Delsarte–Goethals code

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 ${\displaystyle 2^{m}}$ , size ${\displaystyle 2^{r(m-1)+2m}}$  and minimum distance ${\displaystyle 2^{m-1}-2^{m/2-1+r}}$ 

The code sits between the Kerdock code and the second-order Reed–Muller codes. More precisely, we have

${\displaystyle K(m)\subseteq DG(m,r)\subseteq RM(2,m)}$ 

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(${\displaystyle 2^{3m-1},2^{m},\mathbb {Z} _{2},7)}$ .^[4][5]

References

1. "Delsarte-Goethals code - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2017-05-22.
2. Hazewinkel, Michiel (2007-11-23). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN 9780306483738.
3. Leducq, Elodie (2012). "A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes - ScienceDirect" (PDF). Finite Fields and Their Applications. 18 (3): 581–586. doi:10.1016/j.ffa.2011.12.003.
4. Schürer, Rudolf. "MinT - Delsarte–Goethals Codes". mint.sbg.ac.at. Retrieved 2017-05-22.
5. Hazewinkel, Michiel (2007-11-23). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN 9780306483738.