Delsarte–Goethals code

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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 2m, size 2r(m−1)+2m and minimum distance 2m−1−2m/2−1+r

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

K(m)⊆DG(m,r)⊆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(23m−1,2m,ℤ2,7).[4][5]

References

  1. ↑ "Delsarte-Goethals code - Encyclopedia of Mathematics" (in en). https://www.encyclopediaofmath.org/index.php/Delsarte-Goethals_code. 
  2. ↑ Abhyankar, S. S. et al. (2002). 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". http://mint.sbg.ac.at/desc_CDelsarteGoethals.html. 
  5. ↑ Abhyankar, S. S. et al. (2002). 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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