Maximum Weight Spectrum Codes

TL;DR

This paper proves a conjecture about the maximum number of distinct non-zero weights in linear codes over finite fields, providing two different construction-based proofs and discussing code length bounds.

Contribution

It confirms the conjecture that the maximum weight spectrum size is sharp for all prime powers and positive integers, using geometric and algebraic methods.

Findings

The maximum weight spectrum size is exactly rac{q^k-1}{q-1} for all prime powers q and positive integers k.

Two different proofs are provided: one geometric and one algebraic.

New bounds on code lengths satisfying the conjecture are established.

Abstract

In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field $\F_q$ was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by $\frac{q^k-1}{q-1}$. They showed that this bound was sharp in the case $ q=2 $, and in the case $ k=2 $. They conjectured that the bound is sharp for every prime power $ q $ and every positive integer $ k $. In this work quickly establish the truth of this conjecture. We provide two proofs, each employing different construction techniques. The first relies on the geometric view of linear codes as systems of projective points. The second approach is purely algebraic. We establish some lower bounds on the length of codes that satisfy the conjecture, and the length of the new codes constructed here are discussed.