Alternative names: RM\((m-2,m)\) code.
Description
Member of an infinite family of RM\((m-2,m)\) codes with parameters \([2^m,2^m-m-1, 4]\) for \(m \geq 2\) that are extensions of the Hamming codes by a parity-check bit.Cousins
- \([2m+2,m+1]\) Karlin code— The extended Hamming code is equivalent to the Karlin double circulant code for \(m=3\) [4; Ch. 16].
- Universally optimal \(q\)-ary code— Several extended Hamming codes are LP universally optimal codes [5].
- \([2^m,m+1,2^{m-1}]\) First-order RM code— Extended Hamming and first-order RM codes are dual to each other.
- Dual linear code— Extended Hamming and first-order RM codes are dual to each other.
- Incidence-matrix projective code— Columns of an extended Hamming code’s parity-check matrix correspond to points in \(PG(m-1,2)\) that lie in the complement of a hyperplane [6; pg. 182].
- Combinatorial design— Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [7; pg. 89].
- Preparata code— Any code with the same parameters as the Preparata code must be a distance invariant subcode of a (possibly nonlinear) code with the same parameters as the extended Hamming code [8,9].
- \([2^r-1,2^r-r-1,3]\) Hamming code— Extended Hamming codes are extensions of Hamming codes by a parity-check bit. Puncturing extended Hamming codes yields the Hamming codes.
- ZRM code— The weight-four codewords of the binary image of the dual of ZRM\((1,m)\) form a Steiner system that is identical to that formed by the weight-four codewords of an extended Hamming code [10].
- \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code— A Hamming Majorana code is constructed from a first-order RM code (whose dual is the extended Hamming code).
Primary Hierarchy
Reed-Muller (RM) codeLinear code over \(\mathbb{Z}_q\) GRM Evaluation Divisible Linear \(q\)-ary AG LCC LRC Distributed-storage ECC
Parents
Extended Hamming codes are RM\((m-2,m)\) codes.
\([2^m,2^m-m-1,4]\) Extended Hamming code
Children
References
- [1]
- C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
- [2]
- R. W. Hamming, “Error Detecting and Error Correcting Codes”, Bell System Technical Journal 29, 147 (1950) DOI
- [3]
- M. J. E. Golay, “Notes on digital coding”, Proceedings of the IEEE 37, 657 (1949)
- [4]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
- [5]
- H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
- [6]
- R. Hill, A First Course in Coding Theory (Oxford University Press, 1988)
- [7]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [8]
- N. V. Semakov, V. A. Zinoviev, and G. V. Zaitsev, “Uniformly Packed Codes”, Problemy Peredachi Informatsii 7(1), 38–50 (1971); Problems of Information Transmission 7(1), 30–39 (1971)
- [9]
- G. V. Zaitsev, V. A. Zinoviev, and N. V. Semakov, “Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes”, in Proc. 2nd International Symp. Inform. Theory (1973): 257–263
- [10]
- A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
Page edit log
- Victor V. Albert (2022-08-12) — most recent
- Victor V. Albert (2022-03-22)
- Dhruv Devulapalli (2021-12-17)
Cite as:
“\([2^m,2^m-m-1,4]\) Extended Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/extended_hamming