Packing Sets

TL;DR

This paper investigates the construction of large packing sets within finite fields, establishing bounds on their size and providing explicit constructions relevant for error-correcting codes.

Contribution

It proves the existence of large packing sets with optimal bounds and constructs explicit examples for prime fields, advancing coding theory applications.

Findings

Existence of packing sets with size at least (q-1)/|A/A|

Construction of packing sets of size proportional to p/(\lambda \log p)

Results are optimal up to a logarithmic factor

Abstract

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge (q-1)/|A/A|$ and show that this bound is in general optimal. The case that $q=p$ is a prime and $A=\{1,2,\ldots,\lambda\}$ for some positive integer $\lambda$ is particularly interesting in view of the construction of limited-magnitude error correcting codes. Here we construct a packing set $B$ of size $|B|\gg p (\lambda \log p)^{-1}$ for any $\lambda \le c p^{1/2}$ for some explicitly calcuable constant $c$. This result is optimal up to the logarithmic factor.