Hermitian self-dual codes over F ₂ 2m + uF₂2m

We present a method for construction of Hermitian self-dual codes over F₂ 2m + uF₂2m from Hermitian self-dual codes over F ₂2m 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 F₂ 2m+uF₂2m from the information on the equivalence of codes over F₂ 2m. 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 authors' previous work [10].