LP-decodable multipermutation codes

IEEE Trans. Inform. Theory

2016

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In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that may consist of duplicate entries. We first introduce a new class of matrices called multipermutation matrices. We characterize the convex hull of multipermutation matrices. Based on this characterization, we propose a new class of codes that we term LP-decodable multipermutation codes. Then, we derive two LP decoding algorithms. We first formulate an LP decoding problem for memoryless channels. We then derive an LP algorithm that minimizes the Chebyshev distance. Finally, we show a numerical example of our algorithm.

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Section: Remarksmentioning

confidence: 95%

LP-Decodable Multipermutation Codes

IEEE Trans. Inform. Theory

2016

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“…We show in [8] that there is a one-to-one relationship between multipermutation matrices and multipermutations. Furthermore, the convex hull of multipermutation matrices has a simple characterization as shown in Theorem 1.…”

Section: A Linearly Constrained Multipermutation Matricesmentioning

confidence: 98%

“…Definition 1: ( [8]) Given a multiplicity vector r of length m and n = m i=1 r i , we call a m × n binary matrix X a multipermutation matrix parameterized by r if m i=1 X ij = 1 for all j and n j=1 X ij = r i for all i. Denote by M(r) the set of all multipermutation matrices parameterized by r.…”

Section: A Linearly Constrained Multipermutation Matricesmentioning

confidence: 99%

“…Compared to [6] where codes are constructed using linearly constrained permutation matrices, we construct codes using linearly constrained multipermutation matrices, which have a one-to-one mapping to multipermutations. In [8], we show that the convex hull of multipermutation matrices has a characterization analogous to the Birkhoff polytope. In addition, we propose two LP decoding algorithms: one algorithm maximizes the maximum likelihood objective constrained by a relaxed code polytope; the other minimizes the Chebyshev distance constrained within the same polytope, but modified by incorporating an auxiliary variable.…”

Section: Introductionmentioning

confidence: 99%

“…In our previous paper [8], we introduce LP-decodable multipermutation codes. Compared to [6] where codes are constructed using linearly constrained permutation matrices, we construct codes using linearly constrained multipermutation matrices, which have a one-to-one mapping to multipermutations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Encoding and decoding algorithms for LP-decodable multipermutation codes

2015 IEEE Information Theory Workshop (ITW)

2015

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LP-decodable multipermutation codes are a class of multipermutation codes that can be decoded using linear programming (LP). These codes are defined using linearly constrained multipermutation matrices, which are binary matrices that satisfy particular row sum and column sum constraints. Although generic LP solvers are capable of solving the LP decoding problem, they are not efficient in general because they do not leverage structures of the problem. This motivates us to study efficient decoding algorithms.In this paper, we focus on encoding and decoding algorithms for LP-decodable multipermutation codes. We first describe an algorithm that "ranks" multipermutations. In other words, it maps consecutive integers, one by one, to an ordered list of multipermutations. By leveraging this algorithm, we develop an encoding algorithm for a code proposed by Shieh and Tsai. Regarding decoding algorithms we propose an iterative decoding algorithm based on the alternating direction method of multipliers (ADMM), each iteration of which can be solved efficiently using off-the-shelf techniques. Finally, we study decoding performances of different decoders via simulation.

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ADMM decoding of error correction codes: From geometries to algorithms

2015 IEEE Information Theory Workshop (ITW)

2015