Construction of extremal self-dual codes over $${\mathbb {Z}}_{8}$$ Z 8 and $${\mathbb {Z}}_{16}$$ Z 16

Abstract

We present a method of constructing free self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16 which are extremal or optimal with respect to the Hamming weight. We first prove that every (extremal or optimal) free self-dual code over $${\mathbb {Z}}_{2^m}$$Z2m can be found from a binary (extremal or optimal) Type II code for any positive integer $$m \ge 2$$mź2. We find explicit algorithms for construction of self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16. Our construction method is basically a lifting method. Furthermore, we find an upper bound of minimum Hamming weights of free self-dual codes over $${\mathbb {Z}}_{2^m}$$Z2m. By using our explicit algorithms, we construct extremal free self-dual codes over $${\mathbb {Z}}_8$$Z8 and $${\mathbb {Z}}_{16}$$Z16 up to lengths 40.