\([8,4,4]\) extended Hamming code

\([8,4,4]\) extended Hamming code[1–3]

Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code.

\([2^r,2^r-r-1,4]\) Extended Hamming code
\([2^m,m+1,2^{m-1}]\) First-order RM code — The \([8,4,4]\) extended Hamming code is a first-order RM code because it is self-dual and first-order RM codes are dual to extended Hamming codes.
Self-dual linear code — The \([8,4,4]\) extended Hamming code is the smallest doubly even self-dual code.

Divisible code — The \([8,4,4]\) extended Hamming code code is the smallest double-even self-dual code.
\(E_8\) Gosset lattice code — The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice code via Construction A [4; Ex. 10.5.2].
\([7,4,3]\) Hamming code — The Hamming code can be extended by a parity-check bit to yield the \([8,4,4]\) extended Hamming code, the smallest doubly even self-dual code.
Octacode — The octacode reduces modulo-two to the \([8,4,4]\) extended Hamming code [5].
\([[8, 3, 3]]\) Eight-qubit Gottesman code — The \([[8, 3, 3]]\) code is obtained via a modified CSS construction from the \([8,4,4]\) extended Hamming code.

[1]: C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
[2]: R. W. Hamming, “Error Detecting and Error Correcting Codes”, Bell System Technical Journal 29, 147 (1950) DOI
[3]: M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
[4]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[5]: Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI

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“\([8,4,4]\) extended Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamming844