Bounds on $k$-hash distances and rates of linear codes

TL;DR

This paper establishes bounds on the rates of linear codes over finite fields with specific coordinate-distinctness properties among codewords, providing simpler proofs and new bounds for certain parameters, and discusses related open problems.

Contribution

It introduces new bounds on the rate of linear codes with coordinate-distinctness constraints and simplifies existing proofs for the case q=k=3.

Findings

Recovered state-of-the-art bounds for q=k=3 with d_3=1

Derived new bounds for d_3>1

Discussed open problems in list-decoding zero-error capacity

Abstract

In this paper, we bound the rate of linear codes in $\mathbb{F}_q^n$ with the property that any $k\leq q$ codewords are all simultaneously distinct in at least $d_k$ coordinates. For the case of particular interest $q=k=3$ we recover, with a simpler proof, state of the art results in the case $d_3=1$ and new bounds for $d_3>1$. We finally discuss some related open problems on the list-decoding zero-error capacity of discrete memoryless channels.