Delsarte–Goethals code explained
The Delsarte–Goethals code is a type of error-correcting code.
History
The concept was introduced by mathematicians Ph. Delsarte and J.-M. Goethals in their published paper.[1] [2]
A new proof of the properties of the Delsarte–Goethals code was published in 1970.[3]
Function
The Delsarte–Goethals code DG(m,r) for even m ≥ 4 and 0 ≤ r ≤ m/2 − 1 is a binary, non-linear code of length
, size
and minimum distance
The code sits between the Kerdock code and the second-order Reed–Muller codes. More precisely, we have
K(m)\subseteqDG(m,r)\subseteqRM(2,m)
When r = 0, we have DG(m,r) = K(m) and when r = m/2 − 1 we have DG(m,r) = RM(2,m).
For r = m/2 − 1 the Delsarte–Goethals code has strength 7 and is therefore an orthogonal array OA(
.
[4] [5] Notes and References
- Web site: Delsarte-Goethals code - Encyclopedia of Mathematics. www.encyclopediaofmath.org. en. 2017-05-22.
- Book: Hazewinkel, Michiel. Encyclopaedia of Mathematics, Supplement III. 2007-11-23. Springer Science & Business Media. 9780306483738. en.
- A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes - ScienceDirect. en. 10.1016/j.ffa.2011.12.003. 18. 3. Finite Fields and Their Applications. 581–586 . Leducq . Elodie. 2012. free.
- Web site: MinT - Delsarte–Goethals Codes. Schürer. Rudolf. mint.sbg.ac.at. 2017-05-22.
- Book: Hazewinkel, Michiel. Encyclopaedia of Mathematics, Supplement III. 2007-11-23. Springer Science & Business Media. 9780306483738. en.