Skip to main content
SpringerLink
Log in

Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics

  • Chapter
Orthogonal Polynomials

Part of the book series: NATO ASI Series ((ASIC,volume 294))

Abstract

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.

This material is based upon research supported by the National Science Foundation under grant number DMS-8703075.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • R. Askey and J. Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54(1985), No. 319.

    Google Scholar 

  • E. Bannai (1977a): On perfect codes in the Hamming scheme H(n,q) with qarbitrary, J. of Combinatorial Theory (A), 23 (1977), 52 – 67.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Bannai (1977b): On tight designs, Quart. J. Math. (Oxford), 28 (1977), 433 – 448.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Bannai and R. M. Damerell (1979): Tight spherical designs I, J. Math. Soc. Japan, 31 (1979), 199 – 207.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Bannai and R. M. Damerell (1980): Tight spherical designs II, J. of London Math. Soc., 21 (1980), 13 – 30.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • E. Bannai and S.G. Hoggar (1989): Tight designs and squarefree integers, Europ. J. of Combinatorics, 10 (1989), 113 – 135.

    MathSciNet  MATH  Google Scholar 

  • E. Bannai and T. Ito (1984): Algebraic Combinatorics I, Association Schemes, Benja- min/Cummings, Menlo Park. California, 1984.

    MATH  Google Scholar 

  • E. Bannai and T. Ito (1986): Current research on algebraic combinatorics, Graphs and Combinatorics, 2 (1986), 287 – 308.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • M. R. Best (1982): A contribution to the non-existence of perfect codes Ph. D. thesis, Amsterdam, 1982.

    Google Scholar 

  • N. L. Biggs (1973): Perfect codes in graphs, J. of Combinatorial Theory (B), 15 (1973), 289 – 296.

    Article  MathSciNet  MATH  Google Scholar 

  • N. L. Biggs (1974): Algebraic Graph Theory, Cambridge Univ. Press, 1974.

    Google Scholar 

  • N. L. Biggs, A. G. Boshier and J. Shaw-Taylor (1986): Cubic distance-regular graphs, J. London Math. Soc. 33 (1986), 385 – 394.

    Article  MathSciNet  MATH  Google Scholar 

  • A. E. Brouwer, A. Cohen and A. Neumaier (1989): Distance-regular graphs, Springer, 1989.

    Google Scholar 

  • 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 

  • L. Chihara and D. Stanton (1986): Association schemes and quadratic transformations for orthogonal polynomials, Graphs and Combinatorics, 2 (1986), 101 – 112.

    Article  MathSciNet  MATH  Google Scholar 

  • L. Chihara and D. Stanton (to appear): Zeros of generalized Krawtchouk polynomials, to appear in J. Approx. Theory.

    Google Scholar 

  • R. Clayton (to appear): Perfect multiple coverings in metric spaces, to appear in IMA Proceedings.

    Google Scholar 

  • J. H. Conway and N.J.A. Sloane (1988): Sphere Packings, Lattices and Groups, Springer, 1988.

    MATH  Google Scholar 

  • R. M. Damerell (1981): Distance-transitive and distance-regular digraphs, J. of Combinatorial Theory (B), 31 (1981), 46 – 53.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Delsarte (1973): An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. No 10, 1973.

    Google Scholar 

  • 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 

  • P. Delsarte (1978): Hahn polynomials, discrete harmonics, and t-designs, SIAM J. Appl. Math. 34 (1978), 157 – 166.

    MathSciNet  Google Scholar 

  • 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 

  • P. Delsarte, J. M. Goethals and J. J. Seidel (1977): Spherical codes and designs, Geom. Dedicata, 6 (1977), 363 – 388.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Diaconis (1988): Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Lecture Note-Monograph Series, No. 11.

    Google Scholar 

  • C. F. Dunkl (1976): A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25 (1975), 335 – 358.

    MathSciNet  MATH  Google Scholar 

  • C. F. Dunkl (1977): An addition theorem for q-Hahn polynomials, Monats. Math. 85 (1977), 5 – 37.

    MathSciNet  Google Scholar 

  • C. F. Dunkl (1978): An addition theorem for Hahn polynomials: The spherical functions. SIAM J. Math. Anal. 9 (1978), 627 – 637.

    MathSciNet  MATH  Google Scholar 

  • C. F. Dunkl (1979): Discrete quadrature and bounds on t-designs, Mich. Math. J. 26135 (1979), 81 – 102.

    MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • C. F. Dunkl (1988): Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33 – 60.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Enomoto, N. Ito and R. Noda (1979): Tight 4-desings, Osaka J. Math. 16 (1979), 39 – 43.

    MathSciNet  MATH  Google Scholar 

  • J. M. Goethals and H. C. A. van Tilborg (1975), Uniformly packed codes. Philips Res. Rep. 30 (1975), 9 – 36.

    MathSciNet  MATH  Google Scholar 

  • L. Habsieger and D. Stanton (to appear): More zeros of Krawtchouk polynomials, to appear.

    Google Scholar 

  • J. Hemmeter (to appear): A new family of distance-regular graphs, to appear.

    Google Scholar 

  • S. G. Hoggar (1982): t-designs in projectives spaces, Europ. J. Combinatorics 3 (1982), 233 – 254.

    MATH  Google Scholar 

  • S. G. Hoggar (1989): Tight 4- and 5-designs in projectives spaces, Graphs and Combinatorics, 5 (1989), 87 - 94.

    Article  MathSciNet  MATH  Google Scholar 

  • S. G. Hoggar (to appear): t -designs in Delsarte spaces, to appear in IMA Proceedings.

    Google Scholar 

  • 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.

    MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • A. A. Ivanov (1983): Bounding the diameter of distance-regular graphs, Soviet Math. Dokl. 38 (1983), 149 – 152.

    Google Scholar 

  • 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 

  • A. A. Ivanov and S. V. Shpectorov (to appear): A characterization of the association schemes of Hermitian forms, to appear.

    Google Scholar 

  • A. A. Klyachko (1984): Models for the complex representations of the group GL(n,q), Math. USSR Sbornik, 48 (1987), 365 – 379.

    Article  MathSciNet  MATH  Google Scholar 

  • T. H. Koornwinder (1973): The addition formula for Jacobi polynomials and spherical harmonics, SIAM J. Appl. Math. 25 (1973), 236 – 246.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • D. Leonard (1984): Parameters of association schemes that are both P- and Q- polynomial, J. of Combinatorial Theory (A) 36 (1984), 355 – 363.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Leonard (1982): Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal. 13 (1982), 656 – 663.

    MathSciNet  MATH  Google Scholar 

  • D. Leonard (to appear): Non-symmetric, metric, cometric association schemes, to appear in J. of Combinatorial Theory (B).

    Google Scholar 

  • D. Leonard (to appear): Non-symmetric, metric, cometric association schemes are self-dual, to appear.

    Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • S. P. Lloyd (1957): Binary block ciding, Bell System Tech. J., 36 (1957), 517 – 535.

    MathSciNet  Google Scholar 

  • 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 

  • 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 

  • J. F. Mac Williams and N. J. A. Sloane (1977): The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1977.

    MATH  Google Scholar 

  • R. J. McEliece (1977): The theory of infomation and coding, in Encyclopedia of Math, and its Application, Addison-Wesley, 1977.

    Google Scholar 

  • A. Munemasa (to appear): On nonsymmetric P- and Q -polynomial association schemes, to appear.

    Google Scholar 

  • A. Neumaier (1981): Combinatorial configurations in terms of distances, T.H.E. Memorandum 81 1981, No. 09, p. 98, Eindhoven Univ. Tech.

    Google Scholar 

  • C.L. Peterson (1977): On tight 6 -designs, Osaka J. Math., 14 (1977), 417 – 435.

    MATH  Google Scholar 

  • D.K. Ray-Chaudhuri and R. M. Wilson (1975): On t-designs, Osaka J. Math. 12 (1975), 737 – 744.

    MathSciNet  MATH  Google Scholar 

  • H.F. H. Reuvers (1977): Some nonexistence theorems for perfect codes over arbitrary alphabets, Ph. D. thesis, Tech. Univ. Eindhoven, 1977.

    Google Scholar 

  • C. Roos (1982): Some remarks on perfect subsets in distance-regular graphs, Delfte Progress Report, 7 (1982), 90 – 94.

    MathSciNet  MATH  Google Scholar 

  • I. Schur (1930): Gleichungen ohne Affekt; Sitz. der Preuss Acad. Wiss. (1930), 443-449. Also in Gesammelte Abhandlungen, vol. 3, 191 – 197.

    Google Scholar 

  • 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 

  • 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 

  • D. Stanton (1980): Some q-Krawtchouk polynomials in Chevalley groups, Amer. J. Math. 102 (1980), 625 – 662.

    Article  MATH  Google Scholar 

  • D. Stanton (1981): Three addition theorems for some q-Krawtchouk polynomials, Geom. Dedicata 10 (1981), 403 – 425.

    Article  MATH  Google Scholar 

  • 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 

  • D. Stanton (1986): t-designs in classical association schemes, Graphs and Combinatorics, 2 (1986), 283 – 286.

    Article  MathSciNet  Google Scholar 

  • H. Tarnanen, M. J. Aaltonen and J. M. Goethals (1985): On the nonbinary Johnson Scheme, Europ. J. Combinatorics (1985), 279 – 285.

    Google Scholar 

  • P. Terwilliger (1987): A characterization of P- and Q-polynomial association schemes, J. of Combinatorial Theory (A), 45 (1987), 8 – 26.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Terwilliger (to appear): The incidence algebra of a uniform poset, to appear.

    Google Scholar 

  • A. Tietavainen (1973): On the nonexistence of perfect codes over finite fields, SIAM J. Appl. Math. 24 (1973), 88 – 96.

    MathSciNet  Google Scholar 

  • A. Tietavainen (1977): Nonexistence of nontrivial perfect codes in case \(q = p_1^sp_2^t,e \geqslant 3\),

    Google Scholar 

  • Discrete Math. 17(1977), 199–205.

    Google Scholar 

  • L. Vretare (1976): Elementary spherical functions on symmetric spaces. Math. Scand. 39 (1976), 343 – 358.

    MathSciNet  Google Scholar 

  • L. Vretare (1984): Formula for elementary spherical functions and generalized Jacobi polynomials. SIAM J. Math. Anal. 15 (1984), 805 – 833.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, 43210, OH, USA

    Eiichi Bannai

Authors
  1. Eiichi Bannai
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, Ohio, USA

    Paul Nevai

Rights and permissions

Reprints and permissions

Copyright information

© 1990 Kluwer Academic Publishers

About this chapter

Cite this chapter

Bannai, E. (1990). Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In: Nevai, P. (eds) Orthogonal Polynomials. NATO ASI Series, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0501-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-0501-6_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6711-9

  • Online ISBN: 978-94-009-0501-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation