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.

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, 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


 * $$K(m) \subseteq DG(m,r) \subseteq RM(2,m)$$

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