Construction of self-dual codes with an automorphism of order $p$

Hyun Jin Kim; Heisook Lee; June Bok Lee; Yoonjin Lee

doi:10.3934/amc.2011.5.23

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2011, Volume 5, Issue 1: 23-36. Doi: 10.3934/amc.2011.5.23

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Construction of self-dual codes with an automorphism of order $p$

1.
Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-Gu, Seoul 120-749
2.
Department of Mathematics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul, 120-750

Received: March 31, 2010

Revised: November 30, 2010

Published: January 2011

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Abstract

We develop a construction method for finding self-dual codes with an automorphism of order $p$ with $c$ independent $p$-cycles. In more detail, we construct a self-dual code with an automorphism of type $p-(c,f+2)$ and length $n+2$ from a self-dual code with an automorphism of type $p-(c,f)$ and length $n$, where an automorphism of type $p-(c, f)$ is that of order $p$ with $c$ independent cycles and $f$ fixed points. Using this construction, we find three new inequivalent extremal self-dual $[54, 27, 10]$ codes with an automorphism of type $7-(7,5)$ and two new inequivalent extremal self-dual $[58, 29, 10]$ codes with an automorphism of of type $7-(8,2)$. We also obtain an extremal self-dual $[40, 20, 8]$ code with an automorphism of type $3-(10, 10)$, which is constructed from an extremal self-dual $[38, 19, 8]$ code of type $3-(10,8)$, and at least 482 inequivalent extremal self-dual $[58,29,10]$ codes with an automorphism of type $3-(18,4)$, which is constructed from an extremal self-dual $[54, 27, 10]$ code of type $3-(18,0);$ we note that the extremality is preserved.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 11T71.

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