%0 Journal Article
%T On Maximum Lee Distance Codes
%A Tim L. Alderson
%A Svenja Huntemann
%J Journal of Discrete Mathematics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/625912
%X Singleton-type upper bounds on the minimum Lee distance of general (not necessarily linear) Lee codes over are discussed. Two bounds known for linear codes are shown to also hold in the general case, and several new bounds are established. Codes meeting these bounds are investigated and in some cases characterised. 1. Introduction The Lee metric was introduced by Lee [1] in 1958 as an alternative to the Hamming metric for certain noisy channels. It found application and in particular was later developed for certain noisy channels (primarily those using phase-shift keying modulation [2]). The past decade has witnessed a burst of new and varied applications for codes defined in the Lee metric (Lee codes) including constrained and partial-response channels [3], interleaving schemes [4], orthogonal frequency-division multiplexing [5], multidimensional burst-error correction [6], and error correction for flash memories [7]. These recent applications give increased interest in questions surrounding optimal Lee codes. Similar to the case of the Hamming metric, it is desirable to investigate upper bounds on the minimum Lee distance of a code given the code size, code length, and alphabet size. Codes meeting these bounds are of special interest as they are optimal in the sense that their minimum distance is largest. Under the Hamming metric, such codes are referred to as maximum distance separable (MDS) codes. Under the Lee metric, such codes may be referred to as Maximum Lee Distance Separable (MLDS) codes. Here, we will present several upper bounds similar to the Singleton bound and investigate the existence question of MLDS codes. In certain cases, we are able to completely characterize MLDS codes. 2. Preliminaries An block code is a collection of -tuples (codewords) over an alphabet of size such that the minimum (Hamming) distance between any two codewords is (hence, no two codewords have as many as common coordinates). Here, is the dimension of , which need not be an integer. Where context demands, we may also denote the Hamming distance by . The Singleton bound states that and holds for all block codes. Codes meeting this bound with equality are called maximum distance separable (MDS) codes. Research on both linear and nonlinear MDS codes has been extensive (e.g., see [8¨C10] and references therein). 2.1. Lee Codes Let be the set of representatives of the integer equivalence classes modulo . The Lee weight of any element is given by . Given an element , the Lee weight of , denoted , is given by For , the Lee distance between and is defined to be the Lee
%U http://www.hindawi.com/journals/jdm/2013/625912/