Bull. Korean Math. Soc. 2017; 54(1): 29-42
Online first article November 15, 2016 Printed January 31, 2017
https://doi.org/10.4134/BKMS.b150213
Copyright © The Korean Mathematical Society.
Ab\.id\.in Kaya and Nes\.ibe T\"ufek\c{c}\.i
Bursa Orhangazi University, Fatih University
In this work, four circulant and quadratic double circulant (QDC) constructions are applied to the family of the rings $R_{k,m}$. Self-dual binary codes are obtained as the Gray images of self-dual QDC codes over $ R_{k,m}$. Extremal binary self-dual codes of length 64 are obtained as Gray images of $\lambda$-four circulant codes over $R_{2,1}$ and $R_{2,2}$. Extremal binary self-dual codes of lengths 66 and 68 are constructed by applying extension theorems to the $\mathbb{F}_{2}$ and $R_{2,1}$ images of these codes. More precisely, 10 new codes of length 66 and 39 new codes of length 68 are discovered. The codes with these weight enumerators are constructed for the first time in literature. The results are tabulated.
Keywords: extremal codes, codes over rings, Gray maps, quadratic double-circulant codes
MSC numbers: Primary 94B05, 94B60, 94B65
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