Mutual Conversion Between Parity Check Matrix and Generator Matrix

In coding theory, parity check matrices and generator matrices are two fundamental representations of linear block codes that can be mutually converted over Galois fields. While these matrices differ in form, they are essentially equivalent and both completely characterize the properties of a linear code.

The generator matrix is used during the encoding process to transform information bit vectors into codewords. Specifically, when given a k-bit information vector, multiplication with the generator matrix produces an n-bit codeword vector. The row space of the generator matrix constitutes the entire codeword space. In implementation, this can be achieved through matrix multiplication using dedicated functions like numpy.dot() in Python or direct matrix operations in MATLAB.

The parity check matrix is primarily employed for decoding and error detection. A valid codeword must satisfy the condition that its product with the parity check matrix yields a zero vector. The null space of the parity check matrix corresponds exactly to all possible valid codewords. Error detection algorithms typically compute the syndrome vector s = H×c^T to verify codeword validity.

When performing mutual conversion between these matrices over Galois fields, the key lies in understanding their dual relationship. Converting from generator matrix to parity check matrix requires finding the orthogonal complement to the row space of the generator matrix, which can typically be obtained through methods like Gaussian elimination. Conversely, converting from parity check matrix to generator matrix involves determining a basis for the null space of the parity check matrix. Implementation-wise, this can be done using linear algebra libraries such as numpy.linalg in Python or specialized functions like null() in MATLAB for null space computation.

This conversion proves highly useful in coding practice, particularly in scenarios requiring simultaneous utilization of both matrix properties. For instance, encoding using generator matrices is more efficient, while decoding may require parity check matrices for syndrome calculation. Proper implementation of matrix conversion is crucial for building complete encoding and decoding systems, often involving optimized algorithms for large-scale matrices in practical applications.