Parametrization of self-dual codes by orthogonal matrices.

*(English)*Zbl 1138.94389Summary: 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.

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 |

##### Keywords:

orthogonal group; self-dual code; permutation groups; symmetric matrices; weight enumerator; doubly-even codes
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\textit{G. J. Janusz}, Finite Fields Appl. 13, No. 3, 450--491 (2007; Zbl 1138.94389)

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