Description
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 \(r \geq 2\) [3; (3.1)]. These are precisely the nontrivial perfect linear codes over \(\mathbb{F}_q\) [3; Thm. 3.3.1].
The automorphism group is known [4].
Protection
Can detect up to two symbol errors and correct one symbol error.Notes
Example 2.2.2 studies the subfield subcode \(H_{3,2^2}|_{\mathbb{F}_2}\), a \([21,16,3]_2\) code, of a \(q\)-ary Hamming code over an extension field [5; Exam. 2.2.2].Cousins
- Incidence-matrix projective code— Columns of a Hamming parity-check matrix correspond to 1D subspaces of \(\mathbb{F}_q^r\).
- Bose–Chaudhuri–Hocquenghem (BCH) code— When \(\gcd(r,q-1)=1\), \(q\)-ary Hamming codes are narrow-sense BCH codes [6; Exam. 16.4.10][7; Thm. 5.1.4], which are cyclic [8; pg. 194][5; Exam. 2.5.1].
- \([6,3,4]_4\) Hexacode— The hexacode is an extended quaternary Hamming code [7; Exer. 578].
- \(q\)-ary simplex code— \(q\)-ary Hamming and \(q\)-ary simplex codes are dual to each other [9; 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 [10,11]. These codes are related to partial spreads in projective geometry [12].
Primary Hierarchy
Parents
Hamming codes and their once-punctured and once-shortened versions are LP universally optimal codes [13].
\(q\)-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.
References
- [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”, Proceedings of the IEEE 37, 657 (1949)
- [3]
- P. R. J. Östergård, “Construction and Classification of Codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 61-78 DOI
- [4]
- E. V. Gorkunov, “The group of permutation automorphisms of a q-ary hamming code”, Problems of Information Transmission 45, 309 (2009) DOI
- [5]
- C. Ding, “Cyclic Codes over Finite Fields”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 45-60 DOI
- [6]
- W. Willems, “Codes in Group Algebras”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 363-384 DOI
- [7]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [8]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
- [9]
- M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes (Springer Netherlands, 1991) DOI
- [10]
- D. Gottesman, “Pasting Quantum Codes”, (1996) arXiv:quant-ph/9607027
- [11]
- 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
- [12]
- 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
- [13]
- H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2022-08-12)
Cite as:
“\(q\)-ary Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/q-ary_hamming, arXiv:2606.11484