Hermitian self-dual codes over F22m+uF22m

We present a method for construction of Hermitian self-dual codes over F22m+uF22m from Hermitian self-dual codes over F22m via a Gray map we define, where m is a positive integer. For constructing self-dual codes over F2+uF2 with an automorphism of odd order using the decomposition theory, it is necessary to find Hermitian self-dual codes over F22m+uF22m for some appropriate positive integer m. Using the Gray map, we show how to check the equivalence of codes over F22m+uF22m from the information on the equivalence of codes over F22m. We thus classify all Hermitian self-dual codes over F22+uF22 of lengths up to 8. Using these codes, we complete the classification of the Lee-extremal self-dual codes over F2+uF2 of lengths 21 and 22 with a nontrivial automorphism of odd order; these were open cases in the authorsʼ previous work [10].