Two-dimensional Lee-Error-Correcting Codes on Hexagonal Signal Constellations

Abstract

We construct linear codes over odd prime fields for correcting two-dimensional (2-D) Lee-errors on the hexagonal signal constellations. They are obtained by puncturing and enlarging either RS codes or BCH codes. We introduce 2-D Lee-weight on the hexagonal constellations in the same way as the method presented by the first author in ISIT’19, and propose an effective and efficient method for correcting Lee-errors of small weight. The concept of value-locator of an error, which was introduced implicitly by K. Nakamura in the late 1970s and early 1980s and inherited to the ISIT’19 paper, is a key for decoding Lee-error-correcting codes. Our method is based on the Buchberger algorithm for finding Gröbner bases of ideals in the multivariate polynomial ring. A result of simulations shows that our method works well for correcting Lee-errors of small weight.