Metric coset schemes revisited

Paul Camion; Bernard Courteau; André Montpetit

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 3, page 829-859
  • ISSN: 0373-0956

Abstract

top
An Abelian scheme corresponds to a special instance of what is usually named a Schur-ring. After the needed results have been quoted on additive codes in Abelian schemes and their duals, coset configurations, coset schemes, metric schemes and distance regular graphs, partition designs and completely regular codes, we give alternative proofs of some of those results. In this way we obtain a construction of metric Abelian schemes and an algorithm to compute their intersection matrices.

How to cite

top

Camion, Paul, Courteau, Bernard, and Montpetit, André. "Metric coset schemes revisited." Annales de l'institut Fourier 49.3 (1999): 829-859. <http://eudml.org/doc/75366>.

@article{Camion1999,
abstract = {An Abelian scheme corresponds to a special instance of what is usually named a Schur-ring. After the needed results have been quoted on additive codes in Abelian schemes and their duals, coset configurations, coset schemes, metric schemes and distance regular graphs, partition designs and completely regular codes, we give alternative proofs of some of those results. In this way we obtain a construction of metric Abelian schemes and an algorithm to compute their intersection matrices.},
author = {Camion, Paul, Courteau, Bernard, Montpetit, André},
journal = {Annales de l'institut Fourier},
keywords = {association scheme; Schur ring; abelian scheme; additive code; outer distribution matrix; restricted distribution matrix; coset configuration; coset scheme; distance-regular graph; metric coset scheme; distance partition; partition design; covering radius; completely regular code; projective code},
language = {eng},
number = {3},
pages = {829-859},
publisher = {Association des Annales de l'Institut Fourier},
title = {Metric coset schemes revisited},
url = {http://eudml.org/doc/75366},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Camion, Paul
AU - Courteau, Bernard
AU - Montpetit, André
TI - Metric coset schemes revisited
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 3
SP - 829
EP - 859
AB - An Abelian scheme corresponds to a special instance of what is usually named a Schur-ring. After the needed results have been quoted on additive codes in Abelian schemes and their duals, coset configurations, coset schemes, metric schemes and distance regular graphs, partition designs and completely regular codes, we give alternative proofs of some of those results. In this way we obtain a construction of metric Abelian schemes and an algorithm to compute their intersection matrices.
LA - eng
KW - association scheme; Schur ring; abelian scheme; additive code; outer distribution matrix; restricted distribution matrix; coset configuration; coset scheme; distance-regular graph; metric coset scheme; distance partition; partition design; covering radius; completely regular code; projective code
UR - http://eudml.org/doc/75366
ER -

References

top
  1. [1] E. BANNAI, T. ITO, Algebraic Combinatorics, The Benjamin/Cummings Publishing Company, Inc., 1984. Zbl0555.05019MR87m:05001
  2. [2] T. BIER, Hyperplane Codes, Graphs and Combinatorics, 1 (1985), 207-212. Zbl0606.94006MR89d:94040
  3. [3] A.E. BROUWER, A.M. COHEN and A. NEUMAIER, Distance-Regular Graphs, Springer-Verlag Berlin Eidelberg, 1984. Zbl0747.05073
  4. [4] A.R. CALDERBANK and J.M. GOETHALS, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. Zbl0546.94016MR86e:94026
  5. [5] A.R. CALDERBANK and J.M. GOETHALS, On a pair of dual subschemes of the Hamming scheme Hn(q), European J. Combin., 6 (1985), 133-147.z. Zbl0579.05021MR87d:94045
  6. [6] P. CAMION, Linear codes with given automorphism groups, Discrete Mathematics, 3 (1973), 33-45. Zbl0253.94005MR47 #10134
  7. [7] P. CAMION, Codes and Association schemes, Chap. 18 in Handbook of Coding Theory, edited by V.S Pless and W.C. Huffman, Elsevier Amsterdam, 1998. Zbl0978.94048
  8. [8] P. CAMION, B. COURTEAU and P. DELSARTE, On repartition designs in Hamming spaces, Inria Report, 626 (1987). Zbl0756.05036
  9. [9] P. CAMION, B. COURTEAU and P. DELSARTE, On repartition designs in Hamming spaces, Applicable Algebra in Engin. Comm. and Comput., 2 (1992), 147-162. Zbl0756.05036MR96b:94028
  10. [10] P. CAMION, B. COURTEAU, G. FOURNIER and S.V. KANETKAR, Weight distributions of translates of linear codes and genralized Pless identities, Journal of Information & Optimization Sciences, 8 (1987), N01, 1-23. Zbl0633.94018MR88f:94033
  11. [11] P. CAMION, B. COURTEAU and A. MONTPETIT, Weight distribution of cosets of 2-error-correcting binary BCH codes of length 15, 63 and 255 IEEE Trans. Inf. Theory, 38 (1992), No 4, 1353-1357. Zbl0775.94110MR93b:94024
  12. [12] B. COURTEAU, A. MONTPETIT, Dual distances of completely regular codes, Discrete Mathematics, 89 (1991), 7-15. Zbl0725.94009MR92f:94020
  13. [13] P. DELSARTE, An Algebraic Approach to Association Schemes in Coding, Philips Res. Repts Suppl., 10 (1973). Zbl1075.05606
  14. [14] P. DELSARTE, Four fundamental parameters of a code and their combinatorial significance, Inform. Control, 23 (1973), 407-438. Zbl0274.94010MR48 #13453
  15. [15] P. DELSARTE, Bilinear forms over a finite field with applications to coding theory, J. of Combinatorial Theory (A), 25 (1978), 226-241. Zbl0397.94012MR80a:94040
  16. [16] C.D. GODSIL, Equitable partitions, Bolayai society mathematical studies, Combinatorics Paul Erdös is eighty (Vol. 1) Keszthely (Hungary), 1992, 173-192. Zbl0795.05011
  17. [17] C.D. GODSIL, Algebraic Combinatorics, Chapman and Hall, New York, London, 1993. Zbl0784.05001MR94e:05002
  18. [18] C.D. GODSIL and W.J. MARTIN, Quotients of Association Schemes, J. of Combinatorial Theory, Series A, 69 (1995), 185-199. Zbl0813.05070MR95m:05249
  19. [19] J.-M. GOETHALS, Association Schemes, in Algebraic Coding Theory and Applications, edited by G.Longo, CISM courses and Lectures N0. 258, Springer-Verlag Wien, New York, 1979. Zbl0425.94013
  20. [20] D.G. HIGMAN, Coherent configurations, Geom. Dedicata, 4 (1975), 1-32. Zbl0333.05010MR53 #2719
  21. [21] P. HAMMOND and D.H. SMITH, An analog of Lloyd's Theorem for Completely Regular Codes, Proc. 5th British Combinatorial Conf., 1975, 261-267. Zbl0327.94013
  22. [22] D.A. LEONARD, Parameters of Association Schemes that are both P- and Q- Polynomial, J. of Combinatorial Theory, Series A, 36, No 3 (1984), 355-363. Zbl0533.05016MR86d:05014
  23. [23] D.A. LEONARD, Directed Distance-regular Graphs with the Q-Polynomial Property, J. of Combinatorial Theory, Series A, 48, No 2 (1990), 191-196. Zbl0723.05065MR91h:05126
  24. [24] D.A. LEONARD, Non-symmetric, Metric, Cometric Association Schemes are Self-dual, J. of Combinatorial Theory, Series A, 51, No 2 (1991), 244-247. Zbl0754.05076
  25. [25] D.A. LEONARD, The girth of a Directed Distance-regular Graph, J. of Combinatorial Theory, Series A, 58, No 1 (1993), 34-39. Zbl0733.05044MR94e:05269
  26. [26] F.J. MACWILLIAMS, A theorem on the distribution of weights in a systematic code, Bell Syst. Tech. J., 42 (1963), 79-94. 
  27. [27] F.J. MACWILLIAMS and N.J.A. SLOANE, The Theory of Error-Correcting Codes, North-Holland, 1977. Zbl0369.94008
  28. [28] A. MONTPETIT, Codes dans les graphes réguliers, Thèse, Faculté des Sciences, Université de Scherbrooke, 1987. Zbl0632.94009
  29. [29] A. MONTPETIT, Codes et partitions cohérentes, Annales des Sciences Mathématiques du Québec, 14, No 2 (1990), 183-191. Zbl0741.94020MR92b:94026
  30. [30] H.M. MULDER, The Interval Function of a Graph, Mathematical Center Tracts 132, Mathematisch Centrum, Amsterdam (1980). Zbl0446.05039MR82h:05045
  31. [31] A. NEUMAIER, Classification of Graphs by regularity, J. Comb. Theory, Series B, 30 (1981), 318-331. Zbl0429.05058MR84e:05089
  32. [32] A. NEUMAIER, Completely regular codes, Discrete Mathematics, 106/107 (1992), 353-360. Zbl0754.94010MR93g:94028
  33. [33] D.M. CVECTOVIĆ, M. DOOB and H. SACHS, Spectra of Graphs : Theory and Applications, Academic Press, New York, 1979. 
  34. [34] N.V. SEMAKOV, V.A. ZINOVIEV and C.V. ZAITSEV, Uniformly packed codes, Probl. Peredach. Inform., 7 (1971), N0 1, 38-50. Zbl0306.94009
  35. [35] I. SCHUR, Zur Theorie der einfach transitiven Permutationsgruppen, S. B. Preuss. Akad. Wiss., Phys.-Math. Kl, 1933, 598-623. Zbl0007.14903JFM59.0151.01
  36. [36] I. SCHUR, Gesammelte Abhandlungen I, II, III, Springer, 1973. 
  37. [37] A.J. SCHWENK, Computing the Characteristic Polynomial of a Graph, Graphs and Combinatorics, Lecture Notes in Mathematics, 406 (1974), Springer, Berlin 153-162. Zbl0308.05121MR52 #7972
  38. [38] P. SOLÉ, A Lloyd theorem in weakly metric association schemes, Europ. J. Combinatorics, 89 (1989), 189-196. Zbl0722.05061MR90b:05029
  39. [39] P. SOLÉ, Completely regular codes and completely transitive codes, Discrete Mathematics, 81 (1990), 193-201. Zbl0696.94021MR91d:94020
  40. [40] P.M. WEICHSEL, On the Distance-Regularity in Graphs, J. Comb. Theory, Series B., 32 (1982), 156-161. Zbl0477.05047MR83g:05051

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.