Metric coset schemes revisited
Annales de l'Institut Fourier, Volume 49 (1999) no. 3, p. 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.

@article{AIF_1999__49_3_829_0,
     author = {Camion, Paul and Courteau, Bernard and Montpetit, Andr\'e},
     title = {Metric coset schemes revisited},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {3},
     year = {1999},
     pages = {829-859},
     doi = {10.5802/aif.1695},
     zbl = {0917.05085},
     mrnumber = {2000g:05150},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1999__49_3_829_0}
}
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/item/AIF_1999__49_3_829_0/

[1] E. Bannai, T. Ito, Algebraic Combinatorics, The Benjamin/Cummings Publishing Company, Inc., 1984. | MR 87m:05001 | Zbl 0555.05019

[2] T. Bier, Hyperplane Codes, Graphs and Combinatorics, 1 (1985), 207-212. | MR 89d:94040 | Zbl 0606.94006

[3] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag Berlin Eidelberg, 1984. | Zbl 0747.05073

[4] A.R. Calderbank and J.M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. | MR 86e:94026 | Zbl 0546.94016

[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 87d:94045 | Zbl 0579.05021

[6] P. Camion, Linear codes with given automorphism groups, Discrete Mathematics, 3 (1973), 33-45. | MR 47 #10134 | Zbl 0253.94005

[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 0978.94048

[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 96b:94028 | Zbl 0756.05036

[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 88f:94033 | Zbl 0633.94018

[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 93b:94024 | Zbl 0775.94110

[12] B. Courteau, A. Montpetit, Dual distances of completely regular codes, Discrete Mathematics, 89 (1991), 7-15. | MR 92f:94020 | Zbl 0725.94009

[13] P. Delsarte, An Algebraic Approach to Association Schemes in Coding, Philips Res. Repts Suppl., 10 (1973). | Zbl 1075.05606

[14] P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Inform. Control, 23 (1973), 407-438. | MR 48 #13453 | Zbl 0274.94010

[15] P. Delsarte, Bilinear forms over a finite field with applications to coding theory, J. of Combinatorial Theory (A), 25 (1978), 226-241. | MR 80a:94040 | Zbl 0397.94012

[16] C.D. Godsil, Equitable partitions, Bolayai society mathematical studies, Combinatorics Paul Erdös is eighty (Vol. 1) Keszthely (Hungary), 1992, 173-192. | Zbl 0795.05011

[17] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, London, 1993. | MR 94e:05002 | Zbl 0784.05001

[18] C.D. Godsil and W.J. Martin, Quotients of Association Schemes, J. of Combinatorial Theory, Series A, 69 (1995), 185-199. | MR 95m:05249 | Zbl 0813.05070

[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 0425.94013

[20] D.G. Higman, Coherent configurations, Geom. Dedicata, 4 (1975), 1-32. | MR 53 #2719 | Zbl 0333.05010

[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 0327.94013

[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 86d:05014 | Zbl 0533.05016

[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 91h:05126 | Zbl 0723.05065

[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 0754.05076

[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 94e:05269 | Zbl 0733.05044

[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 0369.94008

[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 92b:94026 | Zbl 0741.94020

[30] H.M. Mulder, The Interval Function of a Graph, Mathematical Center Tracts 132, Mathematisch Centrum, Amsterdam (1980). | MR 82h:05045 | Zbl 0446.05039

[31] A. Neumaier, Classification of Graphs by regularity, J. Comb. Theory, Series B, 30 (1981), 318-331. | MR 84e:05089 | Zbl 0457.05062

[32] A. Neumaier, Completely regular codes, Discrete Mathematics, 106/107 (1992), 353-360. | MR 93g:94028 | Zbl 0754.94010

[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 0306.94009

[35] I. Schur, Zur Theorie der einfach transitiven Permutationsgruppen, S. B. Preuss. Akad. Wiss., Phys.-Math. Kl, 1933, 598-623. | JFM 59.0151.01 | Zbl 0007.14903

[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 52 #7972 | Zbl 0308.05121

[38] P. Solé, A Lloyd theorem in weakly metric association schemes, Europ. J. Combinatorics, 89 (1989), 189-196. | MR 90b:05029 | Zbl 0722.05061

[39] P. Solé, Completely regular codes and completely transitive codes, Discrete Mathematics, 81 (1990), 193-201. | MR 91d:94020 | Zbl 0696.94021

[40] P.M. Weichsel, On the Distance-Regularity in Graphs, J. Comb. Theory, Series B., 32 (1982), 156-161. | MR 83g:05051 | Zbl 0487.05043