What is a parity-check matrix in linear codes?

A parity-check matrix is a fundamental mathematical tool in coding theory used to define the linear relations that components of a valid codeword must satisfy. It helps determine whether a specific received vector is a valid codeword and plays an essential role in decoding algorithms. For example, in Hamming codes, the parity-check matrix is constructed by listing all non-zero columns of a given length r.

What we know

• A parity-check matrix describes the linear relations that the components of a codeword must satisfy in a linear block code. [4]
• It can be used to decide whether a particular vector is a codeword and is also utilized in decoding algorithms. [4]
• The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero. [2]
• Any two columns in the parity-check matrix of a Hamming code are pairwise linearly independent. [2]

Where to go next

• compared withHow does a generator matrix define a linear code?
• a case in pointWhat are Hamming codes and how do they work?
• the bigger pictureHow do low-density parity-check codes use parity-check matrices?
• whoWho invented Hamming codes and when?
• what followedHow are error-correcting codes used in computer memory?
• an unexpected connectionHow does a parity-check matrix connect to the Simplex code?The dual code of a Hamming code, derived from its parity-check matrix construction, is a shortened Hadamard code known as a Simplex code.

Sources

• [1]Hamming code (Wikidata Q853922) · CC0 1.0
• [2]Hamming code · CC BY-SA 4.0
• [3]Low-density parity-check code · CC BY-SA 4.0
• [4]Parity-check matrix · CC BY-SA 4.0
• [5]Generator matrix · CC BY-SA 4.0