Metric coset schemes revisited
Annales de l'Institut Fourier, Volume 49 (1999) no. 3, pp. 829-859.

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.

Un schéma abélien correspond à un cas particulier de ce qui est habituellement nommé un anneau de Schur. Après un rappel des résultats dont on a besoin sur les codes additifs dans un schéma abélien, et leurs duaux, les schémas de translatés, les schémas métriques et les graphes distance-réguliers, les partitions cohérentes et les graphes complètement réguliers, nous donnons d’autres preuves de certains de ces résultats. De cette manière, nous obtenons une construction de schémas métriques abéliens et un algorithme pour calculer leurs matrices d’intersection.

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Camion, Paul; Courteau, Bernard; Montpetit, André. Metric coset schemes revisited. Annales de l'Institut Fourier, Volume 49 (1999) no. 3, pp. 829-859. doi : 10.5802/aif.1695. http://www.numdam.org/articles/10.5802/aif.1695/

[1] E. Bannai, T. Ito, Algebraic Combinatorics, The Benjamin/Cummings Publishing Company, Inc., 1984. | MR | Zbl

[2] T. Bier, Hyperplane Codes, Graphs and Combinatorics, 1 (1985), 207-212. | MR | Zbl

[3] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag Berlin Eidelberg, 1984. | Zbl

[4] A.R. Calderbank and J.M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. | MR | Zbl

[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. | MR | Zbl

[6] P. Camion, Linear codes with given automorphism groups, Discrete Mathematics, 3 (1973), 33-45. | MR | Zbl

[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. | Zbl

[8] P. Camion, B. Courteau and P. Delsarte, On repartition designs in Hamming spaces, Inria Report, 626 (1987).

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] B. Courteau, A. Montpetit, Dual distances of completely regular codes, Discrete Mathematics, 89 (1991), 7-15. | MR | Zbl

[13] P. Delsarte, An Algebraic Approach to Association Schemes in Coding, Philips Res. Repts Suppl., 10 (1973). | Zbl

[14] P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Inform. Control, 23 (1973), 407-438. | MR | Zbl

[15] P. Delsarte, Bilinear forms over a finite field with applications to coding theory, J. of Combinatorial Theory (A), 25 (1978), 226-241. | MR | Zbl

[16] C.D. Godsil, Equitable partitions, Bolayai society mathematical studies, Combinatorics Paul Erdös is eighty (Vol. 1) Keszthely (Hungary), 1992, 173-192. | Zbl

[17] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, London, 1993. | MR | Zbl

[18] C.D. Godsil and W.J. Martin, Quotients of Association Schemes, J. of Combinatorial Theory, Series A, 69 (1995), 185-199. | MR | Zbl

[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. | Zbl

[20] D.G. Higman, Coherent configurations, Geom. Dedicata, 4 (1975), 1-32. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[25] D.A. Leonard, The girth of a Directed Distance-regular Graph, J. of Combinatorial Theory, Series A, 58, No 1 (1993), 34-39. | MR | Zbl

[26] F.J. Macwilliams, A theorem on the distribution of weights in a systematic code, Bell Syst. Tech. J., 42 (1963), 79-94.

[27] F.J. Macwilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977. | Zbl

[28] A. Montpetit, Codes dans les graphes réguliers, Thèse, Faculté des Sciences, Université de Scherbrooke, 1987.

[29] A. Montpetit, Codes et partitions cohérentes, Annales des Sciences Mathématiques du Québec, 14, No 2 (1990), 183-191. | MR | Zbl

[30] H.M. Mulder, The Interval Function of a Graph, Mathematical Center Tracts 132, Mathematisch Centrum, Amsterdam (1980). | MR | Zbl

[31] A. Neumaier, Classification of Graphs by regularity, J. Comb. Theory, Series B, 30 (1981), 318-331. | MR | Zbl

[32] A. Neumaier, Completely regular codes, Discrete Mathematics, 106/107 (1992), 353-360. | MR | Zbl

[33] D.M. Cvectović, M. Doob and H. Sachs, Spectra of Graphs : Theory and Applications, Academic Press, New York, 1979.

[34] N.V. Semakov, V.A. Zinoviev and C.V. Zaitsev, Uniformly packed codes, Probl. Peredach. Inform., 7 (1971), N0 1, 38-50. | Zbl

[35] I. Schur, Zur Theorie der einfach transitiven Permutationsgruppen, S. B. Preuss. Akad. Wiss., Phys.-Math. Kl, 1933, 598-623. | JFM | Zbl

[36] I. Schur, Gesammelte Abhandlungen I, II, III, Springer, 1973.

[37] A.J. Schwenk, Computing the Characteristic Polynomial of a Graph, Graphs and Combinatorics, Lecture Notes in Mathematics, 406 (1974), Springer, Berlin 153-162. | MR | Zbl

[38] P. Solé, A Lloyd theorem in weakly metric association schemes, Europ. J. Combinatorics, 89 (1989), 189-196. | MR | Zbl

[39] P. Solé, Completely regular codes and completely transitive codes, Discrete Mathematics, 81 (1990), 193-201. | MR | Zbl

[40] P.M. Weichsel, On the Distance-Regularity in Graphs, J. Comb. Theory, Series B., 32 (1982), 156-161. | MR | Zbl

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