A conjectura de Golomb-Welch com raio 2 para infinitas dimensões

Abstract

An important concept in coding theory is the packing radius of a code, which we define as the maximum radius that allows us to cover the space in which the code is contained with disjoint balls centered at the code words. In this sense, perfect codes are codes where the case of a wrong word being outside all the balls centered at the code words does not occur. Whether a code is perfect or not depends on the chosen metric. In this work, we will present a study on perfect linear codes in the Lee metric and the criteria to optimize the radius around the code elements. Finally, we will present the Golomb-Welch Conjecture (1968), which states that there are no perfect Lee codes for certain values of dimension and radius for large alphabets. The general proof of this conjecture still remains an open problem in Coding Theory, however, for radius 2, we will see a demonstration of its validity for infinite dimensions related to a specific set of prime numbers.