Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings

TL;DR

This paper explores the algebraic structure of additive complementary pairs of cyclic codes over finite chain rings, establishing conditions for their formation and demonstrating their relation to trace duals.

Contribution

It provides a necessary and sufficient condition for additive complementary pairs of cyclic codes over finite rings and constructs examples illustrating their trace dual relationships.

Findings

Both codes in a complementary pair are free modules.

A pair forms an additive complementary pair if and only if certain algebraic conditions are met.

One code is permutation equivalent to the trace dual of the other.

Abstract

This paper investigates the algebraic structure of additive complementary pairs of cyclic codes over a finite commutative ring. We demonstrate that for every additive complementary pair of additive cyclic codes, both constituent codes are free modules. Moreover, we present a necessary and sufficient condition for a pair of additive cyclic codes over a finite commutative ring to form an additive complementary pair. Finally, we construct a complementary pair of additive cyclic codes over a finite chain ring and show that one of the codes is permutation equivalent to the trace dual of the other.