Self-dual codes from a block matrix construction characterised by group rings (Q6544464)

A new technique for constructing self-dual codes based on a block matrix whose blocks arising from group rings and orthogonal matrices is proposed. It utilizes the generator matrix \(G=\left(\begin{array}{c|c} I_{2n} & \begin{array}{cc} AC&B\\\N-B^TC&A^T\end{array} \\\N\end{array}\right),\) where \(C\) is an orthogonal matrix, then \(G\) is a generator matrix of a self-dual code of length \(4n\) if and only if \(AA^T+BB^T = -I_n.\) The main modification that the author is applying it by replacing \(A\) and \(B\) with matrices that arise from two group rings. These rings may or may not be the same. Another suggested improvement is when \(C\) is an orthogonal matrix, either arising from another group ring or a \(\mu\)-circulant matrix. This technique can be used to construct self-dual codes over finite commutative Frobenius rings of characteristic 2. In this work, new necessary conditions are given and proven in order for the technique to produce self-dual codes.\N\NBy applying this new technique together with the building-up construction, 27 new binary self-dual codes are constructed. These codes are extremal of lengths 64, 66, and 68, as well as new best-known codes of length 80 with weight enumerator parameters of previously unknown values.