Bulletin of the
Korean Mathematical Society BKMS

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