Delsarte–Goethals code

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

History

The concept was introduced by mathematicians Philippe 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 $2^{m}$ , 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( $2^{3m-1},2^{m},\mathbb {Z} _{2},7)$ .^[4]^[5]

References

^ "Delsarte-Goethals code - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2017-05-22.
^ Abhyankar, S. S.; et al. (2002). Hazewinkel, M. (ed.). Encyclopaedia of Mathematics, Supplement III. doi:10.1007/978-0-306-48373-8. ISBN 978-1-4020-0198-7.^{[page needed]}
^ Leducq, Elodie (May 2012). "A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes". Finite Fields and Their Applications. 18 (3): 581–586. doi:10.1016/j.ffa.2011.12.003.
^ Schürer, Rudolf. "MinT - Delsarte–Goethals Codes". mint.sbg.ac.at. Retrieved 2017-05-22.
^ Abhyankar, S. S.; et al. (2002). Hazewinkel, M. (ed.). Encyclopaedia of Mathematics, Supplement III. doi:10.1007/978-0-306-48373-8. ISBN 978-1-4020-0198-7.^{[page needed]}

[1] "Delsarte-Goethals code - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2017-05-22.

[2] Abhyankar, S. S.; et al. (2002). Hazewinkel, M. (ed.). Encyclopaedia of Mathematics, Supplement III. doi:10.1007/978-0-306-48373-8. ISBN 978-1-4020-0198-7.^{[page needed]}

[3] Leducq, Elodie (May 2012). "A new proof of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords of generalized Reed–Muller codes". Finite Fields and Their Applications. 18 (3): 581–586. doi:10.1016/j.ffa.2011.12.003.

[4] Schürer, Rudolf. "MinT - Delsarte–Goethals Codes". mint.sbg.ac.at. Retrieved 2017-05-22.

[5] Abhyankar, S. S.; et al. (2002). Hazewinkel, M. (ed.). Encyclopaedia of Mathematics, Supplement III. doi:10.1007/978-0-306-48373-8. ISBN 978-1-4020-0198-7.^{[page needed]}

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