On the Construction of Integer Codes with Minimal Signal Point Constellations

We consider the class of integer codes that are capable of correcting single errors in a two-dimensional lattice (H. Morita et al., 2003). Integer codes, defined over integer rings, allow the correction of single errors with distance 1. In this paper, we extend our previous results (H. Morita et al., 2003). We will detail the derivation of an upper bound on the code size and give constructions that attain this bound. Next, several coding algorithms are presented that efficiently transform binary information into a sequence of symbols that are associated with signal points in a two-dimensional lattice. The proposed decoding algorithms can correct any single error with distance 1 in the given signal point constellation and restore the binary information. We will present a systematic technique to design signal point constellations that support the constructed integer codes and that are optimal in terms of the total Mannheim and Euclidean distance