Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Pedro-José Cazorla García

TL;DR

The paper tackles the existence problem of perfect $2$-error correcting codes over non-prime-power alphabets by reducing the perfect-code condition to a generalized Ramanujan--Nagell Diophantine equation. It develops an algorithmic framework based on Mordell curves to solve these RN-type equations and uses Lloyd's theorem to rule out any RN-solution that could yield a perfect code, yielding extensive nonexistence results for many non-prime-power $q$ values and proving finiteness of such codes for any fixed $q$. The results substantially extend the known landscape beyond previous cases with at most three prime factors in $q$, providing concrete tables of solutions and validating the findings with Magma implementations. The work also discusses limitations and outlines future directions, including potential extensions to quantum codes and improvements in number-theoretic tools to handle wider families of $q$.

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 $q$.

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

TL;DR

The paper tackles the existence problem of perfect -error correcting codes over non-prime-power alphabets by reducing the perfect-code condition to a generalized Ramanujan--Nagell Diophantine equation. It develops an algorithmic framework based on Mordell curves to solve these RN-type equations and uses Lloyd's theorem to rule out any RN-solution that could yield a perfect code, yielding extensive nonexistence results for many non-prime-power values and proving finiteness of such codes for any fixed . The results substantially extend the known landscape beyond previous cases with at most three prime factors in , providing concrete tables of solutions and validating the findings with Magma implementations. The work also discusses limitations and outlines future directions, including potential extensions to quantum codes and improvements in number-theoretic tools to handle wider families of .

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 error correcting perfect codes over ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect error correcting codes were found if , and it was conjectured that no perfect error correcting codes exist over any ary alphabet, where . In the 1970s, this was proved for a prime power, for and for only other values of . Almost years later, it is surprising to note that there have been no new results in this regard and the classification of 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 error correcting codes do not exist for new values of which are not prime powers, substantially increasing the values of which are now classified. In addition, we prove that, for any fixed value of , there can be at most finitely many perfect error correcting codes over an alphabet of size .
Paper Structure (10 sections, 4 theorems, 31 equations, 2 tables)

This paper contains 10 sections, 4 theorems, 31 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The main results
  4. The Hamming bound and Ramanujan--Nagell type equations
  5. Resolving the Diophantine equations
  6. Dealing with the outstanding solutions: Lloyd's theorem
  7. Limitations and future lines of work
  8. Limitations of our methodology
  9. Future lines of work
  10. Conclusions

Key Result

Theorem 2

Let $q \ge 6$ be a positive integer which is not a prime power, satisfying any of the following conditions. Then, there are no perfect $2-$error correcting codes over alphabets of size $q$.

Theorems & Definitions (9)

  • Definition 1
  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:finitelymany']}
  • Lemma 3.1
  • proof
  • Theorem 4
  • proof : Proof of Theorem \ref{['thm:main']}