Formally Self-Dual Codes Related to Type II Codes

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1 Formally Self-Dual Codes Related to Type II Codes Koichi Betsumiya Graduate School of Mathematics Nagoya University Nagoya , Japan and Masaaki Harada Department of Mathematical Sciences Yamagata University Yamagata , Japan March 17, 2000 Abstract It is known that shadows play an important role in the study of self-dual codes. In this note, we give constructions of formally self-dual codes using self-dual codes and their shadows. We introduce a class of binary formally self-dual codes related to extremal Type II codes. A binary formally self-dual [24, 12, 7] code is obtained from the extended Golay code. It is shown that there is a unique binary [24, 12, 7] code with dual distance 7, up to equivalence. Keywords: Formally self-dual codes, Type II codes and codes over Z 2k. 1 Introduction Shadows play an important role in the study of self-dual codes. Shadows of binary Type I codes were introduced by Conway and Sloane [3]. They are used to determine upper bounds on the minimum weights of binary Type I codes. Brualdi and Pless [2] gave constructions This work was partially supported by the Grant-in-Aid for Scientific Research (No ), the Ministry of Education, Science, Sports and Culture, Japan, and the Sumitomo Foundation (No ), Japan. 1

2 of binary self-dual codes of lengths n + 2 and n + 4 from Type I codes of length n and their shadows by investigating orthogonality relations between the four cosets of the doubly-even subcode C 0 of a given Type I code C into C 0. Recently Type II codes codes over the ring Z 2k of integers modulo 2k have been introduced in [1] together with shadows for Type I codes over Z 2k. Shadows of Type I codes over Z 2k are widely studied in [5]. For example, shadows are used in [5] in order to construct larger self-dual codes as a generalization of the constructions in [2]. In this note, we study formally self-dual codes over Z 2k constructed from Type I codes with shadows. In Section 3, we give constructions of formally self-dual codes using self-dual codes and their shadows. In Section 4, we introduce a class of binary formally self-dual codes related to extremal Type II codes. In particular, we give the weight enumerator of a formally self-dual code related to an extremal Type II [72, 36, 16] code. The existence of an extremal Type II [72, 36, 16] code is a long-standing open problem. A binary formally self-dual [24, 12, 7] code is obtained from the extended Golay code. In Section 5, it is shown that there is a unique binary [24, 12, 7] code with dual distance 7, up to equivalence. 2 Self-Dual Codes and Shadows We give definitions for self-dual codes over Z 2k (see [1] for a complete description). The case k = 1 is usual binary codes. A code C of length n over Z 2k (or a Z 2k -code of length n) is a Z 2k -submodule of Z n 2k where Z 2k is the ring of the integers modulo 2k. The Hamming weight wt H (x) of a vector x in Z n 2k is the number of non-zero components. The Euclidean weight wt E (x) of a vector x = (x 1, x 2,..., x n ) is n i=1 min{x 2 i, (2k x i ) 2 }. The symmetrized weight enumerator of C is defined as swe C (x 0, x 1,..., x k ) = c C x n 0(c) 0 x n 1(c) 1 x n k 1(c) k 1 x n k(c) k, where n 0 (c), n 1 (c),..., n k 1 (c), n k (c) are the numbers of 0, ±1,..., ±(k 1), k components of c, respectively. For x = (x 1,..., x n ) and y = (y 1,..., y n ), we define the inner-product of x and y in Z n 2k by x y = x 1 y x n y n. The dual code C of C is defined as C = {x Z n 2k x y = 0 for all y C}. C is self-dual if C = C. Self-dual Z 2k -codes with the property that all Euclidean weights are divisible by 4k are called Type II [1]. A self-dual code which is not Type II is called Type I. A code C is a formally self-dual code with respect to the symmetrized weight enumerator if swe C (x 1, x 2,..., x k ) = swe C (x 1, x 2,..., x k ). A self-dual code is formally self-dual. Let C be a Type I Z 2k -code of length n. Let C 0 be the subcode of codewords whose Euclidean weights are a multiple of 4k. C 0 is a subcode of index 2 in C. Let C 2 be the unique nontrivial coset of C 0 into C. Then C0 can be written as a union of cosets of C 0 : 2

3 C0 = C 0 C 2 C 1 C 3. The shadow of C is defined to be S = C 1 C 3 (see [1], [5] for the details). Throughout this note, C i s (i = 0, 1, 2, 3) denote the cosets for C. 3 Construction of Formally Self-Dual Codes In this section, we give constructions of formally self-dual codes over Z 2k using self-dual codes and their shadows. Lemma 3.1 (Dougherty et al. [5]) Suppose that C is a Type I code over Z 2k of length n. If n 2 (mod 4) then Table 1 holds where the symbol in position (i, j) means that x y = 0 and the symbol means that x y = k for any vector x C i and any vector y C j. Table 1: Orthogonality Relations C 0 C 1 C 2 C 3 C 0 C 1 C 2 C 3 Proposition 3.2 Suppose that there are Type I Z 2k -codes C and D of lengths n and m, respectively where n m 2 (mod 4). If swe C1 (x 1, x 2,..., x k ) = swe C3 (x 1, x 2,..., x k ) and swe D1 (x 1, x 2,..., x k ) = swe D3 (x 1, x 2,..., x k ) then the code E = (C 0, D 0 ) (C 1, D 1 ) (C 3, D 2 ) (C 2, D 3 ), is a formally self-dual code with respect to the symmetrized weight enumerator, that is, swe E (x 1, x 2,..., x k ) = swe E (x 1, x 2,..., x k ). Proof. It follows from Lemma 3.1 that the dual code of E is E has symmetrized weight enumerator (C 0, D 0 ) (C 2, D 1 ) (C 1, D 2 ) (C 3, D 3 ). swe E (x 0, x 1,..., x k ) = swe C0 (x 0, x 1,..., x k ) swe D0 (x 0, x 1,..., x k ) +swe C1 (x 0, x 1,..., x k ) swe D1 (x 0, x 1,..., x k ) +swe C3 (x 0, x 1,..., x k ) swe D2 (x 0, x 1,..., x k ) +swe C2 (x 0, x 1,..., x k ) swe D3 (x 0, x 1,..., x k ). 3

4 From the assumption on the symmetrized weight enumerators, E is formally self-dual. Remark. Since (x 1, y 1 ) (C 1, D 1 ) is not orthogonal to (x 3, y 2 ) (C 3, D 2 ), the code E is not self-dual. We consider some Type I codes over Z 4 as an example of the above construction. Examples of binary codes are given in the next section. Let A 1 be the self-dual Z 4 -code of length 1 with generator matrix (2). Let C and D be the direct sums of two and six copies of A 1. It turns out that C 0 = {(00), (22)}, C 2 = (20) + C 0, C 1 = (11) + C 0 and C 3 = (31) + C 0. Similarly, D 0 = {2(a 1, a 2, a 3, a 4, a 5, a 6 ) a i {0, 1}, 6 i=1 a i 0 (mod 2)}, D 2 = (200000) + D 0, D 1 = (111111) + D 0 and D 3 = (311111) + D 0. Hence we have swe C1 (x 0, x 1, x 2 ) = swe C3 (x 0, x 1, x 2 ) and swe D1 (x 0, x 1, x 2 ) = swe D3 (x 0, x 1, x 2 ). By the above proposition, a formally self-dual code of length 8 with respect to the symmetrized weight enumerator is constructed. The formally self-dual code has the following symmetrized weight enumerator x x x 0 x 2 1x x 0 x 6 1x x 2 0x x 3 0x 2 1x x 4 0x x 5 0x 2 1x x 6 0x x 8 0. Let C and D be codes over Z 2k such that the symmetrized weight enumerators of C and D are the same. The two lattices from C and D by Construction A (see [1] for the details) have the identical theta series. Thus the lattice from a formally self-dual code over Z 2k with respect to the symmetrized weight enumerator has the same theta series as the dual lattice. The lattice constructed from the above formally self-dual code over Z 4 has the following theta series q 3/ q q 5/ q 7/ q 4 +, which is the same as the dual lattice. 4 Binary Formally Self-Dual Codes In this section, we give constructions of binary formally self-dual codes. From now on, all codes are binary. These give a class of such codes related to extremal Type II codes. Proposition 4.1 Let C be a binary Type I code of length n with the shadow S = C 1 C 3 where C 0 is the subcode of doubly-even vectors of C. (1) Suppose that n 2 (mod 4). Let C be the code of length n+2 obtained by extending C0 as follows: (0, 0, C 0 ), (1, 0, C 2 ), (0, 1, C 1 ), (1, 1, C 3 ). If W C1 (y) = W C3 (y) then C is a formally self-dual code with weight enumerator W C0 (y) + y(w C1 (y) + W C2 (y)) + y 2 W C3 (y), 4

5 where W D (y) denotes the (Hamming) weight enumerator of D. (2) Suppose that n 0 (mod 4). Let C be the code of length n + 4 generated by (1, 0, 1, 1, 0,..., 0) and the following extension of C 0 : (0, 0, 0, 0, C 0 ), (1, 1, 0, 0, C 2 ), (1, 1, 1, 0, C 1 ), (0, 0, 1, 0, C 3 ). Then C is a formally self-dual code with weight enumerator (1 + y 3 )W C0 (y) + (y 2 + y 3 )(W C1 (y) + W C2 (y)) + (y + y 2 )W C3 (y). Proof. (1) is a special case of Proposition 3.2. The orthogonality relations between the cosets C 0, C 1, C 2 and C 3 were determined in [2, Tables I and II] (see also Table 1). It follows from the relation that D is the dual code of C. Since W C1 (y) = W C3 (y), C is formally self-dual. Similarly, the dual code of C is the code generated by (1, 1, 0, 1, 0,..., 0) and the following extension of C 0 : (0, 0, 0, 0, C 0 ), (0, 0, 1, 1, C 2 ), (0, 1, 1, 1, C 1 ), (0, 1, 0, 0, C 3 ). Thus C is formally self-dual. By the above construction, we give a class of formally self-dual codes related to extremal Type II codes. Corollary 4.2 If there is an extremal Type II [8k, 4k, 4[k/3]+4] code then there is a formally self-dual [8k, 4k, d 4[k/3] + 3] code. Proof. Let B be an extremal Type II [8k, 4k, 4[k/3] + 4] code. Let C be a Type I child of B, that is C is constructed from B by subtracting. Then C is a Type I [8k 2, 4k 1, 4[k/3]+2] code and the shadow has minimum weight 4[k/3] + 3. It follows from [6, Theorem 3.5] that W C1 (y) = W C3 (y). By Proposition 4.1, C is a formally self-dual code of length 8k. The minimum weight of C is min{d(c 2 ) + 1, d(c 3 ) + 1} where d(d) denotes the minimum weight of D. Since d(c 2 ) = 4[k/3] + 2 and d(c 3 ) 4[k/3] + 3, the minimum weight of C is greater than or equal to 4[k/3] + 3. If the above Type II code has generator matrix of the form ( I 4k A ) then the above formally self-dual code has the following generator matrix A.. I 4k

6 Recently Rains [7] shows that the minimum weight of a self-dual code of length n is bounded by 4 [ ] n , if n 22 (mod 24), d 4 [ ] n , otherwise, and that any self-dual [24l, 12l, 4l + 4] code is Type II. By Theorem 3.6 in [6], the child D of an extremal Type II [24l, 12l, 4l + 4] code has the shadow of minimum weight 4l + 3 exactly. Thus the formally self-dual code has smaller minimum weight than an extremal Type II code of the same length, but it has larger minimum weight than any Type I code of the same length. These cases are investigated. l = 1: The weight enumerators of a Type I code and its shadow are given in [3] for lengths up to 64 and length 72. Let C be the unique Type I [22, 11, 6] code, then W C (y) = y y y 10 +, W C1 (y) = 176y y Then C is a formally self-dual [24, 12, 7] code with weight enumerator y y y 9 +. l = 2: Let C be a Type I [48, 24, 10] code, then W C (y) = y y y 14 + W C1 (y) = 3312y y The extended quadratic residue code QR 48 of length 48 is only one known extremal Type II code of length 48. By the above corollary, a formally self-dual [48, 24, 11] code C 48 is constructed from QR 48. The weight enumerator of C 48 is y y y Note that only formally self-dual codes with minimum weight 10 which are not self-dual are previously known. l = 3: Let C be a Type I [70, 35, 14] code. The existence of such a code is not known. The next case is length 72 and related to an extremal Type II [72, 36, 16] code. The existence of an extremal Type II [72, 36, 16] code is a long-standing open problem. A formally self-dual code F 72 of length 72 is constructed from a Type I [70, 35, 14] code C by Proposition 4.1. The weight distribution of F 72 is listed in Table 2. Note that the weight distributions of C and its shadow are given in [4]. Thus we have the following: Corollary 4.3 If there is no linear [72, 36, 15] code with weight distribution given in Table 2, then there is no extremal Type II [72, 36, 16] code. 6

7 Table 2: The weight distribution of F 72 Weight Frequency Weight Frequency [24, 12, 7] Codes with Dual Distance 7 In the previous section, a formally self-dual [24, 12, 7] code is constructed from the extended Golay code. In this section, it is shown that up to equivalence there is a unique [24, 12, 7] code with dual distance 7, that is, the minimum weight of the dual code is 7. We describe how [24, 12, 7] codes with dual distance 7 were classified. We say that a (1, 0)-matrix M is bi-sorted if the rows and columns of M are sorted as binary integers. Lemma 5.1 Any code is equivalent to a code with a generator matrix of the form ( I, A ) where A is bi-sorted. Proof. It is sufficient to show that we obtain a bi-sorted matrix from a given matrix X, by permuting the rows and columns. First we fix some notations used in this proof. Let X be an m n (1, 0)-matrix, x i and x i be the i-th row and column vector of X, respectively. Moreover let z = (z 1, z 2,..., z n ) be a (1, 0)-vector of length n. We may suppose without loss of generality that X = (x p1 x p 1+q 1 x p2 x p 2+q2 x pt x pt+qt ) such that x p k+i = x p k+i+1 for 0 i q k 1, 1 k t, and x p k x p k+1 for 1 k t 1. Let wx (z) k denote the number of non-zero 7

8 components of (z pk, z pk +1,..., z pk +q k ), and w X (z) denote (w X (z) 1, w X (z) 2,..., w X (z) t ). Let R k = {p k, p k + 1,..., p k + q k } and R X = {R 1, R 2,..., R t }. Let X (k) be the k n sub-matrix obtained from the first k rows of X. Suppose that X (k) is bi-sorted and w X(k) (x 1 ) w X(k) (x 2 ) w X(k) (x n ) on the lexicographic order for a fixed k. By permuting columns fixing X (k), X implies X where the each block of x k+1 in R X(k) is (0,..., 0, 1,..., 1). Then it is easy to see that X (k+1) is bi-sorted. Finally, by permuting the rows {k + 2, k + 3,..., n} of X, we obtain a matrix satisfying the assumption for k + 1 from X. By induction, the result follows. Every [24, 12] code is equivalent to a code with generator matrix of the form ( I, A ) where A is a bi-sorted (1, 0)-matrix. Thus we only need to consider the set of bi-sorted (1, 0)-matrices A, rather than the set of generator matrices. The set of bi-sorted matrices A was constructed, row by row, using a back-tracking algorithm under the condition that the first row is ( ) since the minimum weight is 7. Then our computer search shows that there are distinct 602 [24, 12, 7] codes under the condition the dual distance is 7. The C program, which generates the corresponding matrices, is available at Note that the codes have the same weight enumerator as the formally self-dual code of length 24 given in the previous section. Finally, we verified that all the codes are equivalent by Magma. Proposition 5.2 There is a unique [24, 12, 7] code C 24 with dual distance 7, up to equivalence. Remark. Since C 24 is equivalent to the formally self-dual [24, 12, 7] code given in the previous section, the automorphism group is the Mathieu group M 22. Finally we remark that [24, 12, 7] codes are not unique without the assumption that the dual distance is 7. Let B be a matrix obtained by changing a column of A to the all-zero column vector where ( I, A ) is a generator matrix of the extended Golay code. Then ( I, B ) generates a [24, 12, 7] code with dual distance 1. References [1] E. Bannai, S.T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices, invariant rings, IEEE Trans. Inform. Theory 45 (1999) pp [2] R.A. Brualdi and V.S. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory 37 (1991) pp [3] J.H. Conway and N.J.A. Sloane, A new upper bound on the minimal distance of selfdual codes, IEEE Trans. Inform. Theory 36 (1990) pp

9 [4] S.T. Dougherty and M. Harada, New extremal self-dual codes of length 68, IEEE Trans. Inform. Theory 45 (1999) pp [5] S.T. Dougherty, M. Harada and P. Solé, Shadow lattices and shadow codes, Discrete Math., (to appear). [6] G.T. Kennedy and V.S. Pless, A coding theoretic approach to extending designs, Discrete Math. 142 (1995) pp [7] E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory 44 (1998) pp