LP Decodable Permutation Codes based on Linearly Constrained Permutation Matrices

TL;DR

This paper introduces a new class of permutation codes called LP decodable permutation codes, which utilize linear constraints on permutation matrices to enable linear programming-based decoding, with analysis of their properties and performance.

Contribution

It proposes a novel class of permutation codes with LP decodability using linear constraints, and analyzes their properties through structured and random constraints.

Findings

LP decoding performance is characterized by code polytope vertices

Structured constraints enable efficient encoding algorithms

Random constraints allow probabilistic analysis of code properties

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 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. Two types of linear constraints are discussed; one is structured constraints and another is random constraints. The structured constraints such as pure involution lead to an efficient encoding algorithm. On the other hand, the random constraints enable us to use probabilistic methods for analyzing several code properties such as the average cardinality and the average weight distribution.