Research On The Decoding Of Linear Codes Over Rings With Lee Metric

Codes over rings, especially Z₄-linear codes, have received a great deal of interest among coding theorists since 1990s. The decoding of these codes with Lee metric is a very important problem. In this paper, we creatively solve the problem of decoding the (64, 2³², 14) code which is binary nonlinear but Z4-linear, based on much famous work done by other researchers. The binary nonlinear (64, 2³², 14) code is the best (64, 2³²) code that is presently known, and it is an image via Gray map of a specific cyclic codes over Z₄ which can correct all errors with Lee weight ≤6.The research on Lee metric linear codes over finite fields began very early and has never halted, because Lee metric is better than Hamming metric on measuring errors occurred when signals are translated in certain noisy channels. In 1994, Roth and Siegel constructed a class of BCH codes over GF(p) and presented an efficient decoding procedure meanwhile. In this paper, we further generalize their work. We construct a class of BCH codes over Z/(p^k), and we study the minimum Lee distance of them and present a decoding algorithm utilizing the existing work on the decoding of codes over rings with Lee metric.