List Decoding of Quaternary Codes in the Lee Metric

We present a list decoding algorithm for quaternary negacyclic codes over the Lee metric. To achieve this result, we use a Sudan-Guruswami type list decoding algorithm for Reed- Solomon codes over certain ring alphabets. Our decoding strategy for negacyclic codes over the ring ${\mathbb{Z}_4}$ combines the list decoding algorithm by Wu with the Gröbner basis approach for solving a key equation due to Byrne and Fitzpatrick.

I. Introduction

It has been observed in the literature, that cyclic codes over the alphabet equipped with the Lee distance often have a larger minimum distance and a better decoding capability than predicted by their "designed distance". For this reason a list decoding approach is suggested, although technical difficulties due to zero divisors in rings are to be expected.