Overlap and Covering Polynomials with Applications to Designs and Self-Dual Codes
SIAM Journal on Discrete Mathematics
Handbook of Coding Theory
A new upper bound on the minimal distance of self-dual codes
IEEE Transactions on Information Theory
The Rayleigh Quotient of Bent Functions
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
International Journal of Information and Coding Theory
Classification of self dual quadratic bent functions
Designs, Codes and Cryptography
The selfnegadual properties of generalised quadratic Boolean functions
Designs, Codes and Cryptography
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We study the orthogonal group O"m of mxm matrices over the field of two elements and give applications to the theory of binary self-dual codes. We show that O"2"n acts transitively on the self-dual codes of length 2n. The subgroup O"2"n^(^1^), consisting of all elements in O"2"n having every row with weight congruent to 1 mod 4, acts transitively on the set of doubly even self-dual codes of length 2n. A factorization theorem for elements of O"m leads to a result about generator matrices for self-dual codes, namely if G=[I|A] is such a generator matrix with I= identity, then the number of rows of G having weight divisible by 4 is a multiple of 4. This generalizes the known result that a self-dual doubly-even code exists only in lengths divisible by 8. The set of inequivalent self-dual codes is shown to be in one-to-one correspondence with the H-P"m double cosets in O"m for a certain subgroup H. The analogous correspondence is given for doubly-even codes and H^(^1^)-P"m double-cosets in O"m^(^1^) for a certain a group H^(^1^). Thus the classification problem for self-dual codes is equivalent to a classification of double-cosets. The subgroups of O"m generated by the permutation matrices and one transvection are determined in the Generator Theorem. The study of certain transvections leads to two results about doubly-even self-dual codes: (a) every such a code with parameters [2n,n,d] with d=8 is obtained by applying a transvection to a doubly-even code with parameters [2n,n,d-4] which has some special properties related to a vector of weight 6; (b) every such code with minimum distance at least 16 is a neighbor of a singly-even, self-dual code which has a single word of minimum weight 6. A construction is given for such singly-even codes of length 2n based on the existence of codes of length 2n-6 having special properties.