Various constructions for self-dual codes over rings and new binary self-dual codes

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• 作者：Abidin Kaya^a ; ^{abidin.kaya@bou.edu.tr" class="auth_mail" title="E-mail the corresponding author} ; Bahattin Yildiz^b ; ^{byildiz@fatih.edu.tr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Extremal self-dual codes ; Gray maps ; Four circulant codes ; Extension theorems ; Designs
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 February 2016
• 年：2016
• 卷：339
• 期：2
• 页码：460-469
• 全文大小：417 K

文摘

In this work, extension theorems are used for self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F_2^m+uF_2^m${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ for m=1,2$m=1,2$. The duality and distance preserving Gray maps from F₄+uF₄${\mathbb{F}}_4+u{\mathbb{F}}_4$ to (F₂+uF₂)²${({\mathbb{F}}_2+u{\mathbb{F}}_2)}^2$ and ${\mathbb{F}}_2^4$ are used to obtain self-dual codes whose binary Gray images are [64,32,12]$[64,32,12]$-extremal self-dual. An F₂+uF₂${\mathbb{F}}_2+u{\mathbb{F}}_2$-extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ=3$\gamma =3$ and many codes with the rare γ=4,6$\gamma =4,6$ parameters are obtained. In addition to these, two hundred fifty doubly even self dual [96,48,16]$[96,48,16]$-codes with new weight enumerators are obtained from four-circulant codes over F₄+uF₄${\mathbb{F}}_4+u{\mathbb{F}}_4$. New extremal doubly even binary codes of lengths 80 and 88 are also found by the F₂+uF₂${\mathbb{F}}_2+u{\mathbb{F}}_2$-lifts of binary four circulant codes and thus a lower bound on the number of non-isomorphic 3-(80, 16, 665) designs is modified.