that for i = 1, Theorem 3.4 depends only on dH and, implicitly, on N 0 K . Therefore, for a fixed dH , a straightforward generalization of the previous example implies that the cardinality jS0 (1)j of the set S0 (1) can be written as jS0 (1)j = a(N 0 K ) 0 b, for N 0 K dH 0 1. Table I lists the pairs of coefficients (a; b)
for even dH in the range 2 dH 16. For the (24; 12; 8) Golay code, we obtain jS0 (1)j = 170, which represents about 4% of the total number of syndromes. However, the corresponding syndromes are also the most probable to occur. For the (64; 45; 8) extended BCH code, Table I shows that jS0 (1)j = 310. Example 3-3: Consider the (24; 12; 8) Golay code, and any syndrome of weight 2. Recalling that d1 = 8 and d2 = 12 for this code, Theorem 3.4 implies that order-0 is optimum for p2 5, order-1 is optimum for p2 7, and order-2 is optimum for p2 10. B. Computation Savings 1) Partial Ordering: In [8] and [10], a complete ordering of the received sequence corresponding to the joint determination of BK and N 0K is realized in order to implement efficient stopping criteria as derived in Appendix I. However, as in [2], a partial ordering is sufficient to construct the LRB. This requires the search of at most ? + 1 columns N 0 d?H + 1 minima since no more than K 0 dH are switched by the permutation 2 [8], [10]. If we choose not to order the columns of Q, dependent columns can be automatically discarded at each step of the construction of the LRB, so that exactly N 0 K minima are searched. With this method, comparisons are saved but the resource test of Appendix I is no longer applicable and, in general, all collections of columns of Q corresponding to the order of reprocessing i considered have to be searched. However, the syndromes satisfying Theorem 3.4 not only require no additional computation, but also are the more likely syndromes. Therefore, whenever the average decoding cost of the algorithm of Section IIC is dominated by the reordering, both the maximum and average numbers of computations are improved by partial ordering only. Such is the case for medium-rate codes of small dimension, as the (24; 12; 8) Golay code, and high-rate codes of small-to-medium dimensions. 2) Iterative Syndrome Computation: When constructing the LRB by the algebraic procedure described in [8] and [18], N 0 K steps are required. At step j , j = 1; 1 1 1 ; N 0 K , we process the j th dimension of the parity-check matrix H j 01 delivered by step j 0 1 with elementary row operations. The resulting matrix is H j with H 0 = 1 (H~ ) and H N 0K = 02 1 (H ). To each H j corresponds in general a different syndrome s j which can be evaluated iteratively when constructing H j by observing that
srj = srj 01
8 srj01
(13)
r2 th row of H j 01 is added to the r1 th row of H j 01 and srj = srj 01 (14) srj = srj 01 if the r1 th and r2 th rows of H j 01 are switched. Also, when a dependent column is met, this column is ignored until the (N 0 K )th LRI position has been processed. Then the permutation 2 is applied to both H N 0K and 1 (r ), leaving the syndrome sN 0K unchanged. if the
From this iterative construction of the syndrome, we conclude that after a dimension j has been processed, the values sij , i = 1; 1 1 1 ; N 0 K , can only be modified by syndrome values slj with l > j at the following iterations. Therefore, if at iteration j , the syndrome s j is such that slj = 0 for l > j , then s j = s N 0K . If
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TABLE II TABLE LOOKUP FOR LIST DECODING OF THE RM (8; 4; 4) CODE (“Y” IF s BELONGS TO THE LIST l(h ) ASSOCIATED WITH h )
j s
satisfies Theorem 3.4, the decoding stops. This strategy further reduces the number of computations. IV. EFFICIENT DECODING METHODS
Based on the results of the previous section, we present and analyze several decoding methods. Each of these methods is devised for a particular class of codes, so that their effectiveness strongly depends on the structure of the codes considered. A. Table Lookup Decoding
For high-rate codes with relatively small N 0 K , the total number of syndromes 2N 0K remains manageable. For such codes, we associate with each possible sum s2 of i columns of Q a list of syndromes to be tested by phase-i reprocessing. These syndromes are not guaranteed to be covered by the particular sum s2 under consideration, hence all the possible phase-i reprocessing improvements over order-0 reprocessing are contained in the union of the lists. These (possibly empty) lists are stored in a two-entry lookup table. Based on the algebraic properties of the code, certain (N 0 K )tuples can be discarded as valid column candidates of Q. Obviously, for phase-i reprocessing, the columns to be considered are only those of weights smaller by i than one of the weights contained in the weight distribution of the code. We illustrate the construction of such a table for a particular class of codes. Consider the (2m ; 2m 0 m 0 1; 4) extended Hamming codes, also called RM(r; m) Reed–Muller (RM) codes with r = m02, for which any check matrix H contains all the columns h with odd Hamming weight. Table II describes the lookup table for the (8; 4; 4) RM code. For this code, Example 3.2 implies that 8 out of 16 syndromes are eliminated from the table. The remaining syndromes are tested on the basis of the covering properties described in Section III-A and must appear in at least one list. Generalization of this example to any RM(m 0 2; m) code is straightforward and provides a lookup table of size [2(2m 0 (m + 1))] 2 [2m 0 (m + 1)]. Based on Table I, 2(N 0 K ) = 2(m + 1) syndromes belong to S0 (1). Decoding by this lookup table can easily be accommodated with the reduced-list decoding presented in [2]. However, the effectiveness of combining these two methods depends on the chosen objective. Identification of the N 0 K 0 1 LRI positions is sufficient in [2], whereas Theorem 3.4 requires the full knowledge of N 0K . As a ? m02 +m02 comparisons are result, by the method of [2] d? 2 0dH = 2 saved [10], where d2? is the second member in the list of generalized Hamming weights of the dual of the code under consideration. Therefore, the decoding of [2] remains the most efficient method to minimize the worst case number of computations for MLD. However, for practically optimum decoding, order-1 reprocessing is sufficient for codes with m 7 [8]. Hence the reduced list of [2] can be purged of all patterns corresponding to reprocessing orders larger than 1 and K + 1 patterns survive in the worst case. From this viewpoint, the
393
TABLE III COMPUTATION COST FOR EXTENDED HAMMING CODES
complete knowledge of N 0K provides an additional means which can be exploited to accomplish practically optimum or suboptimum efficient decoding algorithms. Also this additional information can be utilized to reduce the average number of computations of both ML and practically optimum or suboptimum decoding algorithms. For m 7, Table III summarizes the computation costs of the reduced-list decodings, both the ordinary [2] and when combined with order-1 reprocessing. In this table, we also exhibit the comparison between order-1 reprocessing based on the partial ordering required to construct the LRB and order-1 reprocessing with a complete ordering, as described in Section II-C. For m 6 only, the restricted list based on [2] can be significantly reduced by considering only the 2m 0 m patterns associated with order-1 reprocessing. We observe that as m increases, the percentage of computation savings achieved by a partial ordering also increases. Indeed, the resource test of Section II-C is no longer applicable but for high-rate codes, the average number of computations Nave is dominated by the ordering cost [8]. Consequently, a partial ordering is sufficient for order-1 reprocessing of these codes. Also, at high SNR, Nave is further reduced significantly by considering the iterative syndrome computation described in Section III-B2. Order-1 table lookup decoding of other high-rate codes follows similar lines although, in general, column entries are no longer all contained in any form of Q. For example, the number of column entries of all versions of Q for the (16; 7; 6), (32; 21; 6), and (64; 51; 6) extended BCH codes are, respectively, 163, 848, and 3797. These numbers still allow fast access to the data after the construction of the table, which has to be realized only once for a given code. Generalization to any order-i table lookup decoding is straightforward. B. Restricted-List Decoding
As N 0 K increases, building the lookup table as described above may become an enormous task. Also, for such codes, at least order-2 reprocessing is required to achieve practical optimality. Whereas decoding with the aid of a complete table is prohibitive in many cases, a restricted table corresponding to the most likely syndromes with only few searches may still offer significant computation savings. For these categories of codes, our generalized syndrome decoding approach can be implemented as a preliminary test for any decoding algorithm. This test particularly fits the reprocessing algorithm described in Section II-C, as well as any other algorithm based on a reordering of the received sequence. However, it can also be efficiently accommodated to other types of algorithms. In general, the restricted list syndrome decoding is a code-dedicated decoding approach. However, once such list is constructed, it allows a significant average speedup of various decoding methods. For example, for the (24; 12; 8) Golay code, both optimality and a small average number of computations can be achieved when using this test before applying the algorithm of [19]. Also, if we construct an optimum list for all syndromes s = bH T of Hamming weight
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3, and process order-2 reprocessing when ws 4, practically optimum error performance is achieved with on average 49 real operations at the BER 1002:4 and 13 real operations at the BER 1006 , respectively (in comparison with 107 and 101, respectively, in [8]). ws
V. PHASE-j GENERALIZED SYNDROME DECODING In this section, we generalize the results of Section III and determine, for 0 j1 < j2 , sufficient conditions such that phase-j1 reprocessing is not improved by phase-j2 reprocessing. As before, let
=[ 0
H
IN
2 f1 1 1 1 g]
]=[
hi ; i
K Q
;
;N
where the first N 0 K positions are the LRI positions and the columns of both IN 0K and Q are indexed in nondecreasing order of reliability. Consider an error pattern e = [e 1 e 2 ] of length N and Hamming weight wH (e) = we given by e
=
111
1
=1
hp
g 2 f 0 + 1 111
; pj
N
K
;
;N
0 + 2g K
j
:
f1 1 1 1 1 g ;j
0 + 2 for all 2 f1 1 2 0 1 0 1g + +1 0 0 0 2+ 1+ +1 and either 2 or, in case that 2 + 1, max f + 2( 0 1) + 1g 2f +1 111 g N
pj
j
; ;j
pj
< dj
j
j
;
;w
j
j
we
j
j
j
l
j
(15)
j
j
we
K
pl
j
l
dH
(16)
(17)
then order-j2 reprocessing does not improve order-j1 reprocessing. Theorem 3.4 is equivalent to Theorem 5.1 with j1 = 0. Note also that, as for j1 = 0, position pj +1 is always covered. Remark: Theorem 5.1 does not guarantee that the error pattern e for which the generalized covering test is satisfied corresponds to the optimum decoding solution; rather, it implies that only reprocessing phases j with j < j2 can improve the decoding associated with e . As an example, consider the particular case j1 = 1 and j2 = 2. Then Theorem 5.1 becomes: If p1 N 0 K + 2 and either we 2 or, in case that we 3,
max f + 2( 0 1) + 1g
2f3 111 g
l
;
;w
pl
l
dH
0 N
K
-BASED SYNDROME DECODING
dH
0 10 20 j
j
( 1)
wH s
(19)
:
For practically optimum decoding, the orders j1 and j2 are generally small compared to dH . Consequently, if a syndrome s1 of small Hamming weight has been recorded by the algorithm at the reprocessing phase j1 , then it is unlikely that the decoding cost associated with the corresponding error pattern can be improved. Therefore, (19) suggests a simplification of the order-l reprocessing algorithm of b, described in Section II-C, based on recording the error pattern of smallest weight during the successive reprocessing stages. A. The Algorithm
This follows by the following observations: 1) the reliability values corresponding to positions from N 0 K +1 to N are in nondecreasing order and 2) phase-j2 reprocessing modifies j2 > j1 positions in fN 0 K + 1; 1 1 1 ; N g. Consequently, the positions fp1 ; 1 1 1 ; pj g are covered by j1 of the j2 positions modified by phase-j2 reprocessing. Thus the covering problem of the positions fpj +1 ; 1 1 1 ; pj g is identical to that considered in Section III in regard to the covering of positions fp1 ; 1 1 1 ; pj 0j g of order-0 reprocessing with respect to order-(j2 0 j1 ) reprocessing. Finally, once positions fp1 ; 1 1 1 ; pj g are covered, the proof of Appendix II can be applied to positions fpj +1 ; 1 1 1 ; pw g with i = j2 . We thus conclude the following. 2 Theorem 5.1. Generalized Covering Test: If for all j ;
( 10 )
wH s
where p1 > p2 > 1 1 1 > pw . Notice that this expression is a generalization of (7). Denote supp (e ) = fpj ; j 2 f1; 1 1 1 ; we gg. Assume that the syndrome s1 = e 1 with associated error pattern e 2 of Hamming weight wH (e 2 ) = j1 has been computed for phasej1 reprocessing and consider phase-j2 for j2 > j1 . First, positions fp1 ; 1 1 1 ; pj g of e = [e1 e2 ] are covered only if p ;
VI. MINIMUM-WEIGHT
By the results of the previous sections we observe that based merely on encountering certain error patterns, expressed in terms of the LRB, one may conclude that ML decoding has been attained. In general, such errors patterns have relatively small Hamming weight. Indeed, if [s1 e 2 ] and [s1 0 e 2 0 ] are the error patterns recorded at phase-j1 and phase-j2 , respectively, of order-l reprocessing, then
w
j
f
then order-2 reprocessing does not improve order-1 reprocessing. If an error pattern e corresponding to order-1 reprocessing satisfies (18), then the ML solution is delivered by order-1 reprocessing. However, it still remains to ensure that none of the untested error patterns for order-1 reprocessing improves the decoding given by e . Based on Theorem 5.1, additional tests for optimality are provided. These tests require no real operations and can be easily incorporated within MLD algorithms.
(18)
The algorithm differs from that of Section II-C in modification of the order-l reprocessing of b, as follows: During all phases i of order-l reprocessing, the error pattern em such that
( ) = 2fmin f 1 111 g
wH em
i
;
;l
1
( )g
wH ei
i
l
, record (20)
where e i represents any error pattern associated with phase-i reprocessing. When all the li=0 Ki error patterns have been tested, the delivered solution is b 8 e m . Recall that for phase-i reprocessing, the error pattern e i has i and ( ) 0 i nonzero entries pertaining to the MRB and the LRB, respectively. Therefore, for a given reprocessing phase, the foregoing algorithm minimizes the Hamming weight of the syndrome si which corresponds to the error pattern candidate e i . We name this algorithm the minimum-weight syndrome decoding algorithm (MWSDA). Based on (19), we obtain the following sufficient condition for an error pattern to be the MWSDA solution. Corollary 6.1: Let [ s1 e 2 ] be the error pattern recorded at phase-j1 of order-l reprocessing of the MWSDA. If wH e i
( 1) + 1 2
wH s
j
dH
(21)
then, for j2 j1 , [ s 1 e 2 ] 8 b is not improved by phase-j2 of order-l reprocessing for the MWSDA. If wH (s1 )+ j1 b(dH 0 1)=2c then the solution of the MWSDA is identical to the algebraic (hard) decoding solution. Nevertheless, while any algebraic decoding algorithm is guaranteed to correct up to t = b(dH 0 1)=2c errors, the MWSDA corrects such errors only if the order of reprocessing is chosen to be the covering radius r(C ) of the code. On the other hand, as shown in the next section, there exists an integer rT < r(C ) such that for reprocessing phases-i with i > rT the errors corrected by the MWSDA become less likely than the error pattern recorded up to phase-rT reprocessing. Therefore,
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after a complete ordering and the construction of the LRB, the MWSDA with order-rT reprocessing provides an algebraic decoding method which outperforms any conventional algebraic decoder. The associated number of binary operations is determined by rT . The MWSDA can be viewed as an improved version of the algorithm proposed by Omura [21, p. 119, 175]. In particular: 1) it starts with the most likely syndrome; 2) the search stops before unlikely syndromes of low weights are considered; and 3) in case of ties, the most likely error pattern is almost always kept. These improvements are realized without real operations if quantized values are considered, since the ordering (which is the only soft information required by the MWSDA) is straightforward [14].
395
LRP’s are in error. Assuming that this event dominates the occurrence of false alarm, we obtain i
j =1
N ~ i Q(1) (1 0 Q~ (1))N 0i i
K
)
where
Q~ (x) = (N0 )01=2
Ps (i) Punc (i) + Ps; i : An upper bound on Ps; i has been derived in [8]. Also, satisfies the recursive equation
Punc (i) = Punc (i 0 1) + p(i)
(22)
Punc (i) (23)
with
< p(i) PM (i) + PFA (i)
(24)
where 1) PM (i) is the “miss probability,” namely, i MRP’s are in error and the corresponding error pattern ei with i erroneous bits located at the MRP’s is processed by the algorithm after the processing of some error pattern e with wH (e) wH (ei ) and 2) PFA (i) is the “false alarm probability,” namely, j ; j i MRP’s are in error and the corresponding error pattern e with j erroneous bits at the MRP’s has been recorded during the decoding procedure prior to phase-i reprocessing of some error pattern e i that satisfies wH (ei ) < wH (e). Equations (22) and (23) imply two important conclusions. 1) As long as Punc (i) Ps; i , order-i reprocessing for the MWSDA provides the same practical error performance as the ordinary order-i reprocessing algorithm described in Section II-C. 2) If Punc (i) dominates (22) then additional phases of reprocessing tend to deteriorate the error performance. According to these observations, there exists an optimum order rT of reprocessing for the MWSDA. With respect to phase-0 reprocessing, a false alarm occurs if at least
dH 2
+1
(25)
1 0 e
n =N
x
(26)
dn:
Similarly, with respect to phase-0 reprocessing, a miss occurs during phase-1 reprocessing if one MRP and at least
s(dH ) =
As described in the previous section, in case that two error patterns have the same Hamming weight then the first processed error pattern is retained. Therefore, to consider the best performance achievable by the MWSDA, we assume a complete ordering of the received sequence, i.e., no linear dependence is encountered during the formation of the MRB (stated otherwise, 2 (1) happens to be the identity permutation). We consider processing the error patterns on the basis of the reprocessing strategies of [8] and [10]. Accordingly, for a given phase of reprocessing, the most likely error patterns are treated first. Let Ps (i) be the probability that by the end of phase-i reprocessing of the MWSDA a decoding error is made. Denote by Ps; i the probability that the conventional order-i reprocessing (as described in Section II-C) is in error. Let Punc (i) be the probability that j i transmission errors in MRP’s have occurred and the MWSDA with order-i reprocessing is in error. By applying the union bound, we obtain
f (dH ) =
0
i=f (d
B. Performance Analysis
0
N
PFA (j )
dH
01
2
(27)
LRP’s are in error. Denote these positions by p and j1 ; j2 ; respectively. Then, based on the results of [20], we have Pe (p;
111; j , i
N N 0p 1 Pe (j1; 1 1 1 ; ji ; N ):
j1 ; 1 1 1 ; ji ; N ) Pe (p; N ) 1
(28)
where Pe (n1 ; n2 ; 1 1 1 ; nj ; N ) represents the probability that at least positions n1 ; n2 ; 1 1 1 ; nj are in error after ordering a block of N received symbols based on their reliability values. Assuming that the occurrence of a miss is dominated by the event that one MRP and at least s(dH ) LRP’s are in error, we obtain i
j =1
N N 0i
K
PM (i)
i=1
1
N
0
K
i=s(d
)
Pe (i;
N)
N ~ i Q(1) (1 0 Q~ (1))N 0i : i
(29)
(Recall that by assumption 2 (1) = 1, the identity permutation.) Equations (26) and (29) provide an upper bound for Punc (i). However, this upper bound is in general very loose as it is derived under the assumption that at least f (dH ) LRP’s are in error or one MRP and at least s(dH ) LRP’s are in error and, in addition, there exists a column of Q which enables either a false alarm or a miss. Nevertheless, this bound shows that for dH even, the MWSDA always outperforms an algebraic decoder able to correct up to (t + 1) transmission errors at medium to high SNR since f (dH ) = t + 2 and K
i=1
N=(N 0 i)Pe (i; N )
~ (1)2 as the SNR increases [8]. For decreases much faster than Q long codes with dH even, we conclude that although both decoding methods have the same asymptotic error performance,1 the MWSDA will largely outperform any algebraic decoder able to correct up to (t + 1) transmission errors at practical BER since transmission errors of small weights are very unlikely to cause a false alarm or a miss. Finally, by comparing (26) with (29), we conclude that the occurrence of false alarms dominates the bound at medium to high SNR values. The probability Ph (w 0 1) that a column of weight w 0 1 belongs to Q can be approximated as follows:
Ph (w 0 1)
1 It
Nw N w
may readily be shown [8] that Pe (N
N 0K w01
0 1;
N)
(30)
N
2
p2)
~( Q
:
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Fig. 1. Simulation results for MWSDA for the (24; 12; 8) Golay code.
where Nw is the number of codewords of Hamming weight w. It follows by (30) that for codes whose weight distributions are well approximated by a binomial distribution, such as long BCH codes, Ph (w 0 1) 20(N 0K )
N w
0 01
K
:
Fig. 2. Simulation results for MWSDA for the (128; 64; 22) extended BCH code.
we can state that for BER’s of at least 1006 , the MWSDA always performs within 1.5 dB of the optimum ML error performance with generally one order of reprocessing less than the order of reprocessing required by the practically optimum decoding method of [8].
(31)
For such codes, (31) shows that the occurrences of false alarms or misses caused by low-weight columns in Q are very improbable. Note finally that even if, for example, f (dH ) LRP’s are in error and Q contains a column of weight dH 0 1, it is still not guaranteed that a false alarm will occur. C. Simulation Results In this section, we compare the error performance of the MWSDA described in Section VI-A with both MLD and algebraic decoding. The comparison with algebraic decoding methods is justified by the fact that, except for the ordering of the received sequence, the same kinds of operations as for hard-decision decoding methods are performed by the MWSDA. Fig. 1 depicts the simulation and theoretical results for the (24; 12; 8) Golay code. We observe that for the MWSDA, order-1 reprocessing provides the best performance. Also the bound obtained from (26) is quite tight since for f (dH ) = 5 erroneous LRP’s, a column of Q providing a false alarm is likely to exists for this code. The MWSDA with order-1 reprocessing slightly outperforms an algebraic decoding algorithm capable of correcting up to t + 1 = 4 erroneous hard decisions. Simulation results for the decoding of the (128; 64; 22) extended BCH code are provided by Fig. 2. On this figure, we also represented the complete union bound for MLD, computed from [22]. As expected, for this code the bound of (26) becomes very loose as even if f (dH ) LRP’s are in error, it is very unlikely that there exists a column of Q so that a false alarm occurs. Order-3 reprocessing is required to achieve the best performance for MWSDA. The error performance of an algebraic decoder correcting up to f (dH ) 0 1 = t + 1 errors is also indicated on Fig. 2. At the BER 1006 , order-3 and order2 reprocessings for the MWSDA outperform this algebraic decoding algorithm by 2.0 and 1.65 dB, respectively. All three algorithms have the same asymptotic error performance. We simulated the performance of the MWSDA for several other well-known codes of length N 128. Based on these simulations
D. Hybrid Algorithms from the MWSDA By combining the MWSDA with the reprocessing method described in [8], many suboptimum decoding strategies can be devised. Such decoding procedures perform a limited number or no realvalue operations besides the operations required by the sorting. If Punc (i) Ps; i , then the MWSDA can be substituted for the order-i reprocessing described in Section II-C. For the (128; 64; 22) extended BCH code, we observe a 0.1-dB SNR degradation when processing order-2 with the MWSDA, as shown in Fig. 2. Therefore, at the BER 1006 , the MWSDA with order-2 reprocessing still achieves a 5.5-dB coding gain over uncoded BPSK and provides a good tradeoff between computation complexity, decoding speed, and error performance. If Punc (i) dominates (22), then the error performance associated with the MWSDA can be improved by processing the ordinary algorithm of [8] and [10] for the first few reprocessing phases, and terminating with the MWSDA for the remaining phases. We simulated the procedure in which the MWSDA starts after the conventional phase-1 reprocessing. No improvement with respect to the ordinary MWSDA was observed for the (24; 12; 8) Golay code, whereas about the same error performance as the complete conventional order3 reprocessing is achieved for the (128; 64; 22) extended BCH code. Finally, since in general PM (1) PFA (1), the error performance of the MWSDA can be further improved by computing and testing the resource (as in Section II-C) each time a candidate error pattern of smaller weight than the recorded one is found, in a way that PFA (i) = 0 for every phase-i tested. VII. CONCLUSION An approach to MLD of binary block codes that combines Gspace and H -space considerations has been established. It is based on the duality between the most reliable basis BK and the least reliable basis N 0K , for any given received sequence. Syndrome decoding in the LRB N 0K has then been further investigated. A class of new criteria for optimality has been derived. An important
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feature of these criteria is that, following the determination of the LRB, no further soft information is required. These criteria exploit both the ordering associated with the LRB and the structure of the code considered. Consequently, for some categories of codes, they provide low-complexity optimum or near-optimum decoding methods. Finally, a new syndrome-decoding algorithm, referred to as MWSDA, has been presented. For order-i reprocessing, this algorithm performs O N 0 K K i binary operations in the MRB. At practical BER, the MWSDA outperforms any algebraic decoding algorithm errors. An order of reprocessing of capable of correcting up to t is sufficient for codes of length N . at most i
((
) )
+1
=2
128
APPENDIX I RESOURCE TEST IN THE DUAL SPACE H
=(
)
For each BPSK signal sequence x 3 x13 ; x32 ; 1 1 1 ; x3N repre3 senting the codeword a with associated error pattern e 3 a 3 8 b, the MLD rule maximizes the cost function N
( )=
e3
=
N
(32)
= [e 3e 3 ] the differential decod1
2
ej3 jzj j
j =1 N
x3i zi :
i=1
Equivalently, we associate with e 3 ing cost given by
1(e 3 ) =
=
0
K
j =1
e13; j jzj j
+
K
j =1
(33)
1
+ N (a 3 ) + w d s
(34)
H
since for a code of minimum distance dH , no fewer than dH columns of its parity-check matrix sum to 0. Let consider the collection of the f ; dH 0 i 0 wsg values jzj j’s corresponding to LRI smallest , and denote by Di s the set of such positions j for which sj positions. Then
=0
i
1(e 3 )
()
jz 0 N
x=1
1
K +j
j+ j
jz j
(35)
j
2
D
i
+ N (a 3 ) + k + N (a 0 ) d
H
:
(36)
i
H
0
j
= s. Similarly to (35), we obtain jz 0 j + jz j: i
for order- reprocessing, e 1
1(e 3 )
i
N
x=1
jz 0 N
x=1
K +j
j1
min
2
=1
=0
= fp + 1; p + 2; 1 1 1 ; N 0 K g 0 fp ; p ; 1 1 1 ; p g
SQ
i+1
i+1
1
K +j
j
2
j
D
(37)
0 max j
2
D (s)
jz j; j
j
2
D (a
jz j j
:
)
(38)
i
2
+x+id
(39)
H
then s is covered for phase-i reprocessing. We now show that (39) is equivalent to (11) of Theorem 3.4. If pi+1 px x 0 i 0 , then (39) can be rewritten as
= +
1
+ 2(x 0 1) + 1 d : (40) Otherwise, p > p + x 0 i 0 1. In that case, define S as the set and p such that s = 0. Let of positions j between p px
i+1
H
x
R
i+1
x
j
r1 > r2 > 1 1 1 > rjS
j
be the elements of SR . Then r1 covers px+1 , r2 covers px+2 ; 1 1 1 ; rjS j covers px+jS j , and
= p + x 0 i 0 1 + jS j:
pi+1 If px
=p
x
R
j j + jS j, then (39) is equivalent to p +j j + 2(x + jS j 0 1) + 1 d
x+ S
R
x
R
S
Otherwise, px > px+jS with
px
j + jS
R
=p
H
:
(41)
j and we repeat the same procedure
j j + jS
x+ S
R
j + jS j R
where SR is defined similarly to SR for the position between px and px+jS j . This procedure finally ends since the definition of x guarantees that all positions are covered. Therefore, for ws i , if
+1
max
2f +1 111 g
(a )
2
and label the elements of the set of positions SQ in increasing order q1 < q2 < 1 1 1 < qjS j . Then q1 covers pi+1 , q2 covers pi+2 ; 1 1 1 ; qx0i covers px . Therefore, if pi+1 x 0 i dH 0 i, i.e.,
l
Combining (35) and (37), the following necessary condition to process a 3 is obtained: i
2
=
We define D (a 0 ) to be the collection of LRI positions corresponding to the smallest max f0; d 0 i 0 k 0 N (a 0 )g values jz j’s for which e10 = 0. This definition generalizes the definition of D (s ) since ;j
The first condition of the theorem guarantees, by applying Corollary 3.1, that the positions p1 ; p2 ; 1 1 1 ; pi are covered for phase-i reprocessing. If ws i, then the syndrome s is automatically covered. Otherwise, at least dH 0 i positions of the syndrome other than p1 ; p2 ; 1 1 1 ; pi have to be modified (without loss of generality, we assume dH 0 i < N 0 K , since otherwise optimality is trivial). We consider the covering of the positions from pw to pi+1 . To this end, we construct an ordered list of uncovered pj ’s by reading , the the syndrome s from s1 to sN 0K . Each time we read sj corresponding pl is added to the tail of the list and each time we , the head of the list is deleted. Assume that upon encounter sj reaching pi+1 , the positions pi+1 to px are left to be covered in the list. Denote
(s)
where for x 2 f ; 1 1 1 ; ig, jx is the position of the xth column of Q associated with e 3 . Let a 0 be recorded at phase-k reprocessing, k i, so that 3 6 a0 e0 min . Then, at phase-i reprocessing, for a
1( ) = 1
APPENDIX II PROOF OF THEOREM 3.4
pi+1
e23; j jzN 0K+j j
( )
max 0
It is easy to verify that (38) is equivalent to the resource test described in [8].
+
and record the minimum decoding cost min and its corresponding codeword a 0 . For any codeword candidate a 3 , phase-i reprocessing sums i rows of QT . If ws and N a 3 denote the Hamming weights of s and e 1 3 , respectively, we have
i
397
i
;
;w
fp + 2(l 0 1) + 1g d
H
l
(42)
then (39) is satisfied and phase-i reprocessing cannot improve orderreprocessing. Note also that (39) can be substituted in the formulation of Theorem 3.5 but x depends on each syndrome considered. To complete the proof, we need to show that for any i2 > i, phase-i2 reprocessing cannot improve order- i 0 reprocessing. For i l. Then l , let i2
(i 0 1) 1
( 1)
= + pi+l
d 0 2(i + l) + 1 d 0 1: H
H
(43)
398
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 1, JANUARY 1998
Since no dependent position is permuted before reaching the dH th dimension when constructing the LRB, pi+l is automatically covered for order-(i + l) reprocessing. The proof is completed with an inductive argument as max
++1;111;w
l2fi l
g
f
pl +2(l
0 1)+1g
max
+1;111;w
l2fi
g
f
pl +
2(l
[22] Y. Desaki, T. Fujiwara, and T. Kasami, “The weight distributions of the (128, 64, 22) and (128, 71, 20) extended binary primitive BCH codes,” in Proc. IEEE Int. Symp. on Information Theory and Its Applications (Victoria, BC, Canada, Sept. 1996), pp. 594–597.
0 1) + 1g
:
(44) REFERENCES
On Hybrid Stack Decoding Algorithms for Block Codes L. E. Aguado and P. G. Farrell, Member, IEEE
[1] J. Snyders and Y. Be’ery, “Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes,” IEEE Trans. Inform. Theory, vol. 35, pp. 963–975, Sept. 1989. [2] J. Snyders, “Reduced lists of error patterns for maximum likelihood soft decoding,” IEEE Trans. Inform. Theory, vol. 37, pp. 1194–1200, July 1991. , “Partial ordering of error patterns for maximum likelihood soft [3] decoding,” Algebraic Coding—LNCS Vol. 573. New York: SpringerVerlag, 1992, pp. 120–125. [4] , “On survivor error patterns for maximum likelihood soft decoding,” in Proc. IEEE Int. Symp. on Information Theory (Budapest, Hungary, June 1991), p. 192. [5] N. J. C. Lous, P. A. H. Bours, and H. C. A. van Tilborg, “On maximum likelihood soft-decision decoding of binary linear codes,” IEEE Trans. Inform. Theory, vol. 39, pp. 197–203, Jan. 1993. [6] Y. S. Han, C. R. P. Hartmann, and C. C. Chen, “Efficient priorityfirst search maximum-likelihood soft-decision decoding of linear block codes,” IEEE Trans. Inform. Theory, vol. 39, pp. 1514–1523, Sept. 1993. [7] T. Kaneko, T. Nishijima, H. Inazumi, and S. Hirasawa, “An efficient maximum likelihood decoding of linear block codes with algebraic decoder,” IEEE Trans. Inform. Theory, vol. 40, pp. 320–327, Mar. 1994. [8] M. P. C. Fossorier and S. Lin, “Soft-decision decoding of linear block codes based on ordered statistics,” IEEE Trans. Inform. Theory, vol. 41, pp. 1379–1396, Sept. 1995. , “Computationally efficient soft-decision decoding of linear block [9] codes based on ordered statistics,” IEEE Trans. Inform. Theory, vol. 42, pp.738–750, May 1996. [10] D. Gazelle and J. Snyders, “Reliability-based code-search algorithm for maximum-likelihood decoding of block codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 239–249, Jan. 1997. [11] H. T. Moorthy, S. Lin, and T. Kasami, “Soft-decision decoding of binary linear block codes based on an iterative search algorithm,” IEEE Trans. Inform. Theory, vol. 43, pp. 1030–1040, May 1997. [12] D. J. Taipale and M. B. Pursley, “An improvement to generalizedminimum-distance decoding,” IEEE Trans. Inform. Theory, vol. 37, pp. 167–172, Jan. 1991. [13] T. Kasami, T. Koumoto, T. Takata, T. Fujiwara, and S. Lin, “The least stringent sufficient condition on the optimality of suboptimally decoded codewords,” in Proc. IEEE Int. Symp. on Information Theory (Whistler, BC, Canada, Sept. 1995), p. 470; also, IEEE Trans. Inform. Theory, submitted for publication. [14] B. G. Dorsch, “A decoding algorithm for binary block codes and J -ary output channels,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 391–394, May 1974. [15] V. K. Wei, “Generalized Hamming weights for linear codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 1412–1418, Sept. 1991. [16] T. Y. Hwang, “Efficient optimal decoding of linear block codes,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 603–606, Sept. 1980. [17] Y. S. Han and C. R. P. Hartmann, “Designing efficient maximumlikelihood soft-decision decoding algorithms for linear block using algorithm A3 ,” School of Computer and Information Science, Syracuse University, Tech. Rep. SU-CIS-92-10, June 1992. [18] W. W. Peterson and E. J. Weldon, Jr., Error-Correcting Codes, 2nd ed. Cambridge, MA: MIT Press, 1972. [19] A. Vardy and Y. Be’ery, “More efficient soft decoding of the Golay codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 667–672, May 1991. [20] M. P. C. Fossorier and S. Lin, “First-order approximation of the ordered binary-symmetric channel,” IEEE Trans. Inform. Theory, vol. 42, pp. 1381–1387, Sept. 1996. [21] G. C. Clark and J. B. Cain, Jr., Error-Correction Coding for Digital Communications. New York: Plenum, 1981.
Abstract—This correspondence presents sequential algorithms for softdecision decoding of linear block codes. They use a stack algorithm based on the trellis of the code. We are interested in the trellis as a means to avoid path-decoding repetitions. As well, the possibility of bidirectional decoding offers a chance to increase the likelihood of explored paths. We have developed three successive algorithms that offer a good decrement in the overall complexity, and mainly in the most complex decoding case, while giving near-maximum-likelihood performance. This is important since it determines the maximum buffer size necessary in the decoder. Index Terms—Block codes, soft decision, stack algorithms, trellis decodings.
I. INTRODUCTION The trellis representation of a code offers a direct framework to use for soft-decision decoding. Its successful application to convolutional codes and the maximum-likelihood decoding gain we can obtain with it has led to the predominance of these codes in coding implementations. There are applications, however, where the use of block codes is advantageous. For such, algebraic soft-decision decoders are sometimes proposed, but at the cost of more complex and slower systems, and with some loss of performance [6], [7]. Since the first systematic methods for building the trellis of block codes were presented [3], [11], [24] much work has been done in this area concerning two main problems. The first is to find a simple trellis representation of the code in terms of its structure [10], [11], [15] or number of elements [1]. The second major problem is to actually decode those codes in an efficient way using that trellis to obtain maximum- or near-maximum-likelihood performance. The Viterbi algorithm becomes impractical for medium- and large-size codes due to the great number of states in the trellis and its very irregular structure in the general case. Alternately, reduced-search algorithms like the M -algorithm or sequential algorithms have been proposed that offer a good deal in complexity reduction at the cost of some loss in performance [4], [14]. Here we present a sequential algorithm for block codes. It is a trellis-based stack algorithm with which we try to extract the maximum benefit out of the information embedded in the trellis. This is helpful in reducing the complexity while maintaining a high level of performance, making possible the decoding of long codes. We introduce three variants of the algorithm, so that they can be analyzed independently and the tradeoff between complexity reduction and performance that they offer can be compared in each case. Besides, this permits an easier understanding of how they work and, depending on the implementation needs, adoption of that part that suits best our decoding conditions, but perhaps with a relative increase in complexity. Manuscript received December 14, 1995; revised December 20, 1996. The authors are with the Communications Research Group, School of Engineering, The University of Manchester, Manchester M13 9PL, U.K. Publisher Item Identifier S 0018-9448(98)00117-5.
0018–9448/98$10.00 1998 IEEE
