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.

翻译：本文提出了一种基于特征集(CS)的方法，用于推导二元域上符号矩阵的满秩等价条件。这类满秩条件对于通信与信息理论中众多线性编码问题具有基础性意义。在所提出的CS方法基础上，我们设计了一种名为"满秩二元特征集"(BCSFR)的算法，该算法能够高效地将满秩等价条件表示为一组特征集的零点集合。换言之，BCSFR算法可以表征特定线性编码问题（如线性网络编码和分布式存储编码）中所有可行的线性编码方案，此类问题需对若干符号矩阵施加满秩约束以保证解码性或其他编码性质。所推导的等价条件可用于简化编码方案的优化过程，因为优化问题中难以处理的满秩约束被显式表征为简单的三角形式等式约束。