Self-Dual Negacyclic Codes With Variable Lengths and Square-Root-Like Lower Bounds on the Minimum Distances
Conghui Xie, Hao Chen, Cunsheng Ding, Zhonghua Sun · IEEE Transactions on Information Theory · 2024
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 combinatorialt-designs. It is well-known thatq-ary self-dual cyclic codes exist only whenqis an even prime power andq-ary self-dual negacyclic codes exist for any odd prime powerq. In 2009 a family of binary self-dual cyclic codes with lengthsniand minimum distancesdi≥ 1/2√ni, wherenigoes to the infinity ifigoes to the infinity, was constructed. In this paper, we construct several families ofq-ary self-dual negacyclic codes of lengthsnwith their minimum distances larger than or equal ton1/2for various lengthsnand any given odd prime powerq. Whenq∈ {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.