Conversion Between Check Matrix and Generator Matrix in Galois Fields

This code is primarily designed for performing mutual conversion between check matrices and generator matrices over Galois fields. A Galois field (finite field) contains elements whose quantity is a prime power, providing the mathematical foundation for error-correcting codes. The implementation enables users to transform check matrices into generator matrices and vice versa, which is crucial for both encoding and decoding processes in error correction systems. The algorithm employs Gaussian elimination and matrix operations over finite fields to ensure accurate conversions. Key functions include field arithmetic operations, matrix rank calculation, and basis transformation techniques. By utilizing this code, users can streamline their encoding and decoding operations, thereby improving coding efficiency and reducing error rates. The implementation handles various matrix dimensions and supports different prime field characteristics, making it versatile for diverse coding applications. The conversion process maintains the algebraic relationships between matrices, ensuring that the generated matrices satisfy the fundamental property H×Gᵀ = 0 for orthogonal complement spaces. This mathematical consistency guarantees the reliability of the resulting error-correcting codes.