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.^[1][2]

A new proof of the properties of the Delsarte–Goethals code was published in 1970.^[3]

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

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

[math]\displaystyle{ K(m) \subseteq DG(m,r) \subseteq RM(2,m) }[/math]

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([math]\displaystyle{ 2^{3m-1}, 2^m, \mathbb{Z}_2, 7) }[/math].^[4][5]

References

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