\([7,3,4]\) simplex code 

Also known as RM\(^*(1,3)\) code.

Description

Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping.

Its generator matrix is \begin{align} \left(\begin{array}{ccccccc} 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ \end{array}\right)~. \tag*{(1)}\end{align} The automorphism group of the code is \(GL_{3}(\mathbb{F}_{2})\), the second-smallest simple group.

Parents

Cousins

  • \([7,4,3]\) Hamming code — The \([7,4,3]\) Hamming and \([7,3,4]\) simplex codes are dual to each other.
  • Dual linear code — The \([7,4,3]\) Hamming and \([7,3,4]\) simplex codes are dual to each other.
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: simplex734

Cite as:
\([7,3,4]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/simplex734
BibTeX:
@incollection{eczoo_simplex734, title={\([7,3,4]\) simplex code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/simplex734} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/simplex734

Cite as:

\([7,3,4]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/simplex734

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/easy/dual_hamming/simplex734.yml.