Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Some algebraic tools for error-correcting codes

  • Conference paper
  • First Online:
Algebraic Algorithms and Error-Correcting Codes (AAECC 1985)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 229))

  • 172 Accesses

Abstract

We give several algebraic techniques, useful for the study of error-correcting codes : decomposition of ideals, automorphisms. We also prove that codes with coefficients in a noetherian ring are not better than codes with coefficients in a field.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Abstract of Papers IEEE, Brighton (June 1985)

    Google Scholar 

  2. ANDRE M. “Théorie noethérienne des codes linéaires” Rapport Interne, Ecole Polytechnique Fédérale, Lausanne, (1980)

    Google Scholar 

  3. BACHMAIR L., BUCHBERGER B. “A simplified proof of the characterization theorem for Gröbner bases”, SIGSAM Bul., vol 14 no4 (1980)

    Google Scholar 

  4. BEENKER G.M.F. “On double circulant codes” (1980) Th. Report 80 WSK 04, University of Eindhoven, The Netherlands

    Google Scholar 

  5. BERMANN S.D. “Abélian group codes” Kibernetika, vol 3 no1, pp 31–39 (1967)

    Google Scholar 

  6. BERLEKAMP E.R. “The technology of error correcting codes” Proceedings of IEEE, vol 68 no5, pp 564–593 (1980)

    Google Scholar 

  7. BETH T. “Generalizing the Discrete Fourier Transform” Acts of AAECC-1, Discrete Math., vol 56 no2–3, pp 95–100 (1985)

    Google Scholar 

  8. BOURBAKI N. “Algèbre commutative” chapitres 1, 2, 3, 4, Ed. MASSON (1985)

    Google Scholar 

  9. BOURBAKI N. “Topologie générale”, chapitre 1,2, Ed. HERMANN (1965)

    Google Scholar 

  10. BUCHBERGER B. “A critical pair completion algorithm for finitely generated ideals in ring” Symp. “Rekursive Kombin.” in Munster, Lect. Notes in Math. Springer Verlag (1983)

    Google Scholar 

  11. BUCHBERGER B. “Algebraic simplifications” Comp. Suppl. 4, pp 11–43, Springer Verlag (1982)

    Google Scholar 

  12. BUCHBERGER B. “Some properties of Gröbner bases for polynomial ideals” SIGSAM Bul., vol 10 no4 (1976)

    Google Scholar 

  13. CAMION P. “Etude de codes binaires abéliens modulaires autoduaux de petite longueur”, Revue du CETHEDEC N.S. 79-2, pp 3–24 (1978)

    Google Scholar 

  14. CAMION P. “Un algorithme de construction des idempotents primitifs d'idéaux sur \(\mathbb{F}_q\)”, C.R.A.S. Paris, t 291, série A (1980)

    Google Scholar 

  15. CHARPIN P. “Codes idéaux de certaines algèbres modulairés” Thèse de 3ième cycle, Univ. PARIS VII (1982)

    Google Scholar 

  16. DELCLOS G. “Etude conjointe des codes cycliques sur IF2 et IF4, et généralisation aux algèbres de groupe” Thèse de 3ième cycle, Univ. de Provence (1980)

    Google Scholar 

  17. DELSARTE P. “Automorphisms of abelian codes” Philips Res. Reports, pp 389–403 (1970)

    Google Scholar 

  18. DOLFO M. “Transmission d'images iconiques, en temps réel, protégées par un code algébrique, simulation informatique de l'automate DOLVRON 1” Mémoire d'Ingénieur CNAM, Toulouse (1981)

    Google Scholar 

  19. GENNERO M.C. “Un logiciel de simulation de transmission d'information LOUSTICC” Mémoire d'Ingénieur CNAM en Informatique, Toulouse (1983)

    Google Scholar 

  20. GENNERO M.C. “LOUSTICC simulation software: experimental results” Acts of AAECC-2 (1984), to appear in Springer Verlag

    Google Scholar 

  21. DE GROOTE H.F., HEINTZ J. “Commutative algebras of minimal rank” Fachbereit Math., J. Goethe Univ. Frankfurt, Germany (1982)

    Google Scholar 

  22. HEINTZ J., MORGENSTERN J. “Associative algebra of minimal rank” Acts of AAECC-2 Symp. (1984), to appear in Springer Verlag

    Google Scholar 

  23. HOEVE H., TIMMERMANS J., VRIES L.B. “Error correction and cancealment in the compact disc system” Philips Technical Review, vol 40 no6, pp 166–172 (1982)

    Google Scholar 

  24. IKAI H., KOSAKO H., KOJIMA Y. “Basic theory of two dimensional cyclic codes. Period of ideals and fundamental theorems” Elect. on Comm., vol 59A no3 (1976)

    Google Scholar 

  25. IMAI H. “Two dimensional Fire codes” IEEE Trans. on Inform. Theory, vol IT-19 no6, pp 796–806 (1973)

    Google Scholar 

  26. IMAI H. “Multivariate polynomials in coding theory” Acts of AAECC-2 (1984), to appear in Springer Verlag

    Google Scholar 

  27. IMAI H., ARAKAKI M. “Theory of two dimensional cyclic codes” IECE National Conf. Records, 1415 (1974)

    Google Scholar 

  28. INAMURA K., MORII M., “Two classes of finite fields which have no self complementary normal bases” Abstract of Papers IEEE, Brighton (1985), reference no1

    Google Scholar 

  29. KANDRI-RODY A., SAUNDERS B.D. “Primality of ideals in polynomial rings” Private communication

    Google Scholar 

  30. KASAMI A., LIN S., PETERSON W.W. “Some results of cyclic codes which are invariant under the affine group and their applications” IEEE Inf. and Control, vol 11, pp 475–496 (1967)

    Google Scholar 

  31. LAFON J.P. “Algèbre commutative” Tome 2 Edit. HERMANN (1972)

    Google Scholar 

  32. LAUBIE M. “Codes idéaux de certaines algèbres modulaires et ramification” to appear in Communications in Algebra

    Google Scholar 

  33. LESIEUR L., CROISOT R. “Algèbre noetherienne non commutative” Edit. by GAUTHIERS VILLARS (1963)

    Google Scholar 

  34. LIFERMAN J. “Les méthodes rapides de transformation du signal” Edit. by MASSON (1980)

    Google Scholar 

  35. MacWILLIAMS F.J., SLOANE N.J.A. “The theory of error correcting codes” North Holland Pub. Cie (1977)

    Google Scholar 

  36. MALLOWS C.L., PLESS V., SLOANE N.J.A. “Self dual codes over GF(3)” SIAM J. App. Math., vol 31, pp 649–666 (1976)

    Google Scholar 

  37. MÖLLER H.M., BUCHBERGER B., “The construction of multivariate polynomials with preassigned zeroes” CAMP. Public. no82–22.0 Lecture notes (Proc. EUROCAM 82, LNC 3 144, pp 24–31, (1982)

    Google Scholar 

  38. PASQUIER G. “Etude de codes sur une extension de IF2 et leurs images binaires” Thèse de 3ième cvcle. Univ. de Province (1980)

    Google Scholar 

  39. POLI A. “Codes dans certaines algèbres modulaires” Thèse d'Etat, Univ. P. Sabatier, Toulouse (1978)

    Google Scholar 

  40. POLI A. “Codes stables sous le groupe des automorphismes isométriques de A = IFp [X1, ..., Xn] / (X p1 -1, ..., X pn -1)” C.R.A.S., Paris t 280, série A, pp 1029–1032 (1980)

    Google Scholar 

  41. POLI A. “Construction of primitive idempotents for n variable codes” Acts of AAECC-2 (1984), to appear in Springer Verlag

    Google Scholar 

  42. POLI A. “Multicirculant self dual codes over IFq” Submitted for publication to Communications in Algebra.

    Google Scholar 

  43. POLI A., RIGONI C. “Codes autoduaux 2k-circulants (caractéristique impaire)” Revue de Traitement du Signal, vol 1 no2–2, pp 205–209 (1984)

    Google Scholar 

  44. POLI A., THIONG LY J.A. “Automorphisms of principal nilpotent self dual codes in some modular algebras” Acts of AAECC-1, Disc. Math., vol 56 no2–3, pp 165–174 (1985)

    Google Scholar 

  45. POLI A., THIONG LY J.A. “Codes autoduaux principaux nilpotents dans l'algèbre A =...” Revue du Traitement du Signal, vol 1 no2–2, pp 217–221 (1984)

    Google Scholar 

  46. POLI A., VENTOU M. “Codes autoduaux principaux et groupe d'automorphismes de l'algèbre A= ...” European J. of Comb., Acad. Press, vol 2, pp 179–18 179–183 (1981)

    Google Scholar 

  47. POLI A., GENNERO M.C., RANDRIANANJA D. “Codeur/decodeur des codes recommandés par le CCSDS” Rapport final de contrat CNES/AAECC, 210 pages (1985)

    Google Scholar 

  48. RIGONI C. “Contribution à l'étude des codes correcteurs polynomiaux” Thèse de 3ième cycle, Univ. P. Sabatier, Toulouse (1985)

    Google Scholar 

  49. SAKATA S. “On determining the independent point set for doubly periodic arrays”, IEEE Trans. on Inf. Theory, vol IT-27 no 5, pp 556–565 (1981)

    Google Scholar 

  50. SAMUEL P., ZARISKI O. “Commutative Algebra” Princeton, Van Nostram (1958)

    Google Scholar 

  51. SLOANE N.J.A. “A survey of constructive coding theory, and a table of binary codes of highest known rate” Discrete Math., vol 3, pp 265–294 (1972)

    Google Scholar 

  52. THIONG LY J.A. “A propos du produit semi direct et du produit en couronne de groupes” Thèse de 3ième cycle, Univ. P. Sabatier, Toulouse (1978)

    Google Scholar 

  53. THIONG LY J.A. “Construction d'une famille de codes autoduaux binaires” Revue du Traitement du Signal, vol 1 no2–2, pp 233–237 (1984)

    Google Scholar 

  54. VENTOU M. “Contribution à l'étude des codes correcteurs polynomiaux” Thèse de 3ième cycle, Univ. P. Sabatier, Toulouse (1984)

    Google Scholar 

  55. Van Der WAERDEN B.L. “Modern algebra” (Volume II) Frederick Ungar Publishing Co. (New York — 5th printing — 1964)

    Google Scholar 

  56. WASAN S.K. “On codes over ZZm”, IEEE Trans. on Inf. Th. pp 117–121 (1982)

    Google Scholar 

  57. WOLFMANN J. “A permutation decoding of the (24, 12, 8) Golay code” IEEE Trans. of Inf. Th., vol IT-29 no5, pp 748–750 (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AAECC Lab. Université P. Sabatier, 118 route de Narbonne, 31062, Toulouse cédex, France

    A. Poli

Authors
  1. A. Poli
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jacques Calmet

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Poli, A. (1986). Some algebraic tools for error-correcting codes. In: Calmet, J. (eds) Algebraic Algorithms and Error-Correcting Codes. AAECC 1985. Lecture Notes in Computer Science, vol 229. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16776-5_708

Download citation

  • DOI: https://doi.org/10.1007/3-540-16776-5_708

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16776-1

  • Online ISBN: 978-3-540-39855-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation