Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics

  • Eiichi Bannai
Part of the NATO ASI Series book series (ASIC, volume 294)


This paper surveys the role of orthogonal polynomials in Algebraic Combinatorics, an area which includes association schemes, coding theory, design theory, various theories of group representation, and so on. The main topics discussed in this paper include the following: The connection between orthogonal polynomials and P -polynomial (or Q -polynomial) association schemes. The classification problem for P - and Q -polynomial association schemes and its connection with Askey-Wilson orthogonal polynomials. Delsarte theory of codes and designs in association schemes. The nonexistence of perfect e-codes and tight t-designs through the study of the zeros of orthogonal polynomials. The possible importance of multi-variable versions of Askey-Wilson polynomials in the future study of general commutative association schemes.


Orthogonal Polynomial Association Scheme Character Table Perfect Code Primitive Idempotent 
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  1. G.E. Andrews and R. Askey (1985): Classical orthogonal polynomials, Orthogonal polynomials and applications (Bar-le-Duc, 1984), 36–69, Springer Lecture Note Series 1171, 1985.MathSciNetGoogle Scholar
  2. R. Askey and J. Wilson (1979): A set of orthogonal polynomials that generalize the Racah coefficients of 6 - jsymbols, SIAM J. Math. Anal. 10 (1979), 1008 – 1016.MathSciNetzbMATHGoogle Scholar
  3. R. Askey and J. Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54(1985), No. 319.Google Scholar
  4. E. Bannai (1977a): On perfect codes in the Hamming scheme H(n,q) with qarbitrary, J. of Combinatorial Theory (A), 23 (1977), 52 – 67.MathSciNetzbMATHCrossRefGoogle Scholar
  5. E. Bannai (1977b): On tight designs, Quart. J. Math. (Oxford), 28 (1977), 433 – 448.MathSciNetzbMATHCrossRefGoogle Scholar
  6. E. Bannai (1988): On extremal finite sets in the sphere and other metric spaces; in Algebraic, extremal and metric combinatorics, London Math. Lecture Note Series, No. 131, 1988, 13 – 38.MathSciNetGoogle Scholar
  7. E. Bannai (to appear): Character tables of commutative association schemes, to be published in Proceedings of the conference “Finite Buildings and Related Geometries” held in Pingree Park, Colorado, July, 1988, Oxford Univ. Press.Google Scholar
  8. E. Bannai, A. Blokhuis, P. Delsarte and J. J. Seidel: An addition formula for hyperbolic space, J. of Combinatorial Theory (A), 36 (1984), 332 – 341.MathSciNetzbMATHCrossRefGoogle Scholar
  9. E. Bannai and R. M. Damerell (1979): Tight spherical designs I, J. Math. Soc. Japan, 31 (1979), 199 – 207.MathSciNetzbMATHCrossRefGoogle Scholar
  10. E. Bannai and R. M. Damerell (1980): Tight spherical designs II, J. of London Math. Soc., 21 (1980), 13 – 30.MathSciNetzbMATHCrossRefGoogle Scholar
  11. E. Bannai, S. Hao and S.Y. Song (to appear): Character tables of association schemes of finite orthogonal groups acting on the nonisotripic points, to appear in J. of Combinatorial Theory (A).Google Scholar
  12. E. Bannai and S.G. Hoggar (1989): Tight designs and squarefree integers, Europ. J. of Combinatorics, 10 (1989), 113 – 135.MathSciNetzbMATHGoogle Scholar
  13. E. Bannai and T. Ito (1984): Algebraic Combinatorics I, Association Schemes, Benja- min/Cummings, Menlo Park. California, 1984.zbMATHGoogle Scholar
  14. E. Bannai and T. Ito (1986): Current research on algebraic combinatorics, Graphs and Combinatorics, 2 (1986), 287 – 308.MathSciNetzbMATHCrossRefGoogle Scholar
  15. E. Bannai and T. Ito (1987): The study of distance-regular graphs from the algebraic (i.e., character theoretical) viewpoint, Proc. Symp. in Pure Math. (AMS) 47 (1987), 343 – 349.Google Scholar
  16. E. Bannai, N. Kawanaka and S.Y. Song (to appear): The character table of the Heckealgebra H(GL 2n (F q ), Sp 2n (F q )), to appear in J. of Algebra.Google Scholar
  17. E. Bannai and S.Y. Song (1989): The character tables of Paige’s simple Moufang loops and their relationship to the character tables of PSL(2,q), Proc. London Math. Soc. 58 (1989), 209 – 236.MathSciNetzbMATHGoogle Scholar
  18. M. R. Best (1982): A contribution to the non-existence of perfect codes Ph. D. thesis, Amsterdam, 1982.Google Scholar
  19. N. L. Biggs (1973): Perfect codes in graphs, J. of Combinatorial Theory (B), 15 (1973), 289 – 296.MathSciNetzbMATHCrossRefGoogle Scholar
  20. N. L. Biggs (1974): Algebraic Graph Theory, Cambridge Univ. Press, 1974.Google Scholar
  21. N. L. Biggs, A. G. Boshier and J. Shaw-Taylor (1986): Cubic distance-regular graphs, J. London Math. Soc. 33 (1986), 385 – 394.MathSciNetzbMATHCrossRefGoogle Scholar
  22. A. E. Brouwer, A. Cohen and A. Neumaier (1989): Distance-regular graphs, Springer, 1989.Google Scholar
  23. L. Chihara (1987): On the zeros of the Askey-Wilson polynomials with applications to coding theory, SIAM J. Math. Anal. 18 (1987), 183 – 207.Google Scholar
  24. L. Chihara and D. Stanton (1986): Association schemes and quadratic transformations for orthogonal polynomials, Graphs and Combinatorics, 2 (1986), 101 – 112.MathSciNetzbMATHCrossRefGoogle Scholar
  25. L. Chihara and D. Stanton (to appear): Zeros of generalized Krawtchouk polynomials, to appear in J. Approx. Theory.Google Scholar
  26. R. Clayton (to appear): Perfect multiple coverings in metric spaces, to appear in IMA Proceedings.Google Scholar
  27. J. H. Conway and N.J.A. Sloane (1988): Sphere Packings, Lattices and Groups, Springer, 1988.zbMATHGoogle Scholar
  28. R. M. Damerell (1981): Distance-transitive and distance-regular digraphs, J. of Combinatorial Theory (B), 31 (1981), 46 – 53.MathSciNetzbMATHCrossRefGoogle Scholar
  29. P. Delsarte (1973): An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. No 10, 1973.Google Scholar
  30. P. Delsarte (1974): The association schemes of coding thoery; in Combinatorics (M. Hall, Jr. and J. H. van Lint, eds.), Mathematical Center Tracts 55, Amsterdam, 1974, 139–157.Google Scholar
  31. P. Delsarte (1978): Hahn polynomials, discrete harmonics, and t-designs, SIAM J. Appl. Math. 34 (1978), 157 – 166.MathSciNetGoogle Scholar
  32. P. Delsarte, J. M. Goethals and J. J. Seidel (1975): Bounds for systems of lines, and Jacobi polynomials, Philips Res. Repts 30(1975), 95*–105* (Bouwkamp volume).Google Scholar
  33. P. Delsarte, J. M. Goethals and J. J. Seidel (1977): Spherical codes and designs, Geom. Dedicata, 6 (1977), 363 – 388.MathSciNetzbMATHCrossRefGoogle Scholar
  34. P. Diaconis (1988): Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Lecture Note-Monograph Series, No. 11.Google Scholar
  35. C. F. Dunkl (1976): A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25 (1975), 335 – 358.MathSciNetzbMATHGoogle Scholar
  36. C. F. Dunkl (1977): An addition theorem for q-Hahn polynomials, Monats. Math. 85 (1977), 5 – 37.MathSciNetGoogle Scholar
  37. C. F. Dunkl (1978): An addition theorem for Hahn polynomials: The spherical functions. SIAM J. Math. Anal. 9 (1978), 627 – 637.MathSciNetzbMATHGoogle Scholar
  38. C. F. Dunkl (1979): Discrete quadrature and bounds on t-designs, Mich. Math. J. 26135 (1979), 81 – 102.MathSciNetGoogle Scholar
  39. C. F. Dunkl (1980): Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type, SIAM J. Alg. Disc. Math. 1 (1980), 137 – 151.MathSciNetzbMATHCrossRefGoogle Scholar
  40. C. F. Dunkl (1988): Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33 – 60.MathSciNetzbMATHCrossRefGoogle Scholar
  41. H. Enomoto, N. Ito and R. Noda (1979): Tight 4-desings, Osaka J. Math. 16 (1979), 39 – 43.MathSciNetzbMATHGoogle Scholar
  42. J. M. Goethals and H. C. A. van Tilborg (1975), Uniformly packed codes. Philips Res. Rep. 30 (1975), 9 – 36.MathSciNetzbMATHGoogle Scholar
  43. L. Habsieger and D. Stanton (to appear): More zeros of Krawtchouk polynomials, to appear.Google Scholar
  44. J. Hemmeter (to appear): A new family of distance-regular graphs, to appear.Google Scholar
  45. S. G. Hoggar (1982): t-designs in projectives spaces, Europ. J. Combinatorics 3 (1982), 233 – 254.zbMATHGoogle Scholar
  46. S. G. Hoggar (1989): Tight 4- and 5-designs in projectives spaces, Graphs and Combinatorics, 5 (1989), 87 - 94.MathSciNetzbMATHCrossRefGoogle Scholar
  47. S. G. Hoggar (to appear): t -designs in Delsarte spaces, to appear in IMA Proceedings.Google Scholar
  48. Y. Hong (1984): On the non existence of unknown perfect 6- and 8-codes in Hamming schemes H(n,q) with q arbitrary, Osaka J. Math., 21 (1984), 687 – 700.zbMATHGoogle Scholar
  49. Y. Hong(1987): On the nonexistence of nontrivial perfect e-codes and tight 2e-designs in Hamming schemes H(n,q) with e≥ 3 and q≥ 3, Graphs and Combinatorics, 2 (1986), 145 – 164.MathSciNetCrossRefGoogle Scholar
  50. A. A. Ivanov (1983): Bounding the diameter of distance-regular graphs, Soviet Math. Dokl. 38 (1983), 149 – 152.Google Scholar
  51. A. A. Ivanov, M. E. Muzichuk and V. A. Ustimenko (to appear), On a new family of (P and Q)-polynomial schemes, to appear in Europ. J. Combinatorics.Google Scholar
  52. A. A. Ivanov and S. V. Shpectorov (to appear): A characterization of the association schemes of Hermitian forms, to appear.Google Scholar
  53. A. A. Klyachko (1984): Models for the complex representations of the group GL(n,q), Math. USSR Sbornik, 48 (1987), 365 – 379.MathSciNetzbMATHCrossRefGoogle Scholar
  54. T. H. Koornwinder (1973): The addition formula for Jacobi polynomials and spherical harmonics, SIAM J. Appl. Math. 25 (1973), 236 – 246.MathSciNetzbMATHGoogle Scholar
  55. T. H. Koornwinder (1975): Two variable analogues of the classical orthogonal polynomials; in R. Askey (ed.) Theory and Application of Special Functions, Academic Press, N.Y., 1975, pp. 435 – 495.Google Scholar
  56. D. Leonard (1984): Parameters of association schemes that are both P- and Q- polynomial, J. of Combinatorial Theory (A) 36 (1984), 355 – 363.MathSciNetzbMATHCrossRefGoogle Scholar
  57. D. Leonard (1982): Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal. 13 (1982), 656 – 663.MathSciNetzbMATHGoogle Scholar
  58. D. Leonard (to appear): Non-symmetric, metric, cometric association schemes, to appear in J. of Combinatorial Theory (B).Google Scholar
  59. D. Leonard (to appear): Non-symmetric, metric, cometric association schemes are self-dual, to appear.Google Scholar
  60. R. A. Liebler and R. A. Mena (1988): Certain distance-regular digraphs and related rings of characteristic 4, J. of Combinatorial Theory (A), 47 (1988), 111 – 123.MathSciNetzbMATHCrossRefGoogle Scholar
  61. S. P. Lloyd (1957): Binary block ciding, Bell System Tech. J., 36 (1957), 517 – 535.MathSciNetGoogle Scholar
  62. I. G. Macdonald (1987): Commuting differential operators and zonal spherical functions; in “Algebraic Groups, Utrecht 1986”, Lecture Notes in Mathematics No. 1271, Springer, 1987.Google Scholar
  63. I. G. Macdonald (Chapter VI of the second edition of 1979): Symmetric Functions and Hall polynomials, Oxford Univ. Press, Oxford, 1979. (2nd edition, to appear).Google Scholar
  64. J. F. Mac Williams and N. J. A. Sloane (1977): The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1977.zbMATHGoogle Scholar
  65. R. J. McEliece (1977): The theory of infomation and coding, in Encyclopedia of Math, and its Application, Addison-Wesley, 1977.Google Scholar
  66. A. Munemasa (to appear): On nonsymmetric P- and Q -polynomial association schemes, to appear.Google Scholar
  67. A. Neumaier (1981): Combinatorial configurations in terms of distances, T.H.E. Memorandum 81 1981, No. 09, p. 98, Eindhoven Univ. Tech.Google Scholar
  68. C.L. Peterson (1977): On tight 6 -designs, Osaka J. Math., 14 (1977), 417 – 435.zbMATHGoogle Scholar
  69. D.K. Ray-Chaudhuri and R. M. Wilson (1975): On t-designs, Osaka J. Math. 12 (1975), 737 – 744.MathSciNetzbMATHGoogle Scholar
  70. H.F. H. Reuvers (1977): Some nonexistence theorems for perfect codes over arbitrary alphabets, Ph. D. thesis, Tech. Univ. Eindhoven, 1977.Google Scholar
  71. C. Roos (1982): Some remarks on perfect subsets in distance-regular graphs, Delfte Progress Report, 7 (1982), 90 – 94.MathSciNetzbMATHGoogle Scholar
  72. I. Schur (1930): Gleichungen ohne Affekt; Sitz. der Preuss Acad. Wiss. (1930), 443-449. Also in Gesammelte Abhandlungen, vol. 3, 191 – 197.Google Scholar
  73. P. Seymour and T. Zaslavsky (1984): Averaging sets: a generalization of mean values and spherical designs, Advs. in Math. 52 (1984), 213 – 240.Google Scholar
  74. N. J. A. Sloane: An introduction to association schemes and coding theory; in Theory and Application of Special Functions (R. Askey, ed.), Academic Press, N. Y. 1975, 225 – 260.Google Scholar
  75. D. Stanton (1980): Some q-Krawtchouk polynomials in Chevalley groups, Amer. J. Math. 102 (1980), 625 – 662.zbMATHCrossRefGoogle Scholar
  76. D. Stanton (1981): Three addition theorems for some q-Krawtchouk polynomials, Geom. Dedicata 10 (1981), 403 – 425.zbMATHCrossRefGoogle Scholar
  77. D. Stanton (1984): Orthogonal polynomials and Chevalley groups; in Special Functions: Group Theoretical Aspects and Applications (R. A. Askey et al. eds) Reidel, Boston, 1984, pp. 87 – 125.Google Scholar
  78. D. Stanton (1986): t-designs in classical association schemes, Graphs and Combinatorics, 2 (1986), 283 – 286.MathSciNetCrossRefGoogle Scholar
  79. H. Tarnanen, M. J. Aaltonen and J. M. Goethals (1985): On the nonbinary Johnson Scheme, Europ. J. Combinatorics (1985), 279 – 285.Google Scholar
  80. P. Terwilliger (1987): A characterization of P- and Q-polynomial association schemes, J. of Combinatorial Theory (A), 45 (1987), 8 – 26.MathSciNetzbMATHCrossRefGoogle Scholar
  81. P. Terwilliger (to appear): The incidence algebra of a uniform poset, to appear.Google Scholar
  82. A. Tietavainen (1973): On the nonexistence of perfect codes over finite fields, SIAM J. Appl. Math. 24 (1973), 88 – 96.MathSciNetGoogle Scholar
  83. A. Tietavainen (1977): Nonexistence of nontrivial perfect codes in case \(q = p_1^sp_2^t,e \geqslant 3\),Google Scholar
  84. Discrete Math. 17(1977), 199–205.Google Scholar
  85. L. Vretare (1976): Elementary spherical functions on symmetric spaces. Math. Scand. 39 (1976), 343 – 358.MathSciNetGoogle Scholar
  86. L. Vretare (1984): Formula for elementary spherical functions and generalized Jacobi polynomials. SIAM J. Math. Anal. 15 (1984), 805 – 833.MathSciNetzbMATHGoogle Scholar

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© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Eiichi Bannai
    • 1
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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