\(q\)-ary Hamming code[1]
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 \(r \geq 2\).
Protection
Can detect 1-bit and 2-bit errors, and can correct 1-dit errors.
Parents
- 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].
Children
- Hamming code
- Tetracode — The tetracode is equivalent to the \(r=2\) \(3\)-ary Hamming code.
Cousins
- 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.
References
- [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.
Page edit log
- Victor V. Albert (2022-08-12) — most recent
Cite as:
“\(q\)-ary Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_hamming