Construction of self-dual MDR cyclic codes over finite chain rings

Maximum distance with respect to rank codes, or MDR codes, are a family of optimal linear codes that meet a Singleton-like bound in terms of the length and rank of the codes. In this paper, we study the construction of self-dual MDR cyclic codes over a finite chain ring R. We present a new form for the generator polynomials of cyclic codes over R of length n with the condition that the length n and the characteristic of R are relatively prime. Consequently, sufficient and necessary conditions for cyclic codes over R to be self-dual and self-orthogonal are obtained. As a result, self-dual MDR cyclic codes over the Galois ring GR(p(t), m) with length n >= 2 dividing p(m) - 1 are constructed by using torsion codes.

Automated summary

This paper studies the construction of self-dual MDR cyclic codes over a finite chain ring and provides conditions for the codes to be self-dual and self-orthogonal. By utilizing torsion codes, self-dual MDR cyclic codes over the Galois ring with certain length and divisibility conditions are constructed.