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.
Cousins
- \([7,4,3]\) Hamming code— The \([7,3,4]\) simplex code is the dual of the Hamming code and also its even-weight subcode [1,2].
- \(E_7\) root lattice— The \([7,3,4]\) simplex code yields the \(E_7\) root lattice via Construction A [4][3; Exam. 10.5.3].
Primary Hierarchy
References
- [1]
- W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)
- [2]
- C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI
- [3]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [4]
- J. H. Conway and N. J. A. Sloane, “On the Voronoi Regions of Certain Lattices”, SIAM Journal on Algebraic Discrete Methods 5, 294 (1984) DOI
Page edit log
- Victor V. Albert (2024-04-26) — most recent
Cite as:
“\([7,3,4]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/simplex734