Description
Member of an infinite family of binary linear codes with parameters \([2^r,2^r-r-1, 4]\) for \(r \geq 2\) that are extensions of the Hamming codes by a parity-check bit.
Parents
Child
Cousins
- Universally optimal \(q\)-ary code — Several extended Hamming codes are LP universally optimal codes [4].
- \([2^m,m+1,2^{m-1}]\) 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.
- Combinatorial design — Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [5; pg. 89].
- \([2^r-1,2^r-r-1,3]\) Hamming code — Extended Hamming codes are extensions of Hamming codes by a parity-check bit. Puncturing extended Hamming codes yields the Hamming codes.
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, Proc. IEEE, 37 (1949) 657.
- [4]
- H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
- [5]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
Page edit log
- Victor V. Albert (2022-08-12) — most recent
- Victor V. Albert (2022-03-22)
- Dhruv Devulapalli (2021-12-17)
Cite as:
“\([2^r,2^r-r-1,4]\) Extended Hamming code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/extended_hamming