LP decodable permutation codes based on linearly constrained permutation matrices

Abstract

A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding problem for the proposed class of permutation codes as a linear programming (LP) problem. The main feature of this novel class of permutation codes, called LP decodable permutation codes, is this LP decodability. It is demonstrated that the LP decoding performance of the proposed class of permutation codes is characterized by the vertices of the code polytope of the code. In addition, based on a probabilistic method, several theoretical results for randomly constrained permutation codes are derived.

LP-Decodable Multipermutation Codes

“…In this section, we briefly review the concept of permutation matrices and the code construction approach proposed in [16]. A length-n permutation π is a length-n vector, each element of which is a distinct integer between 1 and n, inclusive.…”

“…In other words, the objective of each decoder is to correct some number of errors in the corresponding distance metric. In order to bring soft decoding to permutation codes, Wadayama and Hagiwara introduce linear programming (LP) decoding of permutation codes in [16]. Although the set of codes that can be decoded by LP decoding is restrictive, the framework is promising for two reasons.…”

“…In fact, the construction of Wadayama and Hagiwara in [16] is defined over multipermutations. However in [16], multipermutations are described using permutation matrices. This results in two notable issues.…”

“…The second issue relates to code non-singularity as defined in [16]. To elaborate, we briefly review some concepts therein.…”

Ulam Ball Size Analysis for Permutation and Multipermutation Codes Correcting Translocation Errors

“…The history of permutation codes dates as far back as the 1960's and 70's, with Slepian, Berger, Blake, and others [2], [3], [19]. However, the application of permutation codes and multipermutation codes for use in non-volatile memory storage systems such as flash memory has received attention in the coding theory literature in recent years [1], [12], [13], [17], [21], [22]. One of the main distance metrics in the literature has been the Kendall-τ metric, which is suitable for correction of the type of error expected to occur in flash memory devices [5], [11], [12], [13], [23].…”