\([2^m-1,m,2^{m-1}]\) simplex code

\([2^m-1,m,2^{m-1}]\) simplex code[1,2]

Also known as Shortened Hadamard code, RM\(^*(1,m)\) code, Maximum-length feedback-shift-register code.

Description

A member of the code family that is dual to the \([2^m,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping.

A punctured simplex code is known as a MacDonald code [3], with parameters \([[2^m-2^u,m,2^{m-1}-2^{u-1}]]\) for \(u \leq m-1\) [4].

The automorphism group of the code is \(GL_{m}(\mathbb{F}_{2})\) [5].

Decoding

Serial orthogonal decoder [6,7]Quantum decoder [8].

Parents

Linear binary code
\(q\)-ary simplex code — \(q\)-ary simplex codes reduce to simplex codes for \(q=2\).

Child

\([7,3,4]\) simplex code

Cousins

Extended GRS code — Simplex codes are extended RS codes [9].
\([2^r-1,2^r-r-1,3]\) Hamming code — Hamming and simplex codes are dual to each other.
Dual linear code — Hamming and simplex codes are dual to each other.
Simplex spherical code — Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the antipodal mapping [10; Sec. 6.5.2][11; pg. 18]. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.
\([2^m,m+1,2^{m-1}]\) First-order RM code — First-order RM codes and simplex codes are interconvertible via shortening and lengthening [5; pg. 31]. Punctured first-order RM codes and simplex codes are interconvertible via expurgation and augmentation [5; pg. 31].
Gold code — Simplex codes are used to make gold codes. The dual of a Gold code is the interesection of the duals of the simplex codes used to construct it [12].
Hadamard code — The \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)).
Repetition code — The simplex code for \(m=2\) reduces to a four-bit repetition code.
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Quantum Hamming codes result from applying the CSS construction to Hamming codes and their duals the simplex codes.

References

[1]: R. A. FISHER, “THE THEORY OF CONFOUNDING IN FACTORIAL EXPERIMENTS IN RELATION TO THE THEORY OF GROUPS”, Annals of Eugenics 11, 341 (1941) DOI
[2]: C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal 27, 379 (1948) DOI
[3]: J. E. MacDonald, “Design Methods for Maximum Minimum-Distance Error-Correcting Codes”, IBM Journal of Research and Development 4, 43 (1960) DOI
[4]: A. Patel, “Maximal&lt;tex&gt;q&lt;/tex&gt;-nary linear codes with large minimum distance (Corresp.)”, IEEE Transactions on Information Theory 21, 106 (1975) DOI
[5]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[6]: R. R. Green, "A serial orthogonal decoder," JPL Space Programs Summary, vol. 37–39-IV, pp. 247–253, 1966.
[7]: A. Ashikhmin and S. Litsyn, “Simple MAP decoding of first order Reed-Muller and Hamming codes”, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674) 18 DOI
[8]: A. Barg and S. Zhou, “A quantum decoding algorithm for the simplex code”, in Proceedings of the 36th Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, 23–25 September 1998 (UIUC 1998) 359–365
[9]: Rudolf Schürer and Wolfgang Ch. Schmid. “Simplex Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CSimplex.html
[10]: Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
[11]: T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[12]: M. des Noes, V. Savin, J. M. Brossier, and L. Ros, “Iterative decoding of Gold sequences”, 2015 IEEE International Conference on Communications (ICC) 4840 (2015) DOI

Page edit log

Victor V. Albert (2022-08-09) — most recent
Yi-Ting (Rick) Tu (2022-03-28)

Cite as:

“\([2^m-1,m,2^{m-1}]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/simplex

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/easy/dual_hamming/simplex.yml.