How many weights can a linear code have ?

TL;DR

This paper investigates the maximum number of nonzero weights in linear codes over finite fields, providing exact results for specific cases and bounds for the general case, along with nonlinear analogues.

Contribution

It completely determines the maximum weights for binary codes and codes of dimension two, and introduces bounds and refinements for the general case and nonlinear variants.

Findings

Exact maximum weights for binary codes ($q=2$)

Maximum weights for codes of dimension 2 over any $q$

Bounds and refinements for the general case with $k,q \\ge 3$

Abstract

We study the combinatorial function $L(k,q),$ the maximum number of nonzero weights a linear code of dimension $k$ over $\F_q$ can have. We determine it completely for $q=2,$ and for $k=2,$ and provide upper and lower bounds in the general case when both $k$ and $q$ are $\ge 3.$ A refinement $L(n,k,q),$ as well as nonlinear analogues $N(M,q)$ and $N(n,M,q),$ are also introduced and studied.