Parametrization of self-dual codes by orthogonal matrices

Abstract

We study the orthogonal group O m of m × m 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 2 n. 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 2 n. 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 [ 2 n , n , d ] with d ⩾ 8 is obtained by applying a transvection to a doubly-even code with parameters [ 2 n , 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 2 n based on the existence of codes of length 2 n − 6 having special properties.