An Algebraic Method for Full-Rank Characterization in Binary Linear Coding

TL;DR

This paper introduces a novel algebraic method using characteristic sets to efficiently derive full-rank conditions in binary linear coding, aiding optimization and analysis of coding schemes.

Contribution

It develops a characteristic set-based approach and the BCSFR algorithm to explicitly characterize full-rank conditions in binary linear coding problems.

Findings

The BCSFR algorithm efficiently derives full-rank equivalence conditions.

The method simplifies optimization by providing explicit algebraic constraints.

Applicable to linear network coding and distributed storage coding.

Abstract

In this paper, we develop a characteristic set (CS)-based method for deriving full-rank equivalence conditions of symbolic matrices over the binary field. Such full-rank conditions are of fundamental importance for many linear coding problems in communication and information theory. Building on the developed CS-based method, we present an algorithm called Binary Characteristic Set for Full Rank (BCSFR), which efficiently derives the full-rank equivalence conditions as the zeros of a series of characteristic sets. In other words, the BCSFR algorithm can characterize all feasible linear coding schemes for certain linear coding problems (e.g., linear network coding and distributed storage coding), where full-rank constraints are imposed on several symbolic matrices to guarantee decodability or other properties of the codes. The derived equivalence conditions can be used to simplify the optimization of coding schemes, since the intractable full-rank constraints in the optimization problem are explicitly characterized by simple triangular-form equality constraints.