Bulletin of the
Korean Mathematical Society BKMS

In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders $2,3,4,$ and $5$, and by applying the construction over the binary field and the ring $F_{2}+uF_{2}$, we obtain extremal binary self-dual codes of various lengths: $12, 16, 20, 24, 32, 40,$ and $48$. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code $[24,12,8]$ and the unique Extended Quadratic Residue $[48,24,12]$ Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in [10] by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.

Keywords: Extremal binary self-dual codes, group rings, reverse circulant matrices, bordered constructions

MSC numbers: Primary 94B05, 16S34