Non-RS type cyclic MDS codes over finite fields via cyclotomic field reduction

TL;DR

This paper introduces a new, simpler method for constructing cyclic MDS codes over finite fields using cyclotomic field reduction, producing many non-RS type codes with flexible parameters.

Contribution

It proposes a novel construction approach for cyclic MDS codes via cyclotomic field reduction, expanding the variety of codes beyond Reed-Solomon types.

Findings

Several new cyclic MDS codes over finite fields were obtained.

Many non-RS type cyclic MDS codes were produced.

The method is simpler and offers flexible parameters compared to existing methods.

Abstract

Cyclic maximum distance separable (MDS for short) codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs and finite geometry. However, the existing construction methods for cyclic MDS codes are mainly focused on strict restrictions on certain parameters or are relatively complex in their construction approaches. In this paper, we investigate this approach further via norm reduction in cyclotomic fields. By converting the verification of the MDS property over a finite field into checking non-zero minors in characteristic zero, we propose a construction method of cyclic MDS codes over finite fields via cyclotomic field reduction. Based on this method, we obtain several cyclic MDS codes over finite fields and many non-RS type cyclic MDS codes are produced. Compared with the existing construction methods, our method is relatively simpler. Moreover, the results of this paper show that the parameters of the obtained non-RS cyclic MDS codes are flexible.