×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albert, A.A., Symmetric and alternate matrices in an arbitrary field, Trans. amer. math. soc., 43, 923-957, (1938)
[2] Conway, J.; Sloane, N.J.A., A new upper bound on the minimum distance for self-dual codes, IEEE trans. inform. theory, 36, 1319-1333, (1990) · Zbl 0713.94016
[3] Huppert, B., Finite groups I, (1967), Springer Berlin
[4] Janusz, G., Overlap and covering polynomials with applications to self-dual codes and designs, SIAM J. discrete math., 13, 2, 154-178, (1999) · Zbl 0997.94022
[5] Janusz, G., Parametrization of self-dual codes, (March 2005), preprint
[6] Lempel, A., Matrix factorization over \(\mathit{GF}(2)\) and trace-orthogonal bases of \(\mathit{GF}(2^n)\), SIAM J. comput., 4, 2, 175-186, (1975) · Zbl 0331.94006
[7] MacWilliams, F., Orthogonal matrices over finite fields, Amer. math. monthly, 76, 152-164, (1969) · Zbl 0186.33702
[8] MacWilliams, F.; Sloane, N.J.A., Theory of error correcting codes, (1993), Elsevier
[9] MacWilliams, F.; Sloane, N.J.A.; Thompson, J., Good self dual codes exist, Discrete math., 3, 153-162, (1972) · Zbl 0248.94011
[10] Pless, V., The number of isotropic subspaces in a finite geometry, Accad. naz. lincei, 39, 418-421, (1965) · Zbl 0136.42002
[11] Pless, V.; Huffman, W.C., The handbook of coding theory, (1998), North-Holland · Zbl 0907.94001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.