Delsarte–Goethals code

 

The Delsarte–Goethals code is a type of .

Contents

History

The concept was introduced by mathematicians Ph. Delsarte and J.-M. Goethals in their published paper.

A new proof of the properties of the Delsarte–Goethals code was published in 1970.

Function

The Delsarte–Goethals code DG(m,r) for even m ≥ 4 and 0 ≤ r ≤ m/2 − 1 is a binary, of length 2^{m}, size 2^{r(m-1)+2m} and 2^{m-1} - 2^{m/2-1+r}

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

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 OA(2^{3m-1}, 2^m, mathbb{Z}_2, 7).

Source

http://wikipedia.org/

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