New nonbinary code bounds based on divisibility arguments

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 $A_5(8,6)\le 65$, $A_4(11,8)\le 60$ and $A_3(16,11)\le 29$. These in turn imply the new upper bounds $A_5(9,6)\le 325$, $A_5(10,6)\le 1625$, $A_5(11,6)\le 8125$ and $A_4(12,8)\le 240$. 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 $A_4(9,6)\le 120$ and $A_4(10,6)\le 480$ from these `symmetric net' codes.

Revisions have been made based on comments of the referees. 13 pages. To appear in Designs, Codes and Cryptography

References in corpus (1)

• Semidefinite bounds for nonbinary codes based on quadruples