Quasi-Perfect Lee Codes from Quadratic Curves over Finite Fields

TL;DR

This paper introduces a new method for constructing quasi-perfect Lee codes using quadratic curves over finite fields, proving their properties and solving a related conjecture, with applications in communication channels and memory storage.

Contribution

It presents a novel constructive approach for quasi-perfect Lee codes based on quadratic curves, extending previous constructions and confirming the Ramanujan property of associated Cayley graphs.

Findings

Constructed two classes of 2-quasi-perfect Lee codes over finite fields.

Proved that the related Cayley graphs are Ramanujan or almost Ramanujan.

Extended the class of quasi-perfect Lee codes and solved an existing conjecture.

Abstract

Golomb and Welch conjectured in 1970 that there only exist perfect Lee codes for radius $t=1$ or dimension $n=1, 2$. It is admitted that the existence and the construction of quasi-perfect Lee codes have to be studied since they are the best alternative to the perfect codes. In this paper we firstly highlight the relationships between subset sums, Cayley graphs, and Lee linear codes and present some results. Next, we present a new constructive method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to derive two classes of $2$-quasi-perfect Lee codes are given over the space $\mathbb{Z}_p^n$ for $n=\frac{p^k+1}{2}$ $(\text{with} ~p\equiv 1, -5 \mod 12 \text{and} k \text{is any integer}, \text{or} p\equiv -1, 5 \mod 12 \text{and} k \text{is an even integer})$ and $n=\frac{p^k-1}{2}$ $(\text{with}p\equiv -1, 5 \mod 12, k \text{is an odd integer} \text{and} p^k>12)$, where $p$ is an odd prime. Our codes encompass the quasi-perfect Lee codes constructed recently by Camarero and Mart\'inez. Furthermore, we solve a conjecture proposed by Camarero and Mart\'inez (in "quasi-perfect Lee codes of radius $2$ and arbitrarily large dimension", IEEE Trans. Inf. Theory, vol. 62, no. 3, 2016) by proving that the related Cayley graphs are Ramanujan or almost Ramanujan. The Lee codes presented in this paper have applications to constrained and partial-response channels, in flash memories and decision diagrams.