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

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

[math]\displaystyle{ K(m) \subseteq DG(m,r) \subseteq RM(2,m) }[/math]

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([math]\displaystyle{ 2^{3m-1}, 2^m, \mathbb{Z}_2, 7) }[/math].[4][5]

## References

1. ↑ "Delsarte-Goethals code - Encyclopedia of Mathematics" (in en). https://www.encyclopediaofmath.org/index.php/Delsarte-Goethals_code.
2. ↑ Hazewinkel, Michiel (2007-11-23) (in en). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN 9780306483738. https://books.google.com/books?id=ujnhBwAAQBAJ&q=delsarte+goethals&pg=PA118.
3. ↑ Leducq, Elodie (2012). "A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes - ScienceDirect" (in en). Finite Fields and Their Applications 18 (3): 581–586. doi:10.1016/j.ffa.2011.12.003. http://hal.archives-ouvertes.fr/docs/00/44/69/13/PDF/poidsminarxiv.pdf.
4. ↑ Schürer, Rudolf. "MinT - Delsarte–Goethals Codes". http://mint.sbg.ac.at/desc_CDelsarteGoethals.html.
5. ↑ Hazewinkel, Michiel (2007-11-23) (in en). Encyclopaedia of Mathematics, Supplement III. Springer Science & Business Media. ISBN 9780306483738. https://books.google.com/books?id=ujnhBwAAQBAJ&q=delsarte+goethals&pg=PA118.

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