\([2^m,2^m-m-1,4]\) Extended Hamming code

\([2^m,2^m-m-1,4]\) Extended Hamming code [1–3]

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

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.
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 [5; pg. 182].
Combinatorial design— Weight-four codewords of the \([2^r,2^r-r-1, 4]\) extended Hamming code support the Steiner system \(S(3,4,2^r)\) [6; 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 [7,8].
\([2^r-1,2^r-r-1,3]\) 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 [9].
\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code— A Hamming Majorana code is constructed from a first-order RM code (whose dual is the extended Hamming code).

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC

Coxeter code

Reed-Muller (RM) codeLinear code over \(\mathbb{Z}_q\) GRM Evaluation Divisible Linear \(q\)-ary AG LCC LRC Distributed-storage ECC

Parents

Reed-Muller (RM) code

Extended Hamming codes are RM\((m-2,m)\) codes.

Small-distance block codeECC

\([2^m,2^m-m-1,4]\) Extended Hamming code

Children

\([8,4,4]\) extended Hamming code

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]
H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
[5]
R. Hill, A First Course in Coding Theory (Oxford University Press, 1988)
[6]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[7]
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)
[8]
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
[9]
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 (2026-10-13) — most recent
Victor V. Albert (2026-06-08)
Victor V. Albert (2022-08-12)
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.), 2026. https://errorcorrectionzoo.org/c/extended_hamming, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/reed_muller/hamming/extended_hamming.yml.