New Self-Dual Codes of Length 68 from a 2 × 2 Block Matrix Construction and Group Rings

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Advances in Mathematics of Communications following peer review. The definitive publisher-authenticated version Bortos, M., Gildea, J., Kaya, A., Korban, A. & Tylyshchak,A. (2020). New self-dual codes of length 68 from a 2 × 2 block matrix construction and group rings. Advances in Mathematics of Communications. is available online at: https://www.aimsciences.org/article/doi/10.3934/amc.2020111 ; Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form G = (In | A); where In is the n x n identity matrix and A is the n x n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F2 +uF2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to con- struct many new binary self-dual [68,34,12]-codes with the rare parameters $\gamma = 7$; $8$ and $9$ in $W_{68,2}$: In particular, we find 92 new binary self-dual [68,34,12]-codes.