Reducible Cyclic Codes Constructed as the Direct Sum of Two Semiprimitive Cyclic Codes

TL;DR

This paper introduces a new family of reducible cyclic codes formed by combining two semiprimitive two-weight irreducible cyclic codes, enabling easier computation of their weight distribution frequencies.

Contribution

It generalizes previous reducible cyclic codes and demonstrates a straightforward method to determine weight distribution frequencies using cyclotomic numbers.

Findings

New family of reducible cyclic codes constructed as direct sums

Simplified computation of weight distribution frequencies

Extension of previous code classes

Abstract

We present a family of reducible cyclic codes constructed as the direct sum of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of B. Wang, {\em et al.} \cite{once}. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in a very easy way.