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 New extremal binary self-dual codes of lengths 66 and 68 from codes over $R_{k,m}$ Bull. Korean Math. Soc. 2017 Vol. 54, No. 1, 29-42 https://doi.org/10.4134/BKMS.b150213Published online January 31, 2017 Ab\.id\.in Kaya and Nes\.ibe T\"ufek\c{c}\.i Bursa Orhangazi University, Fatih University 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 $\lambda$-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 Full-Text :