The number of inequivalent binary self-orthogonal codes of dimension 6. (English) Zbl 1156.94013
Shaska, T. (ed.) et al., Advances in coding theory and cryptography. Selected papers based on the presentations at the Vlora conference in coding theory and cryptography, Vlora, Albania, May 26–27, 2007 and a special session on coding theory as part of the applications of computer algebra conference, Rochester, MI, USA, July 19–22, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-270-701-7/hbk). Series on Coding Theory and Cryptology 3, 244-256 (2007).
A brief account for the determination of an explicit formula for the number of inequivalent binary self-orthogonal codes of dimension 6 and arbitrary length is given. For a detailed account, the reader is referred to X.-D. Hou [On the number of inequivalent binary self-orthogonal codes, IEEE Trans. Inf. Theory 53, No. 7, 2459–2479 (2007; Zbl 1155.94403)]. Tables are provided for: (1) elementary divisors as well as the cardinality of the centralizer of rational canonical forms in \(\text{GL}(6, \mathbb F_2)\); (2) formulas for the cardinality of \(\text{Fix}(A,P_{\lambda}) =\{X\in M_{k\times n} | AX=XP_{\lambda}\}\) for a permutation \(P_\lambda \in {\mathfrak S}_n\) of cycle type \(\lambda\), whereby \(A\in \text{GL}(k, \mathbb F_2)\) and \(M_{k,n}\) denotes the binary \(k\times n\) matrices; (3) the explicit number of inequivalent binary self-orthogonal codes of dimension \(\leq 6\) and each length up to 40 (once for a fixed dimension and once more for the sum of the number of codes of a given length with dimension up to 6).
For the entire collection see [Zbl 1120.94004].
For the entire collection see [Zbl 1120.94004].
Reviewer: Katherine Roegner (Berlin)
