Perfect codes over non-prime power alphabets: an approach based on Diophantine equations

TL;DR

This paper investigates the existence of perfect 2-error correcting codes over non-prime power alphabets, using number theory techniques to classify new non-existence results and establish finiteness for fixed alphabet sizes.

Contribution

It applies Diophantine equations and computational number theory to prove the non-existence of perfect 2-error correcting codes for 172 new non-prime power alphabet sizes and shows finiteness for fixed alphabet sizes.

Findings

No perfect 2-error correcting codes for 172 new non-prime power q values.

Finiteness of perfect 2-error correcting codes for any fixed alphabet size q.

Extended classification of perfect codes beyond prime power alphabets.

Abstract

Perfect error correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of $e-$error correcting perfect codes over $q-$ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect $e-$error correcting codes were found if $e \ge 3$, and it was conjectured that no perfect $2-$error correcting codes exist over any $q-$ary alphabet, where $q > 3$. In the 1970s, this was proved for $q$ a prime power, for $q = 2^r3^s$ and for only $7$ other values of $q$. Almost $50$ years later, it is surprising to note that there have been no new results in this regard and the classification of $2-$error correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of generalised Ramanujan--Nagell equation and from modern computational number theory to show that perfect $2-$error correcting codes do not exist for $172$ new values of $q$ which are not prime powers, substantially increasing the values of $q$ which are now classified. In addition, we prove that, for any fixed value of $q$, there can be at most finitely many perfect $2-$error correcting codes over an alphabet of size $q$.