mum-length or PN-sequences, are obtained (see [l]). We shall analyze the class of binary sequences [b,] derived from nonzero [a,]' by nonuniform decimation. (2).
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On the Linear Complexity of Nonuniformly Decimated PN-Sequences JOVAN DJ. GOLIC AND MIODRAG v. ~IVKOVIC Abstract-A lower bound is derived on the probability that, when a PN-sequence of period N = 2" - 1 is nonuniformly decimated by means of a sequence whose period divides M , the decimated sequence will have maximum linear complexity nM. It is shown that by choosing M and n appropriately we can make this probability arbitrarily close to one with nM arbitrarily large.
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have maximum linear complexity nM and period MN if gcd( N, K ) = 1 and every prime factor of M divides N. The problem we are concerned with here is to derive the probability that a sequence from S h , K has maximum linear complexity provided that [a,] and [d,] are chosen at random according to the uniform probability distribution in the general case for any f, M, and K. 11. ANALYSIS
We first examine the structure of the sequences in S h , K . Generalizing the ideas in [3], from (2) and ( 3 ) we have
I. INTRODUCTION $+",=a( ' i l d , + K t ) , t=0,1,2;..,j=O,l;..,M-l. Consider a binary linear feedback shift register (LFSR) with i=O primitive feedback, Le., generating polynomial f ( x ) = Ey=of,~', (4) fo =A, = l . Given an initial state ( a o , a',. . . , a , - , ) E GF(2)", the LFSR generates the output sequence [a,] according to the linear This means that [b,] consists of M interleaved sequences, each of recurrence which is obtained by the uniform K-decimation of an approprin ate cyclic shift of [a,]. It follows from [4], for example, that for a,= Cfra,+,, t = n , n + l , . . . . ( 1 ) any j = O , l ; . . , M-1, MGP[b,+M,] is equal either to 1, when 1-1 [ b, + ,,I is the zero sequence, or to the irreducible polynomial For 2" -1 different nonzero initial states, 2" - 1 distinct and h ( x ) , which is the minimum polynomial of a K , provided that cyclically equivalent sequences of period N = 2" - 1, called maxi- f ( x ) is the minimum polynomial of a, where a is a primitive element of GF(2"). The exponent of h ( x ) is mum-length or PN-sequences, are obtained (see [l]). We shall analyze the class of binary sequences [b,] derived from nonzero [a,]' by nonuniform decimation while the degree p of h ( x ) is given by 02( P), where for any odd natural number e, 0 2 ( e ) is the minimal natural number k such that e12k - 1, i.e., the multiplicative order of 2 modulo e, ord,(2). where, for given integers M 2 1 and 0 I K I N - 1, the difference , -1, t = 0,1,2; . ., is a peri- It follows that pln. decimation sequence [ d , ] ,0 ~ dI N Thus in general S h q K is a subset of the set of all 2"" odic integer sequence with period dividing M that satisfies sequences consisting of M interleaved sequences whose minimal M-1 generating polynomials divide h ( x ) . This is the same as the cyclic ( 3 ) vector space Q ( h ( x M ) )of all 2PM sequences generated by the binary LFSR with the feedback polynomial h ( x M ) , that is, Thus [b,] is a function of [ a , ]and [d,]. Let us denote the defined whose minimal generating polynomials divide h( x"). Thus in the class of binary sequences by Sh, K . Since there are N different generating functions domain we have sequences [a,] and NM-' different sequences [d,], there are at most N" sequences in Sh. K . 3( h( x")) = : deg( r( x)) < (6) Here we are interested in the linear complexity of the binary sequences in Sh, K . Recall that, for any periodic binary sequence Note that the degree and exponent of h(x") are equal to p M [ S I , a unique LFSR of minimal length exists that generates it (see and PM, respectively. This means that each sequence in S h , K [2]) called the minimal LFSR of [SI (MLFSR[s]). The minimal has linear complexity not greater than p M and period dividing generating polynomial of [SI (MGP[s]) is the feedback polyno- PM. mial of MLFSR[s]; the linear complexity of [ S I (L[s]) is the The class S h , K can be determined exactly in the following length of MLFSR[s], that is, the degree of MGP[s]; and the case. period of [s] (Per [SI) is equal to the exponent of MGP[s].~ Lemma I: If A particular subclass of Sh, was considered in [3]. Indeed, it N was assumed in [ 3 ] that a difference decimation sequence is formed by means of another binary LFSR with a primitive 02(gcd(N, K ) ) = n ' feedback polynomial and a nonzero initial state. It was proved there that all the sequences from the considered subclass of Sh, then S h , K is the set of all N" sequences consisting of M interleaved sequences with minimal generating polynomials equal to h ( x ) . (2)
($
Manuscript received August 21, 1987. This work was presented in part at the Yugoslav Conference ETAN in Marine, Zadar, 1987. The authors are with the Institute of Applied Mathematics and Electronics, Belgrade, Yugoslavia. J. Dj. GoliC is also with the Faculty of Technical Sciences, University of Novi Sad, and the Faculty of Electrical Engineering, University of Belgrade, Bulevar Revolucije 73, 11OOO Beograd, Yugoslavia. IEEE Log Number 8824065. 'Throughout this correspondence symbols like u, and u ( r ) are equivalent. *As usual, it is assumed that if j < I , then P(.) = 0. 3The exponent of a binary polynomial f ( x ) such that f ( 0 ) + O is the multiplicative order of x modulo /(x).
PM).
Proof: First note that one-to-one correspondence exists between the set of all NM-' distinct difference decimation sequences [d,] and the set of all decimation sequences [ D,] given by
D,=
(:I: )
d, modN.
(8)
It follows that Do= 0 and DM = K. On the other hand, provided that (7) holds, it is easy to prove that the uniform K-decimation defines a one-to-one correspondence between the set of all N
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sequences whose minimal generating polynomials are equal to f ( x ) and the set of all N sequences whose minimal generating polynomials are equal to h ( x ) . This implies that (4) defines a one-to-one correspondence between the set of all NM distinct pairs ( [ a , ] ,[D,]), i.e., ( [ a , ] ,[ d , ] )and the set of all NM distinct M-tuples of sequences whose minimal generating polynomials are equal to h ( x ) . In addition, any two distinct M-tuples from the latter yield two distinct sequences by interleaving. Assume now that condition (7) is satisfied. In view of the fact that SL, c a ( h ( x ' ) ) , if (7) holds then each sequence from sh,K has linear complexity not greater than nM and period dividing PM. Our interest here is to derive the relative number, Q, = Q ( S L ,K ) r of sequences in SL, having maximum linear complexity nM, provided that (7) is true. According to Lemma 1, if (7) is true, then Q, is exactly the probability that a sequence from SL, has maximum linear complexity provided that [ a , ] and [ d , ] are chosen at random according to the uniform probability distribution. It is difficult to derive Q, exactly, except in some special cases, for example, the one investigated in [3]. Indeed, from [3] it follows that if gcd(N, K ) =1 and every prime factor of M divides N , then each sequence from Sh, obtained by [ d , ] with period exactly equal to M has maximum linear complexity nM. This results in an exact formula for Q,. In the general case, we shall satisfy ourselves with the appropriate lower bounds on Q,. For this purpose we first establish the relative number Q, = Q ( s 2 ( h ( x M ) ) )of sequences in s2(h(x')) having maximum linear complexity nM. Then we relate Q, to Q, in accordance with the following result, which can be proved easily. Lemma 2:
From (6), the number of sequences in s 2 ( h ( x M ) having ) maximum linear complexity nM is equal to the number of polynomials with degree less than nM that are coprime to h ( x M ) .Taking this into consideration, it is not difficult to obtain the following result. Lemma 3: T
Q(s2(h(x')))
=
n
r=l
(1-2-"1)
(10)
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g c d ( M , N ) = l . Case 3 ( $ , , = g c d ( M , N ) > l ) occurs if gcd( N , K ) > 1, gcd( M , N ) > 1, and PM'JN.According to Lemma 3 and Propositions 1 and 2 we then obtain the following result, yielding a lower bound on Q,. Lemma 4: Let M = 2"'M,, with m a nonnegative integer and M, an odd natural number, and M, = MiM," where gcd( Md', P ) = 1 and every prime factor of M,j divides P, P = N/gcd( N, K). Provided that 02( P) = n , it then follows that
where I ) , , = O if g c d ( N , K ) = l and g c d ( M , N ) > l or if = gcd(N, K ) >1and PM; % N; I), = 1 if gcd(M, N) = l ;and g c d ( M , N ) > l if g c d ( N , K ) > l , g c d ( M , N ) > l , and PM,'(N. Combining Lemmas 2 and 4, we obtain the following main result, giving a lower bound on Q,. Theorem: Under the conditions defined in Lemma 4, it is true that
where $,, is defined in Lemma 4. It remains to perform some algebraic manipulations on the lower bound (12). In view of the inequality (1- x)' 21- ax, x E (0,1), for an arbitrary nonnegative integer a , (12) reduces to
1 Q ( S h , K ) 21-
(1 - 2 - 0 )
(I),2-n +( M, - I),)2-2"-1 '"
Further, provided that (14)
MI^",
(13) reduces to Q( SL. K
)
2 1- ( e + S,t)( (2$,,
+ 1)2-"-'
-
t+b,,2-2n-') ( 1 5 )
where
,"
4 2 8, = ( 1- 2 -',) - e 2 0, n = 1,2, . . . . ( 16) where T is the number of distinct irreducible factors of h ( x M ) and tr, , i = 1,. . ., T , are their degrees. For $, =0, $,, =1, and I),, =gcd(M, N), (15) becomes The problem is thus reduced to the factorization of the polynomial h ( x M ) ,where the multiplicities of the factors are irrelevant. Q ( sL~ ) 2 1- ( e + s,,) 2-'1- l , (17) Recall that h ( x ) is an irreducible polynomial of degree n and Q( Sj,,,) > l ( e + S,)(3.2-"-' -2-2'1-1), (18) exponent P = N/gcd( N, K ) , where N = 2" - 1. If in addition gcd( N, K ) =1, then h ( x ) is primitive. We use here the following Q( Sh.K ) 2 1- ( e + &,,I((2gcd( M , N ) + 1)2-"-' three elementary assertions. Propositions 1 and 2 are essentially known (see, for example, [5, chs. V.5, V.91 and [6, Theorem 6.23]), -gcd( M , N)2-,"-l), (19) whereas Proposition 3 can be proved on the basis of [6, Theorem respectively. It is then obvious that by appropriate choice of N, 6.231. Note that condition (7) is assumed to hold. Proposition I : Let M = 2"M0 where m is a nonnegative inte- M , and K we can easily arrange to have the relative number of ger and M, an odd natural number. Then h ( x M )= h ( ~ ' " o ) ~ " ' ,sequences in SL,. with maximum linear complexity arbitrarily close to one. For example, to achieve Q ( S L , , ) 21 -lo-' it is where h( x'o) has no multiple irreducible factors. sufficient to choose M I 2", n 2 30, for I),, = 0 or t+b,z = 1 and Proposition 2: Let M be an odd natural number and M = M'M" where gcd(M", P) = 1 and every prime factor of M' M 5 2"12, n 2 60, for I),l > 1. Recall that these results are true if N/gcd( N, K)) = n , N = 2" - 1. Note that divides P. Then the exponent e and degree d of any irreducible K is chosen so that 02( factor of h ( x M ) satisfy PM'lelPM and n 1 0 2 ( P M ' ) J d ( 0 2 ( P M ) ,the total number q,, of distinct values of K, 0 I K 5 N - 1, satisfying this condition can be significantly greater than the total respectively . Proposition 3: Assume that M is odd and M = M'M" where number of distinct values of K that are coprime to N, that is, gcd( M", P ) =1 and every prime factor of M' divides P. If +( N ) (with 9 denoting Euler's function). For example, if n is a PM'JN then h ( x M )has exactly cd(M, N ) irreducible factors of prime number, then q,l = N - 1. In this case we need not worry degree n. If P M ' I N then h ( x ) has no irreducible factors of about the choice of K. Thus, if we choose K at random according to the uniform probability distribution then the probability that a degree n. Letting I),, denote the number of irreducible factors of h ( x M ) sequence from SL,K has maximum linear complexity nM is of degree n when M is odd, there are three possible cases. Case 1 given by (1-(2" -l)-')Q(Sh..). Finally, instead of SL,K we could consider the set VL,k of ($,,=O) occurs if g c d ( N , K ) = l and g c d ( M , N ) > l or if gcd ( N , K ) > 1 and PM' I N. Case 2 (I),, = 1) occurs if only those sequences in Sh,K obtained by a difference decima-
k
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tion sequence [ d , ]with period exactly equal to M. Let Q( Vh,), denote the relative number of sequences in V h , K having maximum linear complexity nM, provided that condition (7) is true. Since every sequence in SL,.. obtained by [ d , ]with period less than M has linear complexlty less than nM, it follows that Q ( V h , , ) 2 Q ( S L , , ) . Thus the lower bounds derived for Q(SL,K ) are true for Q(Vh*K ) as well.
111. CONCLUSION Linear complexity of nonuniformly decimated PN-sequences was analyzed from a probabilistic standpoint. Given a PNsequence of period N = 2“ - 1 and a difference decimation sequence of period dividing M such that the sum modulo N of its M successive values equals K, it is shown that maximum linear complexity nM of the decimated PN-sequence can be obtained only if the multiplicative order of 2 modulo N/gcd( N, K ) is equal to n. Given that the difference decimation sequence and the phase of the PN-sequence are chosen at random according to the uniform probability distribution, a lower bound on the. probability that a decimated PN-sequence has maximum linear complexity nM is then established. Provided M I 2“, it is shown that, by choosing M and n appropriately, this probability can be made arbitrarily close to one with nM arbitrarily large. If the conditions differ from the ones assumed in that the difference decimation function is generated by a binary linear feedback shift register with a primitive feedback polynomial, we conjecture that the analogous results and conclusions regarding the linear complexity of nonuniformly decimated PN-sequences hold. ACKNOWLEDGMENT The authors thank the reviewers for their useful comments.
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1079 I. INTRODUCTION
It is known that Viterbi decoding for a high-rate convolutional code is significantly simplified if the high-rate code is obtained by periodically deleting bits from a low-rate convolutional code, or by puncturing the code. Cain et al. [l] have defined a class of punctured rate ( n - l)/n convolutional codes and clarified the Viterbi decoding procedure for the punctured code. Yasuda et al. [2], [3] have described an optimum soft-decision Viterbi decoding scheme and the configuration of a variable-rate Viterbi decoder where the various rates were obtained by deleting bits from a fixed-rate 1/2 code in different ways. Hardware experiments and theoretical calculations showed that punctured convolutional coding and soft Viterbi decoding enable reliable communication over band-limited satellite channels. The purpose of this correspondence is to present good rate ( n - l)/n punctured convolutional codes derived from rate 1/2 codes to be used with soft Viterbi decoding. The new tabulated codes are short-constraint-length binary punctured convolutional codes with maximum free distance d f , minimum information weight W ( d ), rate (n-l)/n for n=5,6,7,8, and constraint length k = d3;. .,6. Cain et al. [l] have derived the best rate 2/3,3/4 codes.
11. PRELIMINARIES AND NOTATION A rate ( n - l)/n punctured convolutional code is obtained by periodically deleting bits from a rate 1/2 code according to a deleting bit map. The original rate 1/2 code with constraint length k is defined by two generator polynomials G ’ ( D )= gh g i D . . gk Dk,i =1,2, where k is the number of delay elements in the shift register. The rate 1/2 code is conveniently represented by a 2 x (k + 1) binary matrix G , called the generator matrix:
+
+
+
REFERENCES [l] [2] [3] (41 [5] [6]
S. W. Golomb, Shifr Regisfer Sequences. San Francisco, CA: HoldenDay, 1967. N. Z; der, “Linear recurring sequences,” SIAM J., vol. 7, pp. 31-48, Mar. 1959. W. G. Chambers and S. M. Jennings, “Linear equivalence of certain BRM shift-register sequences,” Electron. Left., vol. 20, pp. 1018-1019, Nov. 1984. F. Surbock and H. Weinrichter, “Interlacing properties of shift register sequences with generator polynomials irreducible over GF( p),” IEEE Trans. Inform.Theory, vol. IT-24, pp. 386-389, May 1978. A. A. Albert, Fundumentul Concepfs of Higher Algebra. Chicago, IL: Univ. of Chicago Press, 1956. E. R. Berlekamp, Algebruic Coding Theory. New York: McGraw-Hill, 1968.
The deleting bit map, defined by Yasuda et al. [2], [3] is represented by a 2 x ( n -1) matrix, wherein the leftmost column contains two 1’s and each of the remaining columns contains one 1 and one 0:
[’ 1
... 01. 1
0
A simple punctured rate 2/3 convolutional encoder with k = 2 is s h o r n in Fig. 1. The output bits from the rate 1/2 encoder are
New Short Constraint Length Rate ( N - 1)/N Punctured Convolutional Codes for Sof &Decision Viterbi Decoding WELL JBRGEN HOLE Abstract -New punctured convolutional codes with good performance are reported for rates ( n - l)/n with n = 5,6,7,8 and constraint lengths k = 2,3,. . .,6. The tabulated codes are nonsystematic binary convolutional codes with maximum free distance df and minimum information weight W ( d , ) for use with soft-decision Viterbi decoding.
transmitting deleting (every fourth bit)
1
dMeq:tPnfg bits
0
-.
xi
Original + coded bits
Manuscript received August 31, 1987; revised December 21, 1987. The author was with the Department of Informatics, University of Bergen, Bergen, Norway. He is now with the Center for Magnetic Recording Research, R-001, University of California-San Diego, La Jolla, CA 92093. IEEE Log Number 8824058.
...
+ --*
x; x: x;
u
x:
...
~
Punctured code block
Fig. 1. Punctured rate 2/3 convolutional encoder. (a) Encoder for original rate 1/2 code with k = 2 and G ’ ( D ) = l + D + 0’.G 2 ( D )= 1 + D2 (b) Bit selection.
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