Three-weight codes and the quintic construction

TL;DR

This paper constructs new three- and five-weight codes over a specific ring, computes their weight distributions, and applies them to secret sharing schemes, advancing algebraic coding theory.

Contribution

It introduces new classes of three- and five-weight codes over a ring related to the quintic construction, with explicit weight distributions and applications.

Findings

Constructed three-Lee-weight and five-Lee-weight codes over ring R.

Derived binary abelian codes with few weights via Gray map.

Identified a class of optimal three-weight codes.

Abstract

We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R=\mathbb{F}_2 +v\mathbb{F}_2 +v^2\mathbb{F}_2 +v^3\mathbb{F}_2 +v^4\mathbb{F}_2,$ where $v^5=1.$ The same ring occurs in the quintic construction of binary quasi-cyclic codes. %The length of these codes depends on the degree $m$ of ring extension. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class of three-weight codes which are optimal. Finally, an application to secret sharing schemes is given.