Quasi-Perfect Codes From Cayley Graphs Over Integer Rings

The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which the existence of these codes over Lee metric spaces was considered. Since perfect codes are not very common, the problem of searching for quasi-perfect codes is also of great interest. In this aspect, also quasi-perfect Lee codes have been considered for 2-D and 3-D Lee metric spaces. In this paper, constructive methods for obtaining quasi-perfect codes over metric spaces modeled by means of Gaussian and Eisenstein-Jacobi integers are given. The obtained codes form ideals of the integer ring thus preserving the property of being geometrically uniform codes. Moreover, they are able to correct more error patterns than the perfect codes which may properly be used in asymmetric channels. Therefore, the results in this paper complement the constructions of perfect codes previously done for the same integer rings. Finally, decoding algorithms for the quasi-perfect codes obtained in this paper are provided and the relationship of the codes and the Lee metric ones is investigated.