The Delsarte–Goethals code is a type of error-correcting code.

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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]

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

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

${\displaystyle K(m)\subseteq DG(m,r)\subseteq RM(2,m)}$

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