New extremal binary self-dual codes of length 68

Abstract

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring $R := \mathbb{F}_2 + u\mathbb{F}_2 + u^2\mathbb{F}_2$, where $u^3 = 1$. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with $\beta = 117, 120, 133$ in $W_{68,1}$ and with $\gamma = 1$, $\beta = 49, 57, 59$ and codes with $\gamma = 2$, $\beta = 69, 81$ in $W_{68,2}$ are constructed for the first time in the literature. The binary generators of these codes are available online at [7]. In addition to these, some known codes are reconstructed via this extension. The results are tabulated.

Received: 9 July 2014 | Accepted: 19 August 2014