LP-Decodable Multipermutation Codes

In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are the permutations of a multiset that generally consist of duplicate entries. We first introduce a class of binary matrices called multipermutation matrices, each of which corresponds to a unique and distinct multipermutation. By enforcing a set of linear constraints on these matrices, we define a new class of codes that we term linear program (LP)-decodable multipermutation codes. In order to decode these codes using an LP, thereby enabling soft decoding, we characterize the convex hull of multipermutation matrices. This characterization allows us to relax the coding constraints to a polytope and to derive two LP decoding problems. These two problems are, respectively, formulated by relaxing the maximum likelihood decoding problem and the minimum Chebyshev distance decoding problem. Because these codes are non-linear, we also study efficient encoding and decoding algorithms. We first describe an algorithm that maps consecutive integers, one by one, to an ordered list of multipermutations. Based on this algorithm, we develop an encoding algorithm for a code proposed by Shieh and Tsai, a code that falls into our class of LP-decodable multipermutation codes. Regarding decoding algorithms, we propose an efficient distributed decoding algorithm based on the alternating direction method of multipliers. Finally, we observe from the simulation results that the soft decoding techniques we introduce can significantly outperform hard decoding techniques that are based on quantized channel outputs.