\(q\)-ary Hamming code

\(q\)-ary Hamming code[1,2]

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 \(r \geq 2\).

The automorphism group is known [3].

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 [4].
Small-distance block code

Incidence-matrix projective 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 ([5], 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 ([6], Thm. 5.1.4).
Generalized RM (GRM) code — \(q\)-ary Hamming codes are dual to first-order GRM codes [7; pg. 45].
Maximum distance separable (MDS) code — The \((4,9,3)_3\) Hamming code is a unique MDS code [8,9].
\(E_8\) Gosset lattice code — The \([4,2,3]_3\) ternary Hamming code can be used to obtain the \(E_8\) Gosset lattice code [10; Ex. 10.5.5].
Hexacode — The hexacode is an extended quaternary Hamming code [6; Exer. 578].
\(q\)-ary simplex code — \(q\)-ary Hamming and \(q\)-ary simplex codes are dual to each other.

[1]: R. A. FISHER, “A SYSTEM OF CONFOUNDING FOR FACTORS WITH MORE THAN TWO ALTERNATIVES, GIVING COMPLETELY ORTHOGONAL CUBES AND HIGHER POWERS”, Annals of Eugenics 12, 283 (1943) DOI
[2]: M. J. E. Golay, Notes on digital coding, Proc. IEEE, 37 (1949) 657.
[3]: E. V. Gorkunov, “The group of permutation automorphisms of a q-ary hamming code”, Problems of Information Transmission 45, 309 (2009) DOI
[4]: H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
[5]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[6]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[7]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[8]: Taussky, Olga, and John Todd. "Covering theorems for groups." Bulletin of the American Mathematical Society. Vol. 54. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1948.
[9]: J. G. Kalbfleisch and R. G. Stanton, “A Combinatorial Problem in Matching”, Journal of the London Mathematical Society s1-44, 60 (1969) DOI
[10]: 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