The Hadamard code is dual to the extended Hamming Code.
- Dual linear code — The Hadamard code is the dual of the extended Hamming Code.
- Hamming code — The Hadamard code is the dual of the extended Hamming Code. Conversely, the shortened Hadamard code is the dual of the Hamming Code.
- Reed-Muller (RM) code — For any Hamming code \([2^m,2^m-m-1,3]\), the dual Hadamard code, when augmented with a bit that is always 0, gives the RM\((1,m)\) code. In general, RM\((1,m)\) is related to the duals of the Hamming code, and when RM\((1,m)\) is self dual, it is directly related to the Hamming code.
Zoo code information
“Hadamard code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hadamard