Self-dual codes over \(\mathbb F_2 +u\mathbb F_2\) with an automorphism of odd order
Consider the ring \(\mathbb{F}_2 +u \mathbb{F}_2\), \(u^2=0\). Let \(x\in (\mathbb{F}_2 +u \mathbb{F}_2)^n\). Then the \textit{Lee weight} of \(x\) is \(n_1(x)+2n_2(x)\), with \(n_1(x)\) the number of components of \(x\) equal to 1 and with \(n_2(x)\) the number of components of \(x\) equal to \(u\). For self-dual codes over \(\mathbb{F}_2 +u \mathbb{F}_2\), there are upper bounds on their minimum Lee-weight. A self-dual code over \(\mathbb{F}_2 +u \mathbb{F}_2\) is called \textit{Lee-extremal} if it meets this bound. If no Lee-extremal self-dual code exists for a given length, then a self-dual code of that length with the highest attainable minimum Lee weight is called \textit{Lee-optimal}. All self-dual codes over \(\mathbb{F}_2 +u \mathbb{F}_2\) of lengths 1 to 8 have been classified in \textit{S.T. Dougherty, P. Gaborit, M. Harada, A. Munemasa}, and \textit{P. Solé} [IEEE Trans. Inf. Theory 45, No. 7, 2345--2360 (1999; Zbl 0956.94025)] and \textit{S. T. Dougherty, P. Gaborit, M. Harada}, and \textit{P. Solé} [IEEE Trans. Inf. Theory 45, No. 1, 32--45 (1999; Zbl 0947.94023)]. The author completes in this article the classification of all Lee-extremal and Lee-optimal self-dual codes over \(\mathbb{F}_2 +u \mathbb{F}_2\) of lengths 9 to 20 that have a non-trivial odd order automorphism, started in \textit{W. C. Huffman} [Finite Fields Appl. 13, No. 3, 681--712 (2007; Zbl 1120.94011)]. As an extra result, all the Lee-extremal self-dual codes over \(\mathbb{F}_2 +u \mathbb{F}_2\) of lengths 9 to 11 are determined.
scientific article; zbMATH DE number 5554711
scientific article