Association Schemes and Coding Theory

This paper contains a survey of association scheme theory (with its algebraic and analytical aspects) and of its applications to coding theory (in a wide sense). It is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code. Special emphasis is put on the linear programming method, inspired by the MacWilliams transform. This produces upper bounds for the size of a code with a given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain well-defined "polynomial properties;" this leads one into the realm of orthogonal polynomial theory. In particular, some "universal bounds" are derived for codes and designs in polynomial type association schemes. Throughout the paper, the main concepts, methods, and results are illustrated by two examples that are of major significance in classical coding theory, namely, the Hamming scheme and the Johnson scheme. Other topics that receive special attention are spherical codes and designs, and additive codes in translation schemes, including Z/sub 4/-additive binary codes.

[1]  A. Robert Calderbank,et al.  On a Pair of Dual Subschemes of the Hamming Scheme Hn(q) , 1985, Eur. J. Comb..

[2]  Claude Carlet,et al.  On Correlation-Immune Functions , 1991, CRYPTO.

[3]  D. Stanton.  Orthogonal Polynomials and Chevalley Groups , 1984 .

[4]  J. J. Seidel,et al.  Orthogonal Matrices with Zero Diagonal , 1967, Canadian Journal of Mathematics.

[5]  Jaume Pujol,et al.  Translation-invariant propelinear codes , 1997, IEEE Trans. Inf. Theory.

[6]  S. G. Hoggar,et al.  t-Designs in Projective Spaces , 1982, Eur. J. Comb..

[7]  H. F. Blichfeldt.  The minimum value of quadratic forms, and the closest packing of spheres , 1929 .

[8]  O. Tamaschke.  Zur Theorie der Permutationsgruppen mit regulärer Untergruppe. I , 1962 .

[9]  Vladimir I. Levenshtein,et al.  Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces , 1995, IEEE Trans. Inf. Theory.

[10]  A. NEUMAIER,et al.  Duality in coherent configurations , 1989, Comb..

[11]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[12]  Gérard D. Cohen,et al.  Covering radius - Survey and recent results , 1985, IEEE Trans. Inf. Theory.

[13]  R. C. Bose.  Strongly regular graphs, partial geometries and partially balanced designs. , 1963 .

[14]  J. Seidel,et al.  Spherical codes and designs , 1977 .

[15]  Vera Pless,et al.  On the coveting radius of extremal self-dual codes , 1983, IEEE Trans. Inf. Theory.

[16]  Richard Askey,et al.  A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or 6 - j Symbols. , 1979 .

[17]  Tatsuro Ito,et al.  Current research on algebraic combinatorics , 1986, Graphs Comb..

[18]  I. J. Schoenberg,et al.  An extremum problem for polynomials , 1960 .

[19]  Dennis Stanton.  An Introduction to Group Representations and Orthogonal Polynomials , 1990 .

[20]  Robert J. McEliece,et al.  A low-rate improvement on the Elias bound (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[21]  Vladimir I. Levenshtein.  Split Orthogonal Arrays and Maximum Independent Resilient Systems of Functions , 1997, Des. Codes Cryptogr..

[22]  W. Haemers,et al.  Association schemes , 1996 .

[23]  Douglas R. Stinson,et al.  Association Schemes for Ordered Orthogonal Arrays and (T, M, S)-Nets , 1999, Canadian Journal of Mathematics.

[24]  Douglas R. Stinson,et al.  Bounds for Resilient Functions and Orthogonal Arrays , 1994, CRYPTO.

[25]  Danyo Danev,et al.  Upper bounds on the minimum distance of spherical codes , 1996, IEEE Trans. Inf. Theory.

[26]  R. C. Bose,et al.  On a class of partially balanced incomplete block designs , 1965 .

[27]  Simon Litsyn,et al.  On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes , 1998, Des. Codes Cryptogr..

[28]  J. Macwilliams.  A theorem on the distribution of weights in a systematic code , 1963 .

[29]  D. G. Higman.  Coherent configurations , 1975 .

[30]  P. Delsarte.  Hahn Polynomials, Discrete Harmonics, and t-Designs , 1978 .

[31]  D. A. Leonard.  Orthogonal Polynomials, Duality and Association Schemes , 1982 .

[32]  Andries E. Brouwer,et al.  The triply shortened binary Hamming code is optimal , 1977, Discret. Math..

[33]  P. Boyvalenkov,et al.  ON LINEAR PROGRAMMING BOUNDS FOR CODES IN POLYNOMIAL METRIC SPACES , 1998 .

[34]  Heeralal Janwa,et al.  Some new upper bounds on the covering radius of binary linear codes , 1989, IEEE Trans. Inf. Theory.

[35]  Tannaka-Krein duality for association schemes , 1982 .

[36]  N. J. A. Sloane,et al.  Inequalities for covering codes , 1988, IEEE Trans. Inf. Theory.

[37]  C. Shannon.  Probability of error for optimal codes in a Gaussian channel , 1959 .

[38]  V M Sidel'nikov.  NEW BOUNDS FOR DENSEST PACKING OF SPHERES IN n-DIMENSIONAL EUCLIDEAN SPACE , 1974 .

[39]  Thomas Siegenthaler,et al.  Correlation-immunity of nonlinear combining functions for cryptographic applications , 1984, IEEE Trans. Inf. Theory.

[40]  D. Vere-Jones.  FINITE BIVARIATE DISTRIBUTIONS AND SEMIGROUPS OF NON-NEGATIVE MATRICES , 1971 .

[41]  Charles F. Dunkl,et al.  Discrete quadrature and bounds on $t$-designs. , 1979 .

[42]  V. Levenshtein.  Designs as maximum codes in polynomial metric spaces , 1992 .

[43]  E. Bannai.  Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics , 1990 .

[44]  Richard M. Wilson,et al.  On $t$-designs , 1975 .

[45]  Alexander Schrijver,et al.  A comparison of the Delsarte and Lovász bounds , 1979, IEEE Trans. Inf. Theory.

[46]  R. Rankin.  The Closest Packing of Spherical Caps in n Dimensions , 1955, Proceedings of the Glasgow Mathematical Association.

[47]  S. P. Lloyd.  Binary block coding , 1957 .

[48]  A. A. Tietavainen.  An upper bound on the covering radius as a function of the dual distance , 1990 .

[49]  Patrick Solé.  A Lloyd Theorem in Weakly Metric Association Schemes , 1989, Eur. J. Comb..

[50]  Douglas R. Stinson,et al.  Combinatorial designs and cryptography , 1993 .

[51]  V. Levenshtein.  Equivalence of Delsarte's bounds for codes and designs in symmetric association schemes, and some applications , 1999 .

[52]  R. Gangolli,et al.  Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's brownian motion of several parameters , 1967 .

[53]  Vladimir A. Yudin.  Lower bounds for spherical designs , 1997 .

[54]  Surveys in Combinatorics: Strongly regular graphs , 1979 .

[55]  Matti J. Aaltonen,et al.  A new upper bound on nonbinary block codes , 1990, Discret. Math..

[56]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[57]  Robert J. McEliece,et al.  New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities , 1977, IEEE Trans. Inf. Theory.

[58]  Patrick Solé,et al.  Covering radius, codimension, dual-distance width , 1993, IEEE Trans. Inf. Theory.

[59]  Simon Litsyn,et al.  Upper Bounds on the Covering Radius of a Code with a Given Dual Distance , 1996, Eur. J. Comb..

[60]  C. R. Rao,et al.  Factorial Experiments Derivable from Combinatorial Arrangements of Arrays , 1947 .

[61]  François Jaeger,et al.  On Spin Models, Triply Regular Association Schemes, and Duality , 1995 .

[62]  Gabor Szegö,et al.  [60–6] An Extremum Problem for Polynomials , 1982 .

[63]  N. Sloane,et al.  An Introduction to Association Schemes and Coding Theory , 1975 .

[64]  Xavier L. Hubaut,et al.  Strongly regular graphs , 1975, Discret. Math..

[65]  R. C. Bose,et al.  Classification and Analysis of Partially Balanced Incomplete Block Designs with Two Associate Classes , 1952 .

[66]  Philippe Delsarte,et al.  Four Fundamental Parameters of a Code and Their Combinatorial Significance , 1973, Inf. Control..

[67]  Tom Verhoeff,et al.  An updated table of minimum-distance bounds for binary linear codes , 1987, IEEE Trans. Inf. Theory.

[68]  H. Mattson,et al.  New 5-designs , 1969 .

[69]  Arnold Neumaier,et al.  Distances, Graphs and Designs , 1980, Eur. J. Comb..

[70]  E. Bannai.  ON TIGHT DESIGNS , 1977 .

[71]  D. G. Higman.  Intersection matrices for finite permutation groups , 1967 .

[72]  J. Seidel,et al.  BOUNDS FOR SYSTEMS OF LINES, AND JACOBI POLYNOMIALS , 1975 .

[73]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[74]  Hendrik W. Lenstra,et al.  Two theorems on perfect codes , 1972, Discret. Math..

[75]  Philippe Delsarte,et al.  Extending the T-design Concept , 1993 .

[76]  Elwyn R. Berlekamp,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. II , 1967, Inf. Control..

[77]  Hsien-Chtjng Wang,et al.  TWO-POINT HOMOGENEOUS SPACES , 1952 .

[78]  H. Mattson,et al.  Coding and Combinatorics , 1974 .

[79]  Jean-Marie Goethals,et al.  On the Nonbinary Johnson Scheme , 1985, Eur. J. Comb..

[80]  Alexander A. Ivanov,et al.  Galois correspondence between permutation groups and cellular rings (association schemes) , 1990, Graphs Comb..

[81]  N. J. A. Sloane,et al.  A strengthening of the Assmus-Mattson theorem , 1991, IEEE Trans. Inf. Theory.

[82]  R. C. Bose,et al.  On Linear Associative Algebras Corresponding to Association Schemes of Partially Balanced Designs , 1959 .

[83]  Ron M. Roth,et al.  Author's Reply to Comments on 'Maximum-rank array codes and their application to crisscross error correction' , 1991, IEEE Trans. Inf. Theory.