et al. [12] we denote by A,(n, t) the maximum number of codewords in's code of length n correcting t or ... code, Ai n Aj =0 for i # j, lAil 2 IAi+i( for 1 I i I k - 1, and.
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It is easy to show that the series C,+,“-,r,(t,q)Z” is again uniformly and absolutely convergent for 0 < IZI < +m. Therefore, the order of integration and summation can be exchanged; we have:
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1991
[ll]
P. Pouzet, “Etude en vue de leur traitement numerique des equations integrales de type Volterra,” Reu. Fancais Traitement Information, (Chiffres) vol. 6, pp. 79-112, 1963. [18] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, Inc., 1965.
New Lower Bounds for Asymmetric Unidirectional Codes
and
Tuvi Etzion, Member, IEEE
Obviously, one necessary condition for r(Z, t) to be analytic for IZI I 1 is that the coefficient of Z-’ in the previous series is zero, or,
~(A(t,T)~,(T)r-,(t,T)Represent
r-&,7)
and r-l(t,T)
R,(~,T) = r-2(t,n)jt
P(T)al(T>r-i(f,T>)d77= 0. by
&(f,77) = r-l(t,v);
Abstract-New lower bounds on the sizes of asymmetric codes and unidirectional codes are presented. Various uiethods are used, three of them of special interest. The first is the partitioning method that is a modification of a method used to construct constant weight codes. The second is the combining codes method that is used to obtain a new code from a few others. The third method is shortening by weights that is applied on symmetric codes or on codes generated by the combining codes method. Tables for the sizes of codes of length n s 23 are presented. Index Terms -Asymmetric codes, Golay code, partitions,
we get (2.7) and (2.8).
I. REFERENCES
[l] R. A. DeCarlo, Linear Systems. Englewood Cliffs, NJ: Prentice Hall, 1989. [2] W. Grassmann, “Transient solutions in Markovian queues,” European J. Oper. Res., vol. 1, pp. 396-402, 1977. [3] G. F. Newell, “Queues with time-dependent arrival rates I, II, III,” .I. Appl. Probab., pp. 436-451, 579-590, and 591-606, May 1968. [4] J. Abate and W. Whitt, “Transient behavior of the M/M/l queue: Starting at the origin,” Queueing Syst. 2, pp. 41-65, 1987. [5] K. L. Rider, “A simple approximation to the average queue size in the time-dependent M/M/l queue,” JACK, vol. 23, no. 2, pp. 361-367, Apr. 1976. [6] M. R. Taaffe and K. L. Ong, “Approximating nonstationary Ph(t)/M(t)/s/c queueing systems,” Ann. Oper. Res., vol. 8, pp. 103-116, 1987. [I] M. H. Rothkopf and S. S. Oren, “A closure approximation for the nonstationary M/M/s queue,” Management Sci., vol. 25, no. 6, pp. 522-534, June 1979. [8] W. A. Massey, “Asymptotic analysis of the time dep.endent M/M/l queue,” Math. Oper. Res., vol. 10, no. 2, pp. 305-327, May 1985. [9] G. E. H. Reuter, “Denumerable Markov processes and the associated contraction semi-groups on 1,” Acta Math., vol. 97, pp. l-46, 1957.
“Denumberable Markov processes, II, III,” J. Len. Math. -, Sot., vol. 34, pp. 81-91, 1959; vol. 34, pp. 63-73, 1962. [ll] G. E. H. Reuter and W. Ledemann, “On the differential equations for the transition probabilities of Markov processes with enumerably many states,” Proc. Cambridge Phil. Sot., vol. 49 pp. 247-262, 1953. [12] A. B. Clarke, “A waiting line process of Markov type,” Ann. Math. Statist., vol. 27, pp. 452-459, 1956. [13] E. Hairer, Ch. Lubich, and M. Schlichte, “Fast numerical solution of nonlinear Volterra cbnvolution equations,” SZAM J. Sci. Stat. Comput., vol. 6, no. 3, pp. 532-541, July 1985. [14] H. M. Jones and S. McKee, “Variable step size predictor-corrector schemes for second kind Volterra integral equations,” Math. Comput., vol. 44, no. 170, pp. 391-404, April 1985. [15] J. M. Bownds, “Theory and performance of a subroutine for solving Volterra integral equations,” Computing, vol. 28, pp. 317-332, 1982. [16] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations. Amsterdam: North-Holland, 1986. m
OOlS-9448/91$01.00
codes, combining codes, constant weight shortening codes, unidirectional codes.
INTRODUCTION
The main goal of this correspondence is to present new lower bounds for asymmetric and unidirectional codes of length II 5 23, and some new constructive methods for these codes. Let ZJ= (Ul,~Z,~~ ., un> and v = (v1,v2; . . , ;,> be two binary vectors. Let N(u,v) denote the number of coordinates where u is 0 and v is 1. The well known Hamming distance is defined by d,(u,v) = N(u,v)+ N(v,u). The asymmetric distance is defined by d,(u,v)=max{N(u,v),N(v,u)}. A code C with minimum asymmetric distance A is capable of correcting A - 1 or fewer asymmetric errors. The unidirectional distance is defined by d,(u,v)
= d,(u,v),
if d,(u,v)
= d,(u,v)
and d,(u,v)
= 2d,(u,v)-1,
if d,(u,v)
+ d,(u,v).
This definition came from the fact that was proved by Bose and Rao [l]. A code C is capable of correcting t or fewer pnidiiectional errors iff the following condition holds. For all distinct u, v E C, d,(u, v> = d,(u, v> 2 2t + 1 or d,(u, v) + d,(u,v) 2 t + 1. Hence, a code is able to correct up to t unidirectional errors iff its minimum unidirectional distance is at least 2t + 1. The interest in these codes is due to the fact that for certain applications in optical communication and VLSI we need asymmetric and unidirectional channels [l], [12]. We denote by (n,M,d) code, a code of length n, minimum Hamming distance d, and M codewords. An cw(n, d, w) code is Manuscript received April 3, 1990; revised January 17, 1991. This work was supported in part by the Technion V.P.R. Fund-Albert Einstein Research Fund. This work was presented in part at the IEEE International Symposium on Information Theory, San Diego, CA, January 14-19, 1991. The author is with the Computer Science Department, The Technion -Israel Institute of Technology, Haifa 32000, Israel, IEEE Log Number 9102257. 01991
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a code of length n, constant weight w, and minimum Hamming distance d. An a(n, A) code is a code of length n and minimum asymmetric distance A. An u(n,2t + 1) code is a code of length n and minimum unidirectional distance 2t + 1. Following Weber et al. [12] we denote by A,(n, t) the maximum number of codewords in’s code of length n correcting t or fewer asymmetric errors. A&, t) denote the maximum number of codewords in a code of length n correcting t or fewer unidirectional errors. The known lower bounds on A,(n,t) and A&n, t) are well documented in Weber et al. [12] with some improvements given by Saitoh et al. [6]. In Tables IV and V, we present the new best known bounds for codes of length less than 24 that are capable of correcting at most four errors. We present various methods to obtain new lower bounds on A&, t) and A&, t). In Section II, we present the partitioning method which is used to attain a(n,2> codes. In Section III, we present the combining code methods. We also use the properties of the preparata code to construct a(n,3) codes with n 5 23. In Section IV, we present the variations of shortening by weights. In Section V, we discuss less systematic methods to construct codes with higher distances. In Section VI, we extend some of the methods to obtain new lower bound on A&, t) and we present the tables of all the known lower bounds for n I 23. Throughout this correspondence, we will represent binary codewords in hexadecimal notation II.
THE
PARTITIONING
METHOD
The first method is a modification of the partitioning method for constant weight codes of van Pul and Etzion [9]. A set of codes will be called disjoint if the intersection of any two different members of the set is empty. Let F” denote the set of all binary n-tuples. A partition IT”%> of F” is a set of k subsets, A,,A,; . ., A,, such that each set Ai is an u(n,2) code, Ai n Aj =0 for i # j, lAil 2 IAi+i( for 1 I i I k - 1, and lJ fzlAi = F”. Each partition has an index vector, INDEX (n”“(n>) =(i,,i,; . ., i k ) of length k, with ij = (Ajl. Similarly we define a partition KI(n,w) of the set of all n-tuples with weight w into disjoint cw(n, 4, w) codes. This partition has its own index vector. Let IT(n)= lJ “,=$(n,w) be a partition of Fn into constant weight codes. The correctness of the following lemmas can be easily verified. Lemma 1: If Ai and Bi, 1 by using combmations of Lemmas 2 and 3.
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The following table presents the parameters for the partitions of asymmetric codes that we found either by use of combinatorial and heuristic methods or by applying Construction B. Appendix A presents these partitions. INDEX
n 4 5 6 7 8 9 10 11
(n”“(n))
(4,3,3,3,3) (6,6,6,6,4,4) (12,10,10,10,8,8,6) (18,18,18,18,18,18,16,4) (36,34,34,34,32,31,25,16,14) (62,62,62,62,56,56,52,43,28,21,7,1) (108,106,106,106,106,92,90,90,80,80,34,14,8,4) (180: 180,176,176,168,168,168,168,168,168,164,164)
The partitions for n = 4, 5, and 6, were generated by hand. The partitions for n = 7, 8, and 9, were generated by using some combinatorial arguments and computer search. II”” was generated by using some combinatorial arguments. IIa”(ll) was generated by using Construction B. A partition II with INDEX (II) = (i,; . ., i,) is called optimal if for each other partition for n with index vector (j,, . . . , j,), we have r 2 k, and for each m, m I k, xy=li, 2 ET=1 j,. It is clear that an u(4,2) code of size 4 must include one word of weight 0, two of weight 2, and one of weight 4. Hence, IIas(4) is optimal. It is known [2] that II(6,3) with index vector (4,4,4,4,2,2) is optimal. Based on this fact, we will prove that our ITaS and IIIaS(6) are optimal. For n = 5 we have the number of words with weight two and three in the codes of the partition is not better than 4,4,4,4,2,2, since it can be used to obtain II(6,3). To each code we can add at most one word of weight less than two and at most one word of weight more than three. Hence, IIas(5) with index vector (6,6,6,6,4,4) is optimal. For n = 6, note that an u(6,2) code contains four codewords of weight less than 3 (more than 3) if and only if it contains three words of weight two (four) and one word of weight zero (six). Since there is one word of weight zero (six) the total number of words with weight 0, 1, 2, 4, 5, and 6, in the codes of the partition is not better than 8,6,6,6,6,6,6. Combining these words with the words of the optimal II(6,3), we have that IIas(6) with index vector (12,10,10,10,8,8,6) is optimal. The following table presents the parameters for the partitions of constant weight codes that were used and were obtained by [2], [9]. These partitions are documented in [2]. (n,w>
(6,3) (8,4) (10,4) (10,4) (12,4) (12,4 (12,4)
(12,6) (12,6)
ZNDEX(Il(n,
w))
(4,4,4,4,2,2) (14,14,12,12,10,8) (30,30,30,30,30,22,22,12,2,2) (30,30,30,30,26,25,22,15,2) (51,51,51,51,49,48,48,42,42,4O,i5,7) (51,51,51,51,49,48,48,45,39,36,22,4) (51,51,51,51,49,48,48,45,41,32,22,6) (132,132,120,120,110,94,90,76,36,14) (132,132,120,120,110,94,90,72,42,12)
We also used the well known facts about optimal partitions. rI(n,O) consists of one codeword, II(n, 1) consists of n codes of size 1, II(n,2) consists of n - 1 codes of size n/2 for even n, and ZNDEX(H(n,w)) = ZNDEX(n(n, n - w)). Using these partitions we obtain the following bounds by Construction A.
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1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
A,(ll, 1) 2 A,(12,1) > A,(13,1) 2 A,(14,1) 2 A&15,1) 2 A&16,1) 2 A,(17,1) 2 A&8,1) 2 A&9,1) 2 A&20,1) 2 A,(21,1) 2 A,(22,1) 2 A,(23,1) 2
TRANSACTIONS
180 by considering II(6) and II”‘(s). 336 by considering II(8) and IIas(4). 652 by considering II(s) and IIIaS(5). 1228 by considering II(s) and IIas(6). 2288 by considering II(s) and IIas(7). 4280 by considering II(l0) and IIas(6). 8308 by considering II(l0) and IIns(7). 15762 by considering II and IIa”(8). 29236 by considering II and IIIas(9). 56144 by considering III(12) and IIas(8). 107212 by considering II(12) and IIas(9). 198336 by considering II(12) and IIIaS(lO). 353512 by considering II(12) and ITaS(
Note, that the cartezian products that we used are from II(2k)x II%) for 4 I k I 6 and 2k -4 I i I 2k - 1. We do not have any explanation for this phenomenon. As an illustration we construct the code showing that A,(14,1> 2 1228. From II(8,O) and IIlas(6) we have 1.12 = 12 codewords. From II(8,2) and IIIaS(6) we have 4.12+4.10+4.10+4*10+ 4.8+4.8+4.6 = 256 codewords. From II(8,4) and IIas(6> we have 14.12 + 14.10 + 12.10 + 12. lO+ 10.8+8.8 = 692 codewords. From II(8,6) and IIas(6> we have 4~12+4~10+4~10+4~10+ 4.8+4.8+4.6 = 256 codewords. From II(8,8) and IIas(6) we have 1.12 = 12 codewords. The total is 12 + 256 + 692 + 256 + 12 = 1228 codewords. The same method was found independently by Darwish and Bose [4]. Our partitions and bounds improve on their partitions and bounds. III.
COMBINING
CODES
The second method is a modification of the method of Sloane, Reddy, and Chen [8] for combining codes. We can view the partitioning method as a special case of combining codes. The asymmetric version has two variants. Combining Codes 1: Suppose we are given k +2 codes: an (n,,M,, d,) code C,, an (n,,M, = kM,, d2) code C,, and k pairwise disjoint u(n,, A3) codes B,, B,; . ., B,-,, with the properties that 1) C, is a union of k disjoint cosets of C,: c,=clu(x,+cl)u(x,+c,)u
... u(x,-,+q,
for suitable vectors x1; . . , xk-i (let x0 be the all-zero vector), and 2) the union of the kB,‘s is an (n,, M4, d4) code. Then the new code C consists of all the vectors
Ix; + uI4,
i=O;..,k-1,
UECl,
VE Bi,
i.e., the vectors of the ith coset of C, in C, are appended to the vectors of the ith asymmetric code Bi in all possible ways. Theorem 1: The new code C has length n, + n3, size M1M4 and asymmetric distance
The proof is simple and similar to the one in Sloane et al. [8]. Combining Codes 2: Suppose we are given 2k + 1 codes, k pairwise disjoint u(n,, A1) codes A,, AI; . ., A,-, and k pairwise disjoint u(n,, Aa) codes B,, B,; . ‘, B,-,, with the properties that 1) the union of the kA,‘s is an (n,, M2, dJ code, and 2) the union of the kBi’s is an (n,, M4, d,) code. Then the new
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1991
code C consists of all the vectors
Ml,
i=O;..,k-1,
UEA,,VEB~,
Theorem 2: The new code C has length Cf~tl Ai( IBil, and asymmetric distance
n, + na, size
The proof is similar to the one of Theorem 1. Applications of the two constructions are also similar to the ones in [8]. The only difference is that we have to find the appropriate asymmetric codes. Note, that combining codes 1 is a special case of combining codes 2. We made a separation between the two methods especially since one of the next methods, namely Shortening by Weights, can be applied on all codes obtained from combining codes 1, but not on all codes obtained from combining codes 2. Another asymmetric code is obtained by shuffling codes. Theorem 3: Let C, be a code of length n, with minimum Hamming distance 2d, such that C, is a code with minimum Hamming distance 2d - 2 and form as a union of disjoint cosets of C,, i.e., c*=c1u(x1+c1)u(x2+c1)u
... u(x,-,+C,),
for suitable vectors xi,. . . , xk- r. Then the code Y defined by n IzI y= l&l w*i, i=O
where Wj are all the codewords of weight j from one of the cosets of C,, has asymmetric distance A 2 d. Proof It is obvious that we ~1 E W2j and ~2 E Wzj+zr, d > 2r immediate result from the facts 2d,(q, u2> = d,(u,, a,)+ \~(a,)--
only have to show that for 2 2, d,(ul, u2) 2 d. This is an that d,(u,, u2) 2 2d -2 and w(a,>l2 2d -2+2r > 2d. q
Theorem 3 can be modified to the case where C, has minimum Hamming distance 2d - 1 and C, has minimum Hamming distance 2d - 3. Theorem 4: Let C, be a code of length n, with minimum Hamming distance 2d - 1, such that C, is a code with minimum Hamming distance 2d - 3 and form as a union of disjoint cosets of C,, i.e., C2=C,U(x,+C,)U(X*+C1)u
.‘. u(xk-l+cl),
for suitable vectors xi,. . . , xk- ,. Then the code Y defined by Y=
u
wi,
i=O
where II$ is defined as in Theorem 3, and jWjj = 0 or IWj+ll = 0 for 0 I j I n - 1, has asymmetric distance A 2 cl. Examples of codes with the properties of C, and C, in our constructions are given in [8]. A special interest we have in the following theorem. Theorem 5 [8J The extended Hamming code of length 2” is a union of disjoint cosets of the Preparata code of length 2” for any even integer n 2 4. Now, we give some applications of Theorems l-5. For length 16, the Preparata code coincides with the Kerdock code and the Nordstrom-Robinson code [5]. We generate this code from the
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extended Golay code with the following
VOL.
37, NO. 6, NOVEMBER
generator matrix [5]
100000010000011011000101 010000010000001101100011 001000010000010110110001 000000001000001011111001 000100010000000101011101 000010010000000010101111 000000000100010001110111 000001010000011000011011 OOOOOOOOOOlOOllldOlOllOl 000000000001001110010111 000000000000110111001011 000000110000011111111110
x~=0000001010010001 x/j=0000001011100000 x,=0001000101100000. The weight distribution of the cosets is the same and it is given with the weight distribution of the Preparata code in the following table. Preparata Code
Coset
1 20 48 120 48 20
112 30 112 1 256
256
We take from the Preparata code the codewords of weight 0, 6, 10, and 16 and from the first coset the codewords with weights 4, 8, and 12. By Theorem 3 these codewords form an u(16,3) code A,, and hence, A,(16,2) 2 386. The codewords of weight 6 of the seven cosets are partitioned into five u(16,3) codes of sizes 96, 80, 64, 48, and 48 [2]. The same partition is applied on the codewords of weight 10. Applying a similar argument as in Theorem 3 implies that we can obtain the u(16,3) codes A,, AI;.., A, with the following distribution table.
1 20 112 120 112 20 1 386
B, = {llO}, B, = {loo},
AI
A2
~43
4
4
4
A,
20 96 120 96 20
20 80 120 80 20
20 64 120 64 20
20 48 120 48 20
20 48 120 48 20
120
352
320
288
256
256
160
20
B, = (0001, lllO},
B, = (0100, loll},
B, = (011).
B, = (0010, llOl},
B, = (1000,0111),
B,={llOO},
B, = { lOlO},
B,={OllO}.
By applying combining codes 2 with A,,, A,; . ., A,, we obtain A,(20,2> z 3650. Let B,, B,; . ., B, the following eight u(6,3) codes: B, = {101010,010101},
X,=0000001010000110
0 4 6 8 10 12 16 Total
B, = {lOl},
B, = {0000,1111},
x,=0000000001110001
Ao
B, = (001) ,
By applying combining codes 2 with A,, A,; ., A,, we obtain A&9,2> 2 2404. Let B,, B,; ., B, the following eight u(4,3) codes:
x,=0000000001100110
w
Let B, = {O} and B, = (1). By applying combining codes 2 with A, and A,, we obtain A,(17,2) 2 738. Let B, = (OO}, B, = {Ol}, B, = {lo}, and B, = (11). By applying combining codes 2 with A,, A,, A,, and A,, and adding the word 00007, we obtain A,(18,2) 2 1347. Let B,, B,; . ., B, the following seven a(3,3) codes:
B, = { OlO} ,
x,=0000000000010111
0 4 6 8 10 12 16 Total
1699
B,, = {OOO,lll},
by taking columns 9 through 24 of codewords where in columns 1 through 7 the weight of the codewords is less than 2. We take translates by adding the following vectors
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1991
B, = {000001,111000,110111},
B, = {OOOlOO,llOOlO, llllOl},
B, = {OOOOlO,llOOOl, lllllO},
B,={001000,000111,111011},
B,={100000,010011,101111},
B, = {010000,100011,011111},
B, = {110100,001011}.
We apply combining codes 1 with the shortened Preparata codes and its seven cosets. We add to the constructed code 202 words, given in Appendix B, using the compressed form of Section V, to obtain A,(21,2) 2 5834. Let Do be the code defined by taking columns 9-24 of codewords from the extended Golay code, where in the first eight coordinates there are either two ONES in columns 1 and 2, or four ONES in the first four coordinates. To D, we add the codewords 0000, ODCB, 635A, 6E91, 916E, 9CA5, F234, FFFF, to obtain A,(16,3) 2 72. A code with the same size was found by the computer search of Saitoh et al. [6]. Let D, be the code defined by taking columns 9-24 of codewords from the extended Golay code where in the first coordinates there are either two ONES in columns 1 and 3 or four ONES in the columns 1, 2, 3, and 5. D, and D, are two disjoint u(16,4) codes and Do U D, is a (16,128,6) code. Let B, = (0} and B, = (1). By applying combining codes 2 with Do and D,, and adding the all-zero and the all-one words, we obtain A,(17,3) 2 130. IV.
SHORTENING
BY WEIGHTS
The third method is shortening by weights was first presented by Shiozaki [7] and generalized by Weber et al. [ll]. The method is applied on codes of length n and Hamming distance at least 2d - 1 to obtain u(n - m, d) codes. We modify the method in a way that it can be applied on asymmetric codes which are constructed by combining codes 1. First, we describe the method as appeared in Weber et al. [ll]. Let x = (x’, x”) be the (m, n m) partition of any x E (GF(2))“. Let C, be a code of length n and Hamming distance at least 2d - 1. Define
30
20 30
n - m and s E(GF(~))~, where for w=O,l;*., weight of x. We define a code Y of length n by ?I-??I Y= U Maw), w=o
w(x)
is the
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TABLE I a,
0 4
0 1
5 6
0 0
I
0 1
8 9
0
10 11 12 13 14 16 17
00000
20 64
6
188
7
320 660 960
8 9 10
11111 11110 11100 11000 10000 00000
30 42 46 42 30
18
0 0
960 660 320 188 64 20
18
11111
1
1308
1
for i = O,l,. . ., IZ - m are such that 2 Id, /2] > A with the’same proof as for shortening by weights [ll]. If vi and v2 belong to the same asymmetric code, but they are different then da(vl,vZ) 2 A3 2 A and hence, d,(q,aJ 2 A. The (23,214,5) Wagner code [lo] can be formed with the following generator matrix 01000000000000111110000 00100000000000111001100 00010000000000101010101 00001000000000010101110 10000000000000111000010 00000100000000110101000 00000010000000110010001 00000001000000101001001 00000000100000010100011 00000000010000001010110 00000000001000001101101 00000000000100101111010 00000000000010110110111 00000000000001011011111. We add to this code the 33 words OlOAOO, 440008, 025000,
a,
T&J
0000
1
1110
12
1111 1110 1100 1000
48 12 88 88 72 48 12
18
00000
The code C, of length n-m is now formed by taking all the codewords of Y and then deleting the first m coordinates of each codeword. Weber et al. [ll] proved the following theorem. Theorem 6 [llj
1
11
8450
where ai E (GF(2))”
Tw(a,)
12 14
Total
dh(aj,aj+l)
a,
0 5
1
1408
0 1
22
w
TJa,)
1308
0
18
37, NO. 6, NOVEMBER
APPLYING SHORTENINGBYWEIGHTSONTHEGOLAYCODE
0 0 1 0 1
15
VOL.
TABLE II
APPLYING SHORTENINGBY WEIGHTS ONTHE (23,214, ~)WAGNER CODE w
THEORY,
0000
0000 1000 1111
19
Total
228
1 442
OOAOSO, 128010, OlOOlA, 000460, 103Cl1, 140201, OOC300, 404044,080005,0A0140,080828,300002,092400,046010,201110, 041402, 202204, 0204OC, 05CO61, 131106, 420021, 008806, 102120, 000851, 204CO0, 24OOC0, 581000, 019001, 2A4A40, OOOOB4,and their complements to’obtain A,(23,2) 2 16450. By applying the shortening by weights on the (23,214,5) Wagner code we obtain the code of Weber et al. [ll], with the distribution given in Table I. We add the 166 words, given in Appendix B, using the compressed form of Section V, to obtain A,(22,2) 2 8616. For A 2 4 we use shortening by weights on the Golay code, constructed by deleting the last coordinate of the extended Golay code, and the weight distribution is given in Table II. We obtain the following codes. We use the shortening by weights on the first five columns of the Golay code and add the ten words 02650, 06120, 08284, ODFFE, 11408, 20842, 26FF9, 33EDF, 3E9BF, 3F775, to obtain A,(18,3) 2 238. We use the shortening by weights on the first four columns of the Golay code and add the 16 words OOOA5, 03108, 04812, 079FF, 0824B, 12A44, 212D0, 3EFF9, 44360, 5F6FE, 6FFC7, 71F7F, 78000, 7BlED, 7C8F7, 7F95A, to obtain A,(19,3) 2 458.
V. CODES WITH ASYMMETRIC THAN
DISTANCE
GREATER
FOUR
For A 2 5 the codes have even less mathematical structure than for A = 4. To obtain a lower bound for A,(n, A - 1) we can start with a good cw(n + 1,2A,[n + l/2]) code [2]. We delete its last column and try to add as many codewords as we succeed when we are adding them weight by weight. Using this method we obtain A,(18, 4) 2 39, A,(19,4) z 54, A,(20,4) 2 71, A,(21,4) 2 104, A,(22,4) 2 160, and A,(23,4) 2 240. All the codes with asymmetric distance 5 are presented in Appendix C, in a compressed form mentioned in Brouwer et al. [2]. We generalize this form to represent words added to a code C. In Appendix C, the code C is empty. The words added to the code C are first sorted into lexicographic order, yielding a sequence of words ci, c2,. . . , cM. The compressed notation for these words is (Y~,(Y~;.‘,(Y~. To find cu&l I i I M), let be the list of all vectors, arranged in lexicographic Ul,UZ,“’ order and following ci-i in the lexicographic order, that have distance 2 d from the subcode C U {c,, . . . , ci- 1}. If ci = u, we set (Y~= r - 1. Informally, given C, ci; . ., ci-i, we must skip (Y~ lexicographic words to get ci. We also abbreviate a, a; . ., a(k times) by uk.
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Weber et al. [12] proved that A&,1) is the same as the maximum number of codewords in a code of length n and minimum Hamming distance 3. The most updated table on these bounds is given by Brouwer et al. [2]. For other unidirectional distances the codes have some similarity to the asymmetric codes. The combining codes methods and the shortening by weights method can also be applied to obtain unidirectional codes. We have to perform the following modifications. In combining codes 1, we take ku(n,,2t, + 1) codes instead of ka(n,, As) codes. This method will be called combining codes 1’. We obtain the following theorem. Theorem 1’: The new code C has length nr + n3, size M1M4 and unidirectional distance 2t + 12 min{d,,2t, + l,d, + d4). In combining codes 2 we take k z&z,, 2t, + 1) codes Uk-r, and ku(n,,2t, + 1) codes instead of the asymu,,u,,..*, metric codes. This method will be called combining codes 2’. We obtain the following theorem. Theorem 2’: The new code C has length n, + na, size Cf:t]Q] ]Bi], and unidirectional distance 2t + 12 min(2tr + 1,2t, + 1, d, + d4}. Some more specific variants of combining codes can also be used here. Shuffling codes, Theorems 3 and 4 will have the following form. Theorem 3’: Let C, be a code of length n, with minimum Hamming distance 2d, such that C, is a code with minimum Hamming distance 2d - 2 and form as a union of disjoint cosets of C,, i.e., C,=C,U(x,+C,)u(x,+C,)U
... U(XkplfCl),
for suitable vectors x1,. . . , xk- i. Then the code Y defined by n I?I y= U w*i, i=O
where I+$ is defined as in Theorem 3 and Wzi and W2i+2d-2 are codewords from the same coset, has unidirectional distance 2t+1>2d-1. Theorem 4’: Let C, be a code of length n, with minimum Hamming distance 2d - 1, such that C, is a code with minimum Hamming distance 2d -3 and form as a union of disjoint cosets of C,, i.e., C,=C,U(x,+C,)U(x,+C,)U
... U(X&l+Cl),
for suitable vectors x1,. . . , xk- r. Then the code Y defined by Y=
;
wi,
i=O
where Wj is defined as in Theorem 3, ]I+$] = 0 or lWj+ll = 0 for 0 I j I n - 1, and also Wzi and W2i+2d-2 are codewords from the same coset, has unidirectional distance 2t + 1 2 2d - 1. The shortening by weights is defined similarly with the equations [12] n--m Y= U Tw(uw), w=o where u,E(GF(~))” for i=O,l;..,n-m+2t are such that forOIj be the code defined by taking the words of weights 6, 8, and 10, after deleting the first eight coordinates of the extended Golay code where in the first eight coordinates there are either two ONES in columns 1 (2) and 3, or four ONES in columns 1, 2, 3, and 5 (6). Let V, be the code defined by taking the words of weights 4, 8, and 12, after deleting the first eight coordinates of the extended Golay code where in the first eight coordinates there are four ONES in columns 1, 2, 3, and 7. V,,“V,, V,, and V,, are disjoint u(16,4) codes of sizes 56, 56, 56, and 32, respectively, and U ?=,K is a (16,200,6) code.
P(5): 52436412315423611521364342361425. P(6):
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Let B, = (01 and B, = (lj. By applying combining codes 2’ with V, and VI, and adding the words 00071 and lFF6A, we obtain A&17,3) 2 114. Let B, = (Ol}, B, = {lo}, B, = {ll}, and B, = (00). By applying combining codes 2’ with V,,V,,V,,V,, and adding the word 30203, we obtain A,(18,3) 2 201. All the other codes that we have constructed are given in Appendix D using the compressed form of Section V. Tables IV and V summarize all the bounds that we have obtained.
ACKNOWLEDGMENT
The author wishes to thank N. J. A. Sloane for supplying partitions for constant weight codes. He also thanks the referees for their helpful comments.
APPENDIX
A
This appendix gives the partitions into disjoint asymmetric codes. The notation is as in Brouwer et al. [2]. We ordered the n-tuples in lexicographic order and the disjoint ah, 2) codes are numbered by 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, and so on. If IP(4): 1253412434155231 then there are 16 4-tuples in the partitions. The first 4-tuple (namely 0000) is in code 1, the second (namely 0001) in code 2, the third (namely 0010) in code 5, and so on.
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TABLE V
TABLE III
LOWER BOUNDS ON A?(n.
APPLYING SHORTENINGBY WEIGHTS ON THE (22, 213, 5) CODE
w 5 6 7
8 9
10 11 12 13 14 15 16 17 Total
n
T&J
a, 1 1 0 0 1 1 0 0 1 1 0 0 1 1
4
16
9
49 115 256 448 617 731 736 592 411 241 96 32
10 11 12 13 14 15 16 17
18 19 20 21 22 23
11 4351
n 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
t=l
t=2
18*
t=3
108 180X 336” 652’ 1228x 2288” 4280x 8308” 15762x 29236” 56144x 107212” 198336x 353512”
12*
18* 30 54 98 186 266 386” 738” 1347x 24O4x 3650’ 5834” 8616x 16450”
26 52 92 184 256 384” 736’ 1344” 2080X 3423” 4672’ 8544” 16384
7*
10 16” 24 42 62 114x 201X 376 737 1474 2588 4096
;: 6* s* 10 14” 24’ 37x 51” 69x 102” 154” 229x
t=4
;: ;:
t=4
t)
6* 12* 36* 62*
t=3
10 16
*Tight bounds. “New bounds.
TABLE IV LOWER BOUNDS ON A&L
t=2
t)
2*
;: 6* 8* 14* 18* 30 44 72 130” 23gx 458” 860 1628 3072 4096
2*
2: ;: 4* 6*
s* 12* 16* 26* 39x 54” 71” 104” 160” 240”
*Tight bounds. “New bounds. APPENDIX B
Words added to obtain A,(21,2) 2 5834: 0, 21, 4, 0, 1, 0, 5, 2, 1, 0, 3, 2, 1, 3, 02, 6, 3, 03, 12, 4, 0, 2, 1, 3, 1, 0, 1, 03, 1, 4, 0, 2, 02, 1, 05, 1, 0, 1, 02, 6, 2, 1, 8, 3, 1, 04, 2, 02, 1, 2, 02, 2, 05, 1, 02, 1, 04, 1, 02, 1, 07, 2, 03, 1, 02, 1, 02, 13, 0, 1, 0, 1, 0, 2, 0, 2, 1, 03, 2, 3, 02, 2, 02, 1, 0, 5, 2, 06, 1, 0, 12, 0, 2, 02, 1, 0, 1, 02, 1, 0, 3, 03, 1, 05, 12, 4, 02, 2, 0’0, 1, 04, 1, o”, 1, 0s. Words added to obtain A,(22,2) 2 8616: 1, 3, 5, 1, 0, 4, 02, 1, 04, 12, 04, 1, 05, 1, 02, 1, 04, 2, 08, 1, 03, 2, OS, 12. 0. 1. 0s. 1. 03. 1. 04; 1,‘oi4, i, o$ i, o$, 4, OS, 13, 03, 1, 05, 1, 03, 1, OS, 1, 04, 1, 2, 03, 22, 06. Words added to obtain A,(21,2) 2 4672: 03, 1, 02, 1, 0, 1, 22, 1, 0, 2, 02, 2, 02, 1, 05, 1, 03, 1, 0, 2, 02, 2, 1, 0 2 cl2 2 011 1 0 1 0 1: o;, 2’ 1’0, ; d3 ; ; bs, 1, 05, 1, 0, 2, 02, 2, 1, 0, 1, 03, 2, 011, 2, 02, 1;, i4: o;, 4: 0'2, i, 02, 2, 1, 3*, 04, 1, 8, 0, 10, 03, 5, 14, 3, 0, 3, 8, 12, 02, 2, 1, 15, 32, 0, 3, 0, 2, 1, 0, 2, 02, 2, 1, 4, 3, 03, 13, 0, 1, 3, 0, 3, 2, 0, 1, 03, 32, 2, 0, 7, 0, 5, 03, 1, 04, 2, 1,
0, 9, 07, 1, 02, 1, 03, 4, 0, 3, 0, 2, 1, 0, 3, 1, 5, 02, 2, 04, 1, 2, 02, 4, 0, 6, 12, 2, 4, 02, 9, 0, 3, 5, 22, 03, 4, 1, 0, 2, 1, 2, 1,02, 12, 03, 3, 1, 2, 0, 1, o*, 3, 9, 0, 1, 0, 1, 0,4, 2, 02, 22, 03, 7, 0, 12, 8, 02, 2, 13, 02, 22, 0, 3, 03, 2, 5, 6, 03, 2, 1, 0’. APPENDIX C
A,(18,4) 2 39: 0, 2394, 3106, 1215, 609, 624, 898, 552, 162, 98, 127, 14, 33, 39, 22, 2, 14, 1, 0, 8, 3, 19, 6, 5, 7, 1, 03, 11, 7, 64, 22, 46, 0, 1, 02, 3. A,(19,4) 2 54: 0, 4256, 7835, 1019, 789, 3180, 72, 616, 676, 374, 292, 335, 46, 263, 63, 3, 125, 226, 18, 7, 51, 139, 0, 1, 0, 40, 4, 75, 1, 13, 40, 24, 17, 9, 3, 8, 0, 42, 96, 7, 26, 56, 3, 21, 77, 83, 39, 82, 0, 19, 32, 30, 91, 31. A,(20,4) 2 71: 0, 7098, 8342, 25626, 3226, 6677, 1202, 3661, 39, 1138, 1069, 1464, 19214, 1006, 2143, 789, 462, 109, 4684, 10488, 494, 3392, 12515, 493, 767, 141, 306, 96, 1, 75, 11, 12, 7, 0, 54, O*, 3, 13, 4, 0, 2, 02, 38, 02, 4, 03, 1, 1, 02, 1, 6, 1, O?, 6, 1, 0, 2, 0, 4, 0, 18, 02, 20. A,(21,4) 2 104: 0, 637, 338, 16, 3498, 258, 1755, 989, 20, 1372, 1787, 1368, 576, 323, 29, 391, 151, 12, 38, 258, 214, 175, 34, 336, 89, 267, 521, 305, 161, 5, 128, 335, 123, 117, 524, 42, 436, 173, 285, 1416, 1507, 913, 2000, 0, 611, 2715, 439, 44, 17, 32, 679, 599, 648, 129, 529, 1325, 640, 220, 931, 2052, 195, 374, 75, 116, 21, 11, 3, 8, 3, 2, 07, 7, 2*, 05, 1, O*, 5, 7, 17, 23, 62, 05, 4, 2, 32, 0, 40, 65. A,(22,4) 2 160: 0, 2523, 8398, 1773, 9887, 8139, 1573, 22197, 6817, 1308, 1797, 5100, 2039, 3110, 22, 3313, 416, 1117, 707, 236, 743, 3035, 1101, 514, 628, 659, 355, 254, 531, 727, 721, 83, 243, 304, 21, 140, 61, 112, 69, 58, 537, 103, 342, 21, 172, 16, 78,
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177, 243, 107, 12, 211, 1073, 122, 285, 280, 3, 2, 83, 57, 135,244, 0, 70, 2, 126, 3, 14, 139, 182, 1, 17, 18, 11, 38, 323, 10, 0, 4, 122, 69, 94, 48, 43, 65, 8, 20, 3, 2, 34, 19, 7, 5, 14, 67, 12, 23, 0, 2, 0, 36, 3, 13, 2, 74, 1, 52, 10, 66, 23, 31, 70, 101, 30, 22, 7, 37, 26, 13, 8, 1, 0, 15, 0, 2, 23, 4, 27, 26, 12, 13, 3, 4, 0, 20, OS, 15, 5, 8, 03, 6, 23, 03, 18, 22, 1. A,(23,4) 2 240: 0, 31472, 32163, 11212, 23438, 3403, 2035, 7607, 17759, 2735, 3222,2933, 2490, 61, 4098, 2376, 2547, 1507, 1772, 1209, 7139, 4686, 77, 4696, 2249, 3577, 176, 3215, 784, 198, 1959, 1605,400,7518,67,931,2429,412,1266, 1845,4, 325, 2306, 1101, 217, 2398, 712, 578, 1708, 51, 117, 108, 237, 695, 2368, 2, 729, 71, 3, 0, 8, 214, 1217, 728, 6, 30, 727, 110, 29, 9, 40, 58, 157, 187, 39, 3, 27, 51, 114, 96, 215, 1, 13, 14, 706, 12, 92, 320, 353, 40, 223, 336, 147, 13, 89, 02, 295, 59, 19, 13, 60, 49, 295, 29, 1, 21, 25, 0, 63, 10, 30, 23, 2, 4, 18, 3, 25, 2, 25, 351, 3, 45, 18, 5, 14, 311, 27, 190, 128, 13, 141, 399, 34, 25, 2, 13, 15, 6, 186, 211, 100, 0, 17, 244, 3, 6, 9, 5, 7, 5, i, 0, 6, 5, 217, 54, 71, 0, 2, 5, 4, 16, 6, 0, 6, 148, 2, 02, 1, 10, 0, 3, 8, 12, 50, 0, 2, 3, 2, 1, 6, 8, 02, 1, 0, 2, 12, 0, 3, 5, 2, 02, 1, 0, 1, 7, 0, 8, 02, 55, 48, 03, 1, 03, 1, 06, 1, 0, 1, 0, 2, 0, 10, 15, 13, 02, 2, 5, 3, 20, 0, 29, 0, 5. APPENDIX D A,(13,3)116: 229, 132, 96, 25, 30, 4, 02, 5, 1, 5, 8, 2, 03. A,(16,4)114: 626, 4258, 126, 470, 588, 6, 44, 13, 78, 32, 17, 7, 3, 0. A,(17,4)1 24: 626, 1409, 929, 662, 156, 63, 21, 42, 16, 15, 3, 0, 32, 0, 16, 13, 72, 3, 05. A,(18,4) 2 37: 10006, 1743, 681, 268, 228, 326, 375, 178, 80, 210, 41, 54, 50, 35, 0, 3, 2, 0, 26, 15, 15, 8, 14, 02, 2, 1, 3, 02, 1, 03. A,(19,4) 2 51: 5260, 7763, 951, 739, 3183, 71, 456, 541, 321, 188, 265, 226, 5, 19, 2, 161, 251, 23, 35, 13, 21, 19, 3, 4, 12, 0, 95, 0, 16, 18, 14, 1, 22, 0, 22, 5, 17, 1, 32, 4, O*, 8, 0, 7, 02, 1, 0. A3(20, 4) r 69: 8191, 784, 3842, 1108, 13408, 7384, 1111, 3313, 1294, 960, 1393, 4546, 9852, 841, 7, 2700,445, 111, 4170, 10008, 128, 2730, 10431,699,93,278,96, 1, 69, 9, 11, 8, 0, 5, 2, 0, 4, 12, 3, 05, 4, 04, 1, 02, 1, 0, 1, 0, 1, 0, 1, 2, 08. A,(21,4) r 102: 1023, 2084, 8883, 5119, 3030, 110, 5474, 175, 53, 455, 2796, 446, 175, 7, 21, 153, 26, 7, 10, 665, 33, 237, 2, 194, 14, 51, 29, 117, 157, 39, 25, 120, 189, 158, 240, 36, 277, 2238, 178, 261, 325, 821, 1612, 1009, 92, 147, 23, 14, 12, 554, 348;247, 508, 949, 839, 21, 1882, 507, 79, 322, 64, 101, 27, 13, 10, 8, 0, 1, 3, 02, 2, 06, 2, 04, 1, 2, 04, 12, 02, 2, 3, 07.
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A,(22,4)1154: 15243, 15783, 12153, 1383, 1002, 9789, 6404, 361, 14930, 1324, 3815, 4737, 521, 1356, 12, 943, 4676, 1490, 246, 1162, 1531, 736, 811, 2108, 222, 781, 357, 49, 924, 229, 168, 474, 6, 54, 446, 73, 204, 7, 50, 81, 1241, 276, 79, 12, 63, 83, 104, 623, 35, 416, 2, 65, 36, 98, 9, 86, 175, 20, 7, 129, 7, 10, 79, 2, 103, 2, 1, 30, 7, 6, 102, 16, 27, 0, 2, 155, 19, 63, 69, 81, 8, 10, 34, 8, 52, 1, 18, 21, 7, 9, 44, 28, 22, 16, 19, 1, 12, 14, 1, 11, 1, 12, 2, 26, 72, 22, 10, 11, 0, 1, 0, 24, 15, 04, 5, 1, 12, 15, 4*, 13, 02, 1, 02, 1, 0, 1, 02, 2, o14. A,(23,4)> 229: 14357, 50448, 12388,22442,4151, 2743, 5782, 1127, 14198, 3768, 970, 1676, 2374, 1572, 2583, 7901, 8850, 1426, 2006, 35, 1762, 913, 550, 9, 71, 1531, 27, 616, 240, 1922, 123, 1451, 4251, 1222, 27, 2484, 254, 510, 687, 2, 295, 826, 257, 102, 20, 1155, 219, 456, 59, 46, 83, 158, 262, 198, 25, 33, 170, 41, 458, 550, 449, 433, 429, 29, 2, 80, 212, 182, 23, 12, 72, 10, 31, 271, 2, 17, 26, 87, 9, 143, 11, 24, 71, 9, 50, 258, 16, 124, 42, 453, 5, 80, 39, 46, 59, 71, 114, 6, 2, 7, 1, 16, 26, 25, 1, 20, 1, 2, 9, 15, 94, 39, 666, 3, 70, 35, 19, 49, 11, 0, 35, 21, 27, 8, 0, 4, 77, 1, 3, 4, 19, 2, 1, 22, 0, 6, 03, 15, 33, 7, 0, 32, 4?, 7, 0, 2, 5, 106, 25, 3, 7, 57, 57, 6, 1, 15, 0, 22, 4, 0, 32, 7, 0, 27, 0, 14, 63, 5, 04, 4, 05, 2, 1, 2, 1, 08, 2, 0, 4, 08, 2, 0, 2, 1, 3, 05, 12, 04, 2, 07.
REFERENCES [l] B. Bose and T. R. N. Rao, “Theory of unidirectional error-correcting/detecting codes,” IEEE Trans. Comput., vol. C-31, pp. 521-530, June 1982. [2] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, and W. D. Smith, “A new table of constant weight codes,” IEEE Trans. fnform. Theory, vol. 36, pp. 1334-1380, Nov. 1990. [31 J. H. Conway and N. J. A. Sloane, “Lexicographic codes: error-correcting codes from game theory,” IEEE Trans. Inform. Theory, vol. IT-342, pp. 337-348, May 1986. [4] N. Darwish and B. Bose, “New lower bounds for single asymmetric error correcting codes,” preprint. [5] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [61 Y. Saitoh, K. Yamaguchi, and H. Imai, “Some new binary codes correcting asymmetric/unidirectional errors,” IEEE Trans. Inform. Theory, vol. 36, pp. 645-647, May 1990. [7] A. Shiozaki, “Construction for binary asymmetric error-correcting codes,” IEEE Trans. Inform. Theory, vol. IT-28 pp. 787-789, Sept. 1982. [Bl N. J. A. Sloane, S. M. Reddy, and C. L. Chen, “New binary cades,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 503-510, July 1972. [91 C. L. M. van Pul and T. Etzion, “New lower bounds for constant weight codes,” IEEE Trans. Znform. Theory, vol. 35, pp. 1324-1329, Nov. 1989. [lo] T. J. Wagner, “A search technique for quasi-perfect codes,” Znform. Contr., vol. 9, pp. 94-99, 1966. [ll] J. H. Weber, C. de Vroedt, and D. E. Boekee, “Bounds and constructions for binary codes for length less than 24 and asymmetric distance less than 6,” IEEE Trans. Znform. Theory, vol. 34, pp. 1321-1331, Sept. 1988. [12] -, “Bounds and constructions for codes correcting unidirectional errors,” IEEE Trans. Inform. Theory, vol. 35, pp. 797-810, July 1989.
