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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.

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