New Inequalities for q-ary Constant-Weight Codes

Transcription

1 New Inequalities for q-ary Constant-Weight Codes Hyun Kwang Kim 1 Phan Thanh Toan 1 1 Department of Mathematics, POSTECH International Workshop on Coding and Cryptography April 15-19, 2013, Bergen (Norway

2 Outline 1 Results on binary codes Binary codes A fundamental problem in coding theory Delsarte s linear programming bound for binary codes New upper bounds for binary constant-weight codes 2 q-ary codes Delsarte s linear programming bound for q-ary codes New inequalities for q-ary constant-weight codes New upper bounds for q-ary constant-weight codes

3 Binary codes What is a code? Binary code Let F = {0, 1}. A subset C of F n is called a (binary code of length n. An element of a code C is called a codeword. Minimum distance of a code Hamming distance between two vectors u, v F n, denoted by d(u, v, is the number of coordinates where they differ. Minimum distance of a code C is defined by min{d(u, v u, v C, u v}.

4 A fundamental problem in coding theory A fundamental problem in coding theory Definition Given n and d, define A(n, d = maximum number of codewords in any code of length n and minimum distance d. Remarks Determining the exact values of A(n, d is an extremely difficult problem (for large n. Since A(n, d = A(n + 1, d + 1 if d is odd, we can always assume that d is even. When d is even, all values of A(n, d are known for n 16. For an unknown A(n, d, one may try to find its lower and upper bound.

5 Delsarte s linear programming bound for binary codes Delsarte s linear programming bound Distance distribution of a code Let C be a code of length n. The distance distribution {A i } n i=0 of C is defined by A i = 1 C {(u, v C2 d(u, v = i} for i = 0, 1,..., n. Remark By definition, A 0 = 1 and n i=0 A i = C. Delsarte s linear programming bound For upper bounds on A(n, d, Delsarte s linear programming bound is a powerful bound. Delsarte s linear programming bound is based on the fact that the following linear combinations of the distance distribution {A i } n i=0 are nonnegative (as follows.

6 Delsarte s linear programming bound for binary codes Delsarte s linear programming bound Theorem (Delsarte Let C be a code with distance distribution {A i } n i=0. For k = 1, 2,..., n, n P k (n; ia i 0, where P k (n; x is the Krawtchouk polynomial given by i=0 P k (n; x = k ( ( 1 j x j j=0 ( n x. k j

7 Delsarte s linear programming bound for binary codes Delsarte s linear programming bound Theorem (Delsarte s linear programming bound A(n, d 1 + max(a 1 + A A n, where the maximization is taken over all (A 1, A 2,..., A n satisfying A i 0 for i = 1, 2,..., n and satisfying the above linear constraints. Remark If d is even, then A(n, d is attained by a code with all vectors having even weights. Hence, if d is even, then we can put A i = 0 if i is odd. Also, by definition, A i = 0 if 0 < i < d.

8 Delsarte s linear programming bound for binary codes Delsarte s linear programming bound Theorem (Delsarte s linear programming bound and its improvements Let C be a code with distance distribution {A i } n i=0. For k = 1, 2,..., n, n P k (n; ia i 0. i=0 If M = C is odd, then n P k (n; ia i 1 M i=0 ( n k. If M = C 2 (mod 4, then there exists t {0, 1,..., n} such that n P k (n; ia i 2 [( n ] + P k (n; t. M k i=0

9 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Result 1 We prove simultaneously Delsarte s linear programming bound and its well known improvements. The proof is based on counting the number of 2 k submatrices of C, where C is considered as a C n matrix (each codeword in C is a row. Definition For each k = 1, 2,..., n, we introduce polynomials k ( ( x n x k P k (n; x = and P + k j k j (n; x = j=0 j odd j=0 j even ( x j ( n x. k j Remark It follows that P + k (n; x + P k (n; x = ( n k. The polynomial P k (n; x := P + k (n; x P k (n; x is called the Krawtchouk polynomial.

10 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices The proof immediately follows from the following lemma. Lemma Let C be a code with size M and distance distribution {A i } n i=0 and let t be the number of columns of C containing an odd number of ones. For each k = 1, 2,..., n, n P k (n; ia i 2 M i=1 [ ( n ] N δp + k (n; t k (1 and n i=1 ( P + n k (n; ia i (M [ ( n ] N δp + k (n; t, (2 k M k where{ N and δ are given by M 2 if M is even N = 4 M 2 1 if M is odd 4 and δ = { 1 if M 2 (mod 4 0 otherwise.

11 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma Write C = (c mi, 1 m C, 1 i n. Let S 1 (k be the number of 2 k matrices ( cmi1 c A = mi2 c mik c li1 c li2 c lik such that m l, i 1 < i 2 < < i k, and A contains an odd number of 1 s. The entries of A are on the intersection of two rows and k columns of C.

12 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued For an ordered pair (u, v of different rows of C, to get such a matrix A choose j coordinates (j odd where u and v are differ and choose k j coordinates where u and v are the same.

13 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued Hence, an ordered pair (u, v will contribute n j=0 j odd ( ( d(u, v n d(u, v = P k (n; d(u, v j k j to S 1 (k. Therefore, S 1 (k = = u,v C u v n i=1 = M P k P k (n; d(u, v = n (n; i u,v C d(u,v=i n P k (n; ia i. i=1 1 i=1 u,v C d(u,v=i P k (n; i

14 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued Let u 1, u 2,..., u n be the n columns of C. For k columns u i 1, u i 2,..., u i k, to get such a matrix A choose one row such that the intersection of this row with the k columns has an odd number of 1 s and choose another row such that the intersection of that row with the k columns has an even number of 1 s.

15 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued Hence, S 1 (k = 2 i 1 <i 2 < <i k wt(u i u i k [M wt(u i u i k ]. If M is odd, then δ = 0 by definition. For all i 1 < i 2 < < i k, wt(u i 1 + +u i k [M wt(u i 1 + +u i k ] M 1 M + 1 = M2 1 = N So ( n S 1 (k 2 N = 2N k i 1 <i 2 < <i k [ ( n ] = 2 N δp + k (n; t. k Therefore, (1 is proved if M is odd.

16 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued If M 0 (mod 4, then δ = 0 by definition. For all i 1 < i 2 < < i k, wt(u i u i k [M wt(u i u i k ] M 2 M 2 = M2 4 = N. So ( n [ ( n ] S 1 (k 2N = 2 N δp + k (n; t. k k Therefore, (1 is proved if M 0 (mod 4.

17 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued If M 2 (mod 4, then δ = 1 by definition. Let I be the collection of coordinates i such that the column u i contains an odd number of 1 s. If {i 1, i 2,..., i k } I is odd, then wt(u i u i k [M wt(u i u i k ] M 2 However, if {i 1, i 2,..., i k } I is even, then wt(u i u i k [M wt(u i u i k ] M 2 2 S 1 (k 2 [ = 2 N {i 1,i 2,...,i k } I odd ( n k δp + k (n; t Hence, (1 is proved if M 2 (mod 4. N + ]. M 2 = M2 4 = N. M {i 1,i 2,...,i k } I even = N 1. N 1

18 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Proof of Lemma (continued For (2, one can count (two times of the number of 2 k submatrices A such that A contains an even number of 1 s or just use the equality n P (n; ia i + i=1 n i=1 ( P + n (n; ia i = (M 1. k

19 Delsarte s linear programming bound for binary codes Counting the number of 2 k submatrices Theorem (Delsarte s linear programming bound and its improvements Let C be a code with distance distribution {A i } n i=0. For k = 1, 2,..., n, n ( n P k (n; ia i. k i=1 i=1 If M = C is odd, then n ( n P k (n; ia i + 1 ( n. k M k If M = C 2 (mod 4, then there exists t {0, 1,..., n} such that n ( n P k (n; ia i + 2 [( n ] + P k (n; t. k M k i=1 Proof of Theorem Take sum of inequalities (1 and (2 in the above lemma.

20 New upper bounds for binary constant-weight codes Upper bounds for A(n, d, w Definition Given n, d, and w, define A(n, d, w = maximum number of codewords in any code of length n and minimum distance d such that each codeword has exactly w ones.

21 New upper bounds for binary constant-weight codes Counting the number of 1 k submatrices Proposition (1-row k-column formula Let C be a code of length n and constant-weight w. For each k = 1, 2,..., n, wt(u i u i k = MP k (n; w, i 1 < <i k where the sum is taken over all (i 1, i 2,..., i k such that i 1 < i 2 < < i k. Sketch of proof Count the number of 1 k submatrices of C containing an odd number of 1 s.

22 New upper bounds for binary constant-weight codes Counting the number of 2 k submatrices Result 2 (2-row k-column formula Let C be a code of length n and constant-weight w. For each k = 1, 2,..., n, and w P k (n; 2iA 2i 2 M i=d/2 w P + k (n; 2iA 2i 2 M i=d/2 [(( n k r k q k (M q k +r k (q k + 1(M q k 1] [(( n k r k q k (M q k ( n +r k (q k + 1(M q k 1] (M 1, k where q k and r k are the quotient and the remainder, respectively, when dividing MP k (n; w by ( n k, i.e., MP k (n; w = q ( n k k + rk, with 0 r k < ( n k.

23 New upper bounds for binary constant-weight codes New upper bounds on A(n, d, w Sketch of proof Count (two times of the number of 2 k submatrices of C containing an odd (even number of 1 s. Result 2 gives the following new upper bounds for A(n, d, w, n 28. A(18, 6, (428 A(18, 6, (425 A(20, 6, (1421 A(27, 6, (66079 A(27, 6, (84574 A(27, 6, (91080 A(28, 6, ( A(28, 6, ( A(28, 6, (169740

24 New upper bounds for binary constant-weight codes New upper bounds on A(n, d, w New upper bounds A(24, 10, (171 A(24, 10, (223 A(24, 10, (247 A(26, 10, (214 A(27, 10, (299 A(28, 10, (2629 A(26, 12, (48 A(27, 12, (140 A(27, 12, (156 A(28, 12, (149 A(28, 12, (199 A(28, 12, (245 A(28, 12, (265

25 q-ary codes q-ary codes Results 1 and 2 appeared in [B. G. Kang, H. K. Kim, and P. T. Toan, Delsarte s linear programming bound for constant-weight codes," IEEE Trans. Inf. Theory, vol. 58, no. 9, pp , Sep. 2012]. In the remaining, we generalize Results 1 and 2 to q-ary codes.

26 q-ary codes Definition Definition Let F q be a finite field with q elements. A subset C of F n q is called a q-ary code of length n. Definition Given n and d, define A q(n, d = maximum number of codewords in any q-ary code of length n and minimum distance d. Definition Given n, d, and w, define A q(n, d, w = maximum number of codewords in any q-ary code of length n and minimum distance d such that each codeword has weight w.

27 Delsarte s linear programming bound for q-ary codes Delsarte s linear programming bound for q-ary codes Result 3 We prove simultaneously Delsarte s linear programming bound and its well known improvements for q-ary codes. Theorem (Delsarte s linear programming bound and its improvements Let C be a q-ary code with distance distribution {A i } n i=0. Let M = C. For k = 1, 2,..., n, n i=0 P k (n; ia i 1 ( M r(q r(q n 1k 1, k where r is the remainder when dividing M by q and P k (n; x = k ( ( ( 1 j (q 1 k j i n i j k j j=0 is the Krawtchouk polynomial.

28 Delsarte s linear programming bound for q-ary codes Delsarte s linear programming bound for q-ary codes Idea of proof Write C = (c mi, 1 m C, 1 i n. Consider all 2 k matrices ( cmi1 c A = mi2 c mik c li1 c li2 c lik such that m l, i 1 < i 2 < < i k and all vectors where F q = F q \ {0}. α = (α 1, α 2,..., α k (F q k, Count the number of pairs (A, α such that α 1 c mi1 + + α k c mik α 1 c li1 + + α k c lik.

29 New inequalities for q-ary constant-weight codes Inequalities for q-ary constant-weight codes Result 4 Using the same idea in the proof, we get the following new inequalities for q-ary constant-weight codes. Theorem Suppose that {A i } n i=0 is the distance distribution of a q-ary constant-weight code C of length n and constant-weight w. Let M = C. Then for each k = 1, 2,..., n, n i=1 ( P k (n; ia i (M 1(q 1 k n 2q k (q 1M T (k, where T (k = T 1 (k + T 2 (k + T 3 (k.

30 New inequalities for q-ary constant-weight codes Inequalities for q-ary constant-weight codes Notations [ T 1 (k = T 2 (k = (q 1 k ( n k [ (q 1 k ( n k r k ] (M q k q k + r k (M q k 1(q k + 1, ] [( q 1 t k r k sk 2 2 +(q 1 t k t k s k (s k [( q 1 t T 3 (k = r k k s k 2 + (q 1 t kt ks k(s k where ( tk (s k ], 2 ( t k 2 (s k ], q k and r k are the quotient and the remainder, respectively, when dividing 2(q 1M P q k (n; w by (q ( 1k n k, s k and t k are the quotient and the remainder, respectively, when dividing q k by (q 1, s k and t k are the quotient and the remainder, respectively, when dividing q k + 1 by (q 1.

31 New inequalities for q-ary constant-weight codes Inequalities for q-ary constant-weight codes Notations For each k = 1, 2,..., n, P k (n; x = 1 2 k ( [(q 1 j ( 1 j ](q 1 k j x j j=0 ( n x k j (3 and ( P + k (n; x = (q 1 k n P k (n; x. (4 k

32 New inequalities for q-ary constant-weight codes Inequalities for q-ary constant-weight codes For k = 1, the new inequalities give the following corollary, which was shown by P. R. J. Östergård and M. Svanström in [Ternary constant weight codes, Electron. J. Combin., vol. 9, no. 1, 2002]. Corollary If there exists a q-ary code of length n, constant-weight w, and minimum distance d, then where q 2 M(M 1d 2t q 1 i=0 j=i+1 q 2 M i M j + 2(n t q 1 i=0 j=i+1 M i M j, (5 k and t are the quotient and the remainder, respectively, when dividing Mw by n, M 0 = M k 1, M 0 = M k, M i = (k + i/(q 1, and = (k + i 1/(q 1. M i

33 New upper bounds for q-ary constant-weight codes New upper bounds for q-ary constant-weight codes Example Suppose that q = 3 and (n, d, w = (9, 3, 7. The best known upper bound for A 3 (9, 3, 7 is A 3 (9, 3, Suppose that A 3 (9, 3, 7 = 576. Let C be a code whose size attains the upper bound and let {A i } n i=0 be the distance distribution of C. The above theorem gives the following inequalities. 9A 3 + 6A 4 + 3A 5 3A 7 6A 8 9A A 3 + 6A 4 6A 5 9A 6 3A A A A 3 24A 4 18A 5 + 6A A 7 84A A 3 39A A A 6 21A 7 42A A A A A 5 36A 6 21A A 8 126A A A 4 48A 5 15A A 7 84A A A 3 48A 4 24A A 6 57A A 8 36A A A 5 36A A 7 15A 8 + 9A A A 4 16A 5 + 8A 6 4A 7 + 2A 8 A /3

34 New upper bounds for q-ary constant-weight codes New upper bounds for q-ary constant-weight codes Example (continued Since A 0 = 1 and A 1 = A 2 = 0, A i = i=3 9 A i = C = 576. i=0 Consider the following linear programming (where the A i are considered as variables ( 9 max 1 + A i subject to A i 0, i = 3, 4,..., 9 and subject to the above inequalities. Solving this linear programming, we get the maximum value of i=3 A i is 12094/21, which is less than 576. This contradiction shows that such a code C does not exist. Therefore, i=3 A 3 (9, 3,

35 Thank you!