Skip to main content
Top
Published in: Applicable Algebra in Engineering, Communication and Computing 5/2016

09-02-2016 | Original Paper

Partial permutation decoding for MacDonald codes

Authors: Jennifer D Key, Padmapani Seneviratne

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2016

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We show how to find s-PD-sets of the minimal size \(s+1\) for the \(\left[ \frac{q^n-q^u}{q-1},n,q^{n-1}-q^{u-1}\right] _q \) MacDonald q-ary codes \(C_{n,u}(q)\) where \(n \ge 3\) and \(1 \le u \le n-1\). The construction of [6] can be used and gives s-PD-sets for s up to the bound \(\lfloor \frac{q^{n-u}-1}{(n-u)(q-1)} \rfloor -1\), of effective use for u small; for \(u \ge \lfloor \frac{n}{2} \rfloor \) an alternative construction is given that applies up to a bound that depends on the maximum size of a set of vectors in \(V_u(\mathbb {F}_q)\) with each pair of vectors distance at least 3 apart.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Appendix
Available only for authorised users
Literature
1.
go back to reference Assmus, Jr., E.F., Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992), Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993) Assmus, Jr., E.F., Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992), Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993)
2.
go back to reference Bhandar, M.C., Durairajan, C.: A note on covering radius of MacDonald codes. In: Proceedings of the International Conference on Information Technology: Computers and Communications (ITCC03) 0-7695-1916-4/03, IEEE (2003) Bhandar, M.C., Durairajan, C.: A note on covering radius of MacDonald codes. In: Proceedings of the International Conference on Information Technology: Computers and Communications (ITCC03) 0-7695-1916-4/03, IEEE (2003)
3.
5.
go back to reference Cannon, J., Steel, A., White, G.: Linear codes over finite fields. In: Cannon, J., Bosma, W. (eds.) Handbook of Magma Functions, Computational Algebra Group, V2.13, pp. 3951-4023. Department of Mathematics, University of Sydney (2006) http://magma.maths.usyd.edu.au/magma Cannon, J., Steel, A., White, G.: Linear codes over finite fields. In: Cannon, J., Bosma, W. (eds.) Handbook of Magma Functions, Computational Algebra Group, V2.13, pp. 3951-4023. Department of Mathematics, University of Sydney (2006) http://​magma.​maths.​usyd.​edu.​au/​magma
6.
go back to reference Fish, Washiela, Key, Jennifer D., Mwambene, Eric: Partial permutation decoding for simplex codes. Adv. Math. Commun. 6, 505-516 (2012)MathSciNetCrossRefMATH Fish, Washiela, Key, Jennifer D., Mwambene, Eric: Partial permutation decoding for simplex codes. Adv. Math. Commun. 6, 505-516 (2012)MathSciNetCrossRefMATH
7.
8.
go back to reference Cary Huffman, W.: Codes and groups. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, Volume 2, Part 2, Chapter 17, pp. 1345-1440. Elsevier, Amsterdam (1998) Cary Huffman, W.: Codes and groups. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, Volume 2, Part 2, Chapter 17, pp. 1345-1440. Elsevier, Amsterdam (1998)
9.
go back to reference Key, J.D., McDonough, T.P., Mavron, V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Combin. 26, 665-682 (2005)MathSciNetCrossRefMATH Key, J.D., McDonough, T.P., Mavron, V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Combin. 26, 665-682 (2005)MathSciNetCrossRefMATH
10.
go back to reference Key, J.D., McDonough, T.P., Mavron, V.C.: Information sets and partial permutation decoding for codes from finite geometries. Finite Fields Appl. 12, 232-247 (2006)MathSciNetCrossRefMATH Key, J.D., McDonough, T.P., Mavron, V.C.: Information sets and partial permutation decoding for codes from finite geometries. Finite Fields Appl. 12, 232-247 (2006)MathSciNetCrossRefMATH
11.
12.
go back to reference MacDonald, J.E.: Design methods for maximum minimum-distance error-correcting codes. IBM J. Res. Dev. 4, 43-57 (1960)MathSciNetCrossRef MacDonald, J.E.: Design methods for maximum minimum-distance error-correcting codes. IBM J. Res. Dev. 4, 43-57 (1960)MathSciNetCrossRef
13.
go back to reference MacWilliams, F.J.: Permutation decoding of systematic codes. Bell Syst. Tech. J. 43, 485-505 (1964)CrossRefMATH MacWilliams, F.J.: Permutation decoding of systematic codes. Bell Syst. Tech. J. 43, 485-505 (1964)CrossRefMATH
14.
go back to reference MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)MATH MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)MATH
15.
go back to reference Patel, A.M.: Maximal $q$-ary linear codes with large minimum distance. IEEE Trans. Inform. Theory 21, 106-110 (1975)CrossRefMATH Patel, A.M.: Maximal $q$-ary linear codes with large minimum distance. IEEE Trans. Inform. Theory 21, 106-110 (1975)CrossRefMATH
17.
Metadata
Title
Partial permutation decoding for MacDonald codes
Authors
Jennifer D Key
Padmapani Seneviratne
Publication date
09-02-2016
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 5/2016
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-016-0286-7

Other articles of this Issue 5/2016

Applicable Algebra in Engineering, Communication and Computing 5/2016 Go to the issue

Premium Partner