Delsarte–Goethals code
# 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($2^{3m-1},2^{m},\mathbb {Z} _{2},7)$.

