Delsarte–Goethals code

The concept was introduced by mathematicians Philippe Delsarte and J.-M. Goethals in their published paper.{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Delsarte-Goethals_code|title=Delsarte-Goethals code - Encyclopedia of Mathematics|website=www.encyclopediaofmath.org|language=en|access-date=2017-05-22}}{{Cite book|url=https://books.google.com/books?id=ujnhBwAAQBAJ&q=delsarte+goethals&pg=PA118|title=Encyclopaedia of Mathematics, Supplement III|last=Hazewinkel|first=Michiel|date=2007-11-23|publisher=Springer Science & Business Media|isbn=9780306483738|language=en}}

A new proof of the properties of the Delsarte–Goethals code was published in 1970.{{Cite journal|title=A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes - ScienceDirect|language=en|doi=10.1016/j.ffa.2011.12.003|volume=18|issue=3|journal=Finite Fields and Their Applications|pages=581–586 | last1 = Leducq | first1 = Elodie|year=2012|url=http://hal.archives-ouvertes.fr/docs/00/44/69/13/PDF/poidsminarxiv.pdf|doi-access=free}}

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

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

: $K(m)\; \backslash subseteq\; DG(m,r)\; \backslash subseteq\; RM(2,m)$

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($2^\{3m-1\},\; 2^m,\; \backslash mathbb\{Z\}\_2,\; 7)$.{{Cite web|url=http://mint.sbg.ac.at/desc_CDelsarteGoethals.html|title=MinT - Delsarte–Goethals Codes|last=Schürer|first=Rudolf|website=mint.sbg.ac.at|access-date=2017-05-22}}{{Cite book|url=https://books.google.com/books?id=ujnhBwAAQBAJ&q=delsarte+goethals&pg=PA118|title=Encyclopaedia of Mathematics, Supplement III|last=Hazewinkel|first=Michiel|date=2007-11-23|publisher=Springer Science & Business Media|isbn=9780306483738|language=en}}

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