On the asymptotic number of inequivalent binary self-dual codes
type: Item
instance of: scholarly article
title: On the asymptotic number of inequivalent binary self-dual codes
zbMATH Open document ID: 1145.94015
author: Xiang-Dong Hou
published in: Journal of Combinatorial Theory. Series A
publication date: 2007-05-04T00:00:00Z
review text: The author obtains an asymptotic formula for the number of inequivalent binary self dual codes. By slightly modifying the proof of this result he also obtained a slightly modified asymptotic formula for the number of inequivalent doubly even binary self dual codes. The proof of the asymptotic formulas is based on the Cauchy-Frobenius Lemma (Burnsides Lemma) expressing the number of inequivalent binary self dual codes in terms of the number of codes fixed by a given permutation, summing over all permutations and dividing by the order of the appropriate symmetric group. The last sum is split into the term belonging to the identity permutation and the rest. It is then proved that the term belonging to the identity permutation is the dominant term and the rest is insignificant compared to that term. Moreover the dominant term equals the number of self dual codes divided by the order of the appropriate symmetric group. To prove that the term belonging to the identity is the dominant term is rather hard work, discriminating between a lot of cases and performing lots of nasty calculations and estimates. Nevertheless the proof seems to be sound, however it uses a result of an earlier paper (given in equation 3.7 of the present paper) of this author, which I did not check upon.
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 05A16
zbMATH DE Number: 5149435
zbMATH Keywords: doubly even
zbMATH Keywords: equivalency
zbMATH Keywords: asymptotic enumeration
zbMATH Keywords: invariant subspace
reviewed by: Herman J. Tiersma
MaRDI profile type: Publication
full work available at URL: j.jcta.2006.07.003
OpenAlex ID: W1984060289
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DOI: 10.1016/J.JCTA.2006.07.003
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description: scientific article; zbMATH DE number 5149435