Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖_i| ≤ 7)

Abstract

The Golomb-Welch conjecture states that there is no perfect r-error correcting Lee code of word length n over ℤ for n ≥ 3 and r ≥ 2. This problem has received great attention due to its importance in applications in several areas beyond mathematics and computer sciences. Many results on this subject have been achieved, however the conjecture is only solved for some particular values of n and r, namely: 3 ≤ n ≤ 5 and r ≥ 2; n = 6 and r = 2. Here we give an important contribution for the case n = 7 and r = 2, establishing cardinality restrictions on codeword sets.