An Algebraic Method for Full-Rank Characterization in Binary Linear Coding
This work addresses a fundamental challenge in linear coding for communication and information theory, providing a method to efficiently handle full-rank constraints, though it appears incremental as it builds on existing characteristic set techniques.
The paper tackles the problem of deriving full-rank equivalence conditions for symbolic matrices over the binary field, which is crucial for linear coding problems like network coding and distributed storage, by developing a characteristic set-based method and an algorithm called BCSFR that efficiently characterizes these conditions as zeros of characteristic sets, enabling simplification of optimization problems by replacing intractable full-rank constraints with simple equality constraints.
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.