\(q\)-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.

Incidence-matrix projective code— Columns of a Hamming parity-check matrix correspond to one-dimensional subspaces of \(\mathbb{F}_q^n\).
Cyclic linear \(q\)-ary code— Hamming codes are equivalent to cyclic codes when \(q\) and \(r\) are relatively prime ([4], 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 ([5], Thm. 5.1.4).
\([6,3,4]_4\) Hexacode— The hexacode is an extended quaternary Hamming code [5; Exer. 578].
\(q\)-ary simplex code— \(q\)-ary Hamming and \(q\)-ary simplex codes are dual to each other [6; pg. 45]
Perfect quantum code— For qubits (\(q=2\)), the only nontrivial perfect codes are the stabilizer code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r \geq 2\), obtained from Hamming codes over \(\mathbb{F}_4\) via the Hermitian construction [7,8]. These codes are related to partial spreads in projective geometry [9].

Parents

Hamming codes and their punctured and shortened versions are LP universally optimal codes [10].

\(q\)-ary Hamming code

Children

The \(q\)-ary Hamming codes reduce to the Hamming codes at \(q=2\).

The tetracode is equivalent to the \(r=2\) \(3\)-ary Hamming code.

[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]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[5]: W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
[6]: M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
[7]: D. Gottesman, “Pasting Quantum Codes”, (1996) arXiv:quant-ph/9607027
[8]: A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[9]: J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini, and F. Pambianco, “The geometry of quantum codes”, Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial 6, 53 (2008) DOI
[10]: H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI

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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