\(q\)-ary Hamming code

\(q\)-ary Hamming code[1]

Member of an infinite family of perfect linear \(q\)-ary codes with parameters \([(q^r-1)/(q-1),(q^r-1)/(q-1)-r, 3]_q\) for \(r \geq 2\).

Can detect 1-bit and 2-bit errors, and can correct 1-dit errors.

Linear \(q\)-ary code
Perfect code
Universally optimal \(q\)-ary code — Hamming codes and their punctured and shortened versions are LP universally optimal codes [2].

Projective geometry code — Columns of a Hamming parity-check matrix correspond to one-dimensional subspaces of \(GF(q)^n\).
Cyclic linear \(q\)-ary code — Hamming codes are equivalent to cyclic codes when \(q\) and \(r\) are relatively prime ([3], pg. 194).
Binary BCH code — Some narrow sense BCH codes of length \(n=(q^r-1)/(q-1)\) such that \(\text{gcd}(r,q-1)=1\) are \(q\)-ary Hamming codes ([4], Thm. 5.1.4).
Generalized RM (GRM) code — Hamming codes are dual to first-order GRM codes ([5], pg. 45).
\(E_8\) Gosset lattice code — The \([4,2,3]_3\) ternary Hamming code can be used to obtain the \(E_8\) Gosset lattice code [6; Ex. 10.5.5].
Simplex code — Hamming and simplex codes are dual to each other.

[1]: M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
[2]: H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
[3]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[4]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[5]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[6]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.

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“\(q\)-ary Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_hamming