Two-weight codes and second order recurrences

TL;DR

This paper investigates cyclic codes of dimension 2 over finite fields, establishing they have at most two nonzero weights, extending previous constructions, and providing conditions for these codes to be MDS, with explicit weight distributions.

Contribution

It extends Rao et al's construction, disproves Schmidt-White's conjecture, and characterizes when these codes are MDS based on check polynomial roots.

Findings

Cyclic codes of dimension 2 have at most two nonzero weights.

The paper provides explicit weight distributions for these codes.

Conditions for these codes to be MDS are derived based on roots of check polynomials.

Abstract

Cyclic codes of dimension $2$ over a finite field are shown to have at most two nonzero weights. This extends a construction of Rao et al (2010) and disproves a conjecture of Schmidt-White (2002). We compute their weight distribution, and give a condition on the roots of their check polynomials for them to be MDS.