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Advances in Mathematics of Communications (AMC)
 

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

Pages: 23 - 36, Volume 5, Issue 1, February 2011      doi:10.3934/amc.2011.5.23

 
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Hyun Jin Kim - Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-Gu, Seoul 120-749, South Korea (email)
Heisook Lee - Department of Mathematics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul, 120-750, South Korea (email)
June Bok Lee - Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-Gu, Seoul 120-749, South Korea (email)
Yoonjin Lee - Department of Mathematics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul, 120-750, South Korea (email)

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.

Keywords:  Automorphism, self-dual code, extremal code, weight enumerator.
Mathematics Subject Classification:  Primary: 94B05; Secondary: 11T71.

Received: April 2010;      Revised: December 2010;      Available Online: February 2011.

 References