Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

Abstract

A construction of 2-quasi-perfect Lee codes is given over the space Z n p for p prime, p ≡ ±5 (mod 12) and n = 2[ p 4 ]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee-metric do not exist for dimension n ≥ 3 and radius r ≥ 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between Coding and Graph Theories.

On the nonexistence of lattice tilings of Zn by Lee spheres

“…Note that |S(n, r)| = min{n,r} i=0 2 i n i r i which is proved in [31]. This link was also pointed out by Camarero and Martínez [6]. Let AC(∆, k) be the largest order of abelian Cayley graphs of degree ∆ and diameter k. Then Golomb-Welch conjecture implies the following conjecture.…”

50 Years of the Golomb–Welch Conjecture

“…Thus, the property for a code to be a quasi-perfect code in the Lee metric is still too restrictive. The first construction of QP L(n, e, q)-codes for infinitely many n, based on Cayley graph, has been recently presented in [46] and [47]. In [48] it was shown that these Cayley graphs are in fact Ramanujan graphs.…”

The Cayley Graphs Associated With Some Quasi-Perfect Lee Codes Are Ramanujan Graphs

“…Resolving this conjecture has been one of the main motivations for studying perfect and quasi-perfect Lee codes. Very recently, Camarero and Martínez [2], showed that for every prime number p > 5 such that p ≡ ±5 (mod 12), the Cayley graph G p = Cay(Z p [i], S 2 ), where S 2 is the set of units of Z p [i], induces a 2-quasi-perfect Lee code over Z m p , where m = 2⌊ p 4 ⌋. They also conjectured [2,Conj.…”

“…Very recently, Camarero and Martínez [2], showed that for every prime number p > 5 such that p ≡ ±5 (mod 12), the Cayley graph G p = Cay(Z p [i], S 2 ), where S 2 is the set of units of Z p [i], induces a 2-quasi-perfect Lee code over Z m p , where m = 2⌊ p 4 ⌋. They also conjectured [2,Conj. 31] that the Cayley graph G p = Cay(Z p [i], S 2 ) is a Ramanujan graph for every prime p such that p ≡ 3 (mod 4).…”