On two-weight codes

TL;DR

This paper studies two-weight block codes over finite fields, providing combinatorial constructions, bounds on their size, and exploring the relationship between code parameters, with specific results for small field sizes.

Contribution

It introduces new combinatorial constructions for optimal two-weight codes and establishes conditions linking code parameters, along with bounds and tables for small cases.

Findings

Existence of linear two-weight codes implies certain gcd conditions.

Upper bounds for code sizes derived from linear programming and spherical codes.

Tables of bounds for small q and code length are provided.

Abstract

We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+\delta$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear $2$-weight code with $\delta > 1$ implies the following equality of great common divisors: $(d,q) = (\delta,q)$. Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small $q = 2,3,4$ and $q\,n < 50$ are presented.