New cubic self-dual codes of length 54, 60 and 66

Abstract

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length $$3\ell $$ with the algebraic approach of Ling and Solé (IEEE Trans Inf Theory 47(7):2751–2760, 2001. doi: 10.1109/18.959257 ). In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.