New nonbinary code bounds based on divisibility arguments

TL;DR

This paper introduces new upper bounds for nonbinary codes using divisibility arguments and establishes a correspondence between certain combinatorial designs and codes, leading to improved bounds.

Contribution

It presents novel divisibility-based upper bounds for nonbinary codes and links symmetric nets to code constructions, advancing coding theory bounds.

Findings

New bounds: A_5(8,6) ≤ 65, A_4(11,8) ≤ 60, A_3(16,11) ≤ 29

Derived bounds: A_5(9,6) ≤ 325, A_5(10,6) ≤ 1625, A_5(11,6) ≤ 8125, A_4(12,8) ≤ 240

Established a correspondence between symmetric nets and codes, leading to bounds A_4(9,6) ≤ 120 and A_4(10,6) ≤ 480

Abstract

For $q,n,d \in \mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \leq 65$, $A_4(11,8)\leq 60$ and $A_3(16,11) \leq 29$. These in turn imply the new upper bounds $A_5(9,6) \leq 325$, $A_5(10,6) \leq 1625$, $A_5(11,6) \leq 8125$ and $A_4(12,8) \leq 240$. Furthermore, we prove that for $\mu,q \in \mathbb{N}$, there is a 1-1-correspondence between symmetric $(\mu,q)$-nets (which are certain designs) and codes $C \subseteq [q]^{\mu q}$ of size $\mu q^2$ with minimum distance at least $\mu q - \mu$. We derive the new upper bounds $A_4(9,6) \leq 120$ and $A_4(10,6) \leq 480$ from these `symmetric net' codes.