## Simplex Code

Simplex codes S(*n*, *b*) are projective, linear [*s*, *n*, *b*^{n−1}]-codes over **F**_{b}, with

*s*= .

They exist for all *n* ≥ 1. For *n* = 1 the [1, 1, 1]-code without redundancy or repetition code is obtained.

The *n*×*s* generator matrix of the Simplex code S(*n*, *b*) is constructed by using the *s* pairwise linearly independent vectors of **F**_{b}^{n}. Such a set can always be found, e.g. by choosing all non-zero vectors with leftmost non-zero coordinate equal to 1 from **F**_{b}^{n}.

The columns of the generator matrix correspond to all points in the projective space PG(*n*−1, *b*), thus the simplex code is the longest projective code for given *n*.

The weight distribution of the simplex code is *A*_{0} = 1 and *A*_{bn−1} = *b*^{n} − 1, thus S(*n*, *b*) ∖ { 0} is a constant-weight code. Therefore, if one identifies ℤ_{2}^{s} with vertices of the *s*-dimensional unit cube [0, 1]^{s}, the 2^{n} code words of the simplex code over **F**_{2} are the vertices of a regular (2^{n} − 1)-dimensional simplex with edge length 2^{(n−1)/2} (therefore the name simplex code).

The dual orthogonal array A = S(*n*, *b*)^{⊥} is a linear OA(*b*^{s−n}, *s*,**F**_{b}, *b*^{n−1} − 1). When interpreted as a code, A is the [*s*, *s*−*n*, 3]-Hamming code.

### Optimality

The simplex code meets the Griesmer bound and is therefore a linear code with the lowest possible length *s* for given dimension *n* and distance *b*^{n−1}.

### Special Cases

S(1,

*b*) is the [1, 1, 1]-code without redundancy.S(2,

*b*) is an extended Reed-Solomon code RS_{e}(2,*b*).

### See also

### References

[1] | F. Jessie MacWilliams and Neil J. A. Sloane.The Theory of Error-Correcting Codes.North-Holland, Amsterdam, 1977. |

[2] | Jürgen Bierbrauer.Introduction to Coding Theory.Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004. MR2079734 (2005f:94001) |

### Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.

Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Simplex Code.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2008-04-04.
http://mint.sbg.ac.at/desc_CSimplex.html