Self-Dual Negacyclic Codes With Variable Lengths and Square-Root-Like Lower Bounds on the Minimum Distances

The construction of self-dual codes with large minimum distances has been an active topic in coding theory. The construction and classification of extremal self-dual codes over small fields are related to other fields of mathematics, such as lattices and invariant theory as well as combinatorial t -designs. It is well-known that q -ary self-dual cyclic codes exist only when q is an even prime power and q -ary self-dual negacyclic codes exist for any odd prime power q. In 2009 a family of binary self-dual cyclic codes with lengths ni and minimum distances di ≥ 1/2√ ni, where ni goes to the infinity if i goes to the infinity, was constructed. In this paper, we construct several families of q -ary self-dual negacyclic codes of lengths n with their minimum distances larger than or equal to n 1/2 for various lengths n and any given odd prime power q. When q ∈ 3, 5 and the length is small, the minimum distances of the constructed self-dual negacyclic codes are comparable with these self-dual codes with largest known minimum distances in the literature.