Simplex Code

Simplex codes S(n, b) are projective, linear [s, n, bn−1]-codes over Fb, with

s = $\displaystyle {\frac{{b^{n}−1}}{{b−1}}}$.

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 ℤ2s 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(bs−n, s,Fb, bn−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 bn−1.

Special Cases

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: 2024-09-05. http://mint.sbg.ac.at/desc_CSimplex.html

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