New extremal binary self-dual codes of lengths 66 and 68 from codes over r_k,m

2015/02/24 by Abidin Kaya, Kaya, Abidin, Nesibe Tüfekçi +1
Computer Science · Engineering · Mathematics · #94B05 #94B60 #94B65 #Coding theory and cryptography #FOS: Computer and information sciences #Finite Group Theory Research #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1502.06945

openalex publication_date 2015/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ?-four circulant codes over R₂,1 and R₂,2. Extremal binary self-dual codes of lengths 66 and 68 are constructed by applying extension theorems to the F₂ and R₂,1 images of these codes. More precisely, 11 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.

Citations

• A new upper bound on the minimal distance of self-dual codes
• Shadow bounds for self-dual codes

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