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Parametrization of self-dual codes by orthogonal matrices. (English) Zbl 1138.94389
Summary: We study the orthogonal group $${\mathcal O}_m$$ of $$m\times m$$ matrices over the field of two elements and give applications to the theory of binary self-dual codes. We show that $${\mathcal O}_{2n}$$ source acts transitively on the self-dual codes of length $$2n$$. The subgroup $${\mathcal O}_{2n}^{(1)}$$ consisting of all elements in $${\mathcal O}_{2n}$$ 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 $${\mathcal O}_{m}$$ leads to a result about generator matrices for self-dual codes, namely if $$G=[I|A]$$ is such a generator matrix with $$I=\text{identity}$$, 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 $${\mathcal H} - {\mathcal P}_m$$ double cosets in $${\mathcal O}_{2n}$$ for a certain subgroup $${\mathcal H}$$. The analogous correspondence is given for doubly-even codes and $${\mathcal H}^{(1)}-{\mathcal P_m}$$ double-cosets in $${\mathcal O}^{(1)}_{2n}$$ for a certain a group $${\mathcal H}^(1)$$. Thus the classification problem for self-dual codes is equivalent to a classification of double-cosets.
The subgroups of $${\mathcal O}_{m}$$ 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 $$d\geq 8$$ 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.

##### MSC:
 94B05 Linear codes, general 15B33 Matrices over special rings (quaternions, finite fields, etc.) 20G40 Linear algebraic groups over finite fields
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