Advertisement

Some algebraic tools for error-correcting codes

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

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.

Keywords

Prime Ideal Discrete Fourier Transform Dual Code Primitive Idempotent Algebraic Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abstract of Papers IEEE, Brighton (June 1985)Google Scholar
  2. 2.
    ANDRE M. “Théorie noethérienne des codes linéaires” Rapport Interne, Ecole Polytechnique Fédérale, Lausanne, (1980)Google Scholar
  3. 3.
    BACHMAIR L., BUCHBERGER B. “A simplified proof of the characterization theorem for Gröbner bases”, SIGSAM Bul., vol 14 no4 (1980)Google Scholar
  4. 4.
    BEENKER G.M.F. “On double circulant codes” (1980) Th. Report 80 WSK 04, University of Eindhoven, The NetherlandsGoogle Scholar
  5. 5.
    BERMANN S.D. “Abélian group codes” Kibernetika, vol 3 no1, pp 31–39 (1967)Google Scholar
  6. 6.
    BERLEKAMP E.R. “The technology of error correcting codes” Proceedings of IEEE, vol 68 no5, pp 564–593 (1980)Google Scholar
  7. 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. 8.
    BOURBAKI N. “Algèbre commutative” chapitres 1, 2, 3, 4, Ed. MASSON (1985)Google Scholar
  9. 9.
    BOURBAKI N. “Topologie générale”, chapitre 1,2, Ed. HERMANN (1965)Google Scholar
  10. 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. 11.
    BUCHBERGER B. “Algebraic simplifications” Comp. Suppl. 4, pp 11–43, Springer Verlag (1982)Google Scholar
  12. 12.
    BUCHBERGER B. “Some properties of Gröbner bases for polynomial ideals” SIGSAM Bul., vol 10 no4 (1976)Google Scholar
  13. 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. 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. 15.
    CHARPIN P. “Codes idéaux de certaines algèbres modulairés” Thèse de 3ième cycle, Univ. PARIS VII (1982)Google Scholar
  16. 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. 17.
    DELSARTE P. “Automorphisms of abelian codes” Philips Res. Reports, pp 389–403 (1970)Google Scholar
  18. 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. 19.
    GENNERO M.C. “Un logiciel de simulation de transmission d'information LOUSTICC” Mémoire d'Ingénieur CNAM en Informatique, Toulouse (1983)Google Scholar
  20. 20.
    GENNERO M.C. “LOUSTICC simulation software: experimental results” Acts of AAECC-2 (1984), to appear in Springer VerlagGoogle Scholar
  21. 21.
    DE GROOTE H.F., HEINTZ J. “Commutative algebras of minimal rank” Fachbereit Math., J. Goethe Univ. Frankfurt, Germany (1982)Google Scholar
  22. 22.
    HEINTZ J., MORGENSTERN J. “Associative algebra of minimal rank” Acts of AAECC-2 Symp. (1984), to appear in Springer VerlagGoogle Scholar
  23. 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. 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. 25.
    IMAI H. “Two dimensional Fire codes” IEEE Trans. on Inform. Theory, vol IT-19 no6, pp 796–806 (1973)Google Scholar
  26. 26.
    IMAI H. “Multivariate polynomials in coding theory” Acts of AAECC-2 (1984), to appear in Springer VerlagGoogle Scholar
  27. 27.
    IMAI H., ARAKAKI M. “Theory of two dimensional cyclic codes” IECE National Conf. Records, 1415 (1974)Google Scholar
  28. 28.
    INAMURA K., MORII M., “Two classes of finite fields which have no self complementary normal bases” Abstract of Papers IEEE, Brighton (1985), reference no1Google Scholar
  29. 29.
    KANDRI-RODY A., SAUNDERS B.D. “Primality of ideals in polynomial rings” Private communicationGoogle Scholar
  30. 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. 31.
    LAFON J.P. “Algèbre commutative” Tome 2 Edit. HERMANN (1972)Google Scholar
  32. 32.
    LAUBIE M. “Codes idéaux de certaines algèbres modulaires et ramification” to appear in Communications in AlgebraGoogle Scholar
  33. 33.
    LESIEUR L., CROISOT R. “Algèbre noetherienne non commutative” Edit. by GAUTHIERS VILLARS (1963)Google Scholar
  34. 34.
    LIFERMAN J. “Les méthodes rapides de transformation du signal” Edit. by MASSON (1980)Google Scholar
  35. 35.
    MacWILLIAMS F.J., SLOANE N.J.A. “The theory of error correcting codes” North Holland Pub. Cie (1977)Google Scholar
  36. 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. 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. 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. 39.
    POLI A. “Codes dans certaines algèbres modulaires” Thèse d'Etat, Univ. P. Sabatier, Toulouse (1978)Google Scholar
  40. 40.
    POLI A. “Codes stables sous le groupe des automorphismes isométriques de A = IFp [X1, ..., Xn] / (X1p-1, ..., Xnp-1)” C.R.A.S., Paris t 280, série A, pp 1029–1032 (1980)Google Scholar
  41. 41.
    POLI A. “Construction of primitive idempotents for n variable codes” Acts of AAECC-2 (1984), to appear in Springer VerlagGoogle Scholar
  42. 42.
    POLI A. “Multicirculant self dual codes over IFq” Submitted for publication to Communications in Algebra.Google Scholar
  43. 44.
    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. 45.
    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. 46.
    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. 47.
    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. 48.
    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. 49.
    RIGONI C. “Contribution à l'étude des codes correcteurs polynomiaux” Thèse de 3ième cycle, Univ. P. Sabatier, Toulouse (1985)Google Scholar
  49. 50.
    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. 51.
    SAMUEL P., ZARISKI O. “Commutative Algebra” Princeton, Van Nostram (1958)Google Scholar
  51. 52.
    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. 53.
    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. 54.
    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. 55.
    VENTOU M. “Contribution à l'étude des codes correcteurs polynomiaux” Thèse de 3ième cycle, Univ. P. Sabatier, Toulouse (1984)Google Scholar
  55. 56.
    Van Der WAERDEN B.L. “Modern algebra” (Volume II) Frederick Ungar Publishing Co. (New York — 5th printing — 1964)Google Scholar
  56. 57.
    WASAN S.K. “On codes over ZZm”, IEEE Trans. on Inf. Th. pp 117–121 (1982)Google Scholar
  57. 58.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • A. Poli
    • 1
  1. 1.AAECC Lab. Université P. SabatierToulouse cédexFrance

Personalised recommendations