An Algebraic Method for Full-Rank Characterization in Binary Linear Coding
2026-04-03 • Information Theory
Information Theory
AI summaryⓘ
The authors created a new mathematical method to find easy conditions that tell when certain symbolic binary matrices have full rank, which means they are as informative as possible. They designed an algorithm called BCSFR that uses this method to identify all possible ways to build linear coding schemes that meet these full-rank requirements. This helps simplify complex coding problems by turning hard-to-handle full-rank conditions into simpler equations. Their approach is useful for fields like network coding and distributed storage where data must be reliably encoded and decoded.
characteristic setfull-rank matrixbinary fieldsymbolic matrixlinear codingnetwork codingdistributed storage codingdecodabilitytriangular formoptimization constraints
Authors
Mingyang Zhu, Laigang Guo, Zhenyu Huang, Xingbing Chen, Jue Wang, Tao Guo, Xiao-Shan Gao
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.