## Simplex Code

Simplex codes S(n, b) are projective, linear [s, n, bn−1]-codes over Fb, 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 Fbn. Such a set can always be found, e.g. by choosing all non-zero vectors with leftmost non-zero coordinate equal to 1 from Fbn.

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 A0 = 1 and Abn−1 = bn − 1, thus S(n, b) ∖ { 0} is a constant-weight code. Therefore, if one identifies 2s with vertices of the s-dimensional unit cube [0, 1]s, the 2n code words of the simplex code over F2 are the vertices of a regular (2n − 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(bsn, s,Fb, bn−1 − 1). When interpreted as a code, A is the [s, sn, 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 bn−1.