Orthogonal Polynomials in Information Theory

  • Rudolf Ahlswede
Part of the Foundations in Signal Processing, Communications and Networking book series (SIGNAL, volume 13)


The following lectures are based on the works (Tamm, Communication complexity of sum-type functions, 1991, [115]; Appl Math Lett 7:39–44, 1994, [116]; Inf Comput 116(2):162–173, 1995, [117]; Tamm, Discret Appl Math 61:271–283, 1995, [118]; Proceedings of 1998 IEEE Symposium on Information Theory, MIT, Cambridge, 1998, [120]; Tamm, IEEE Trans Inf Theory 44(5):2003–2009, 1998, [121]; Numbers, Information and Complexity, Special Volume in Honour of Rudolf Ahlswede, 2000, [122]; Proceedings of the Workshop Codes and Association Schemes, 2001, [123]; Electron J Comb 8(A1):31, 2001, [124]; J Stat Plan Interf 2(2):433–448, 2002, [125]) of Tamm.


  1. 1.
    K.A.S. Abdel Ghaffar, H.C. Ferreira, On the maximum number of systematically – encoded information bits in the Varshamov – Tenengolts codes and the Constantin – Rao codes, in Proceedings of 1997 IEEE Symposium on Information Theory, Ulm (1997), p. 455Google Scholar
  2. 2.
    M. Aigner, Catalan-like numbers and determinants. J. Combin. Theory Ser. A 87, 33–51 (1999)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. Aigner, A characterization of the Bell numbers. Discret. Math. 205(1–3), 207–210 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    G.E. Andrews, Plane partitions (III): the weak Macdonald conjecture. Invent. Math. 53, 193–225 (1979)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    G.E. Andrews, Pfaff’s method. I. The Mills-Robbins-Rumsey determinant. Discret. Math. 193(1–3), 43–60 (1998)MATHGoogle Scholar
  6. 6.
    G.E. Andrews, D. Stanton, Determinants in plane partition enumeration. Eur. J. Combin. 19(3), 273–282 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    E.E. Belitskaja, V.R. Sidorenko, P. Stenström, Testing of memory with defects of fixed configurations, in Proceedings of 2nd International Workshop on Algebraic and Combinatorial Coding Theory, Leningrad (1990), pp. 24–28Google Scholar
  8. 8.
    E.A. Bender, D.E. Knuth, Enumeration of plane partitions. J. Combin. Theory Ser. A 13, 40–54 (1972)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    E.R. Berlekamp, A class of convolutional codes. Information and Control 6, 1–13 (1963)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    E.R. Berlekamp, Algebraic Coding Theory (McGraw-Hill, New York, 1968)MATHGoogle Scholar
  11. 11.
    E.R. Berlekamp, Goppa codes. IEEE Trans. Inf. Theory 19, 590–592 (1973)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    R.E. Blahut, Theory and Practice of Error Control Codes (Addison-Wesley, Reading, 1984)Google Scholar
  13. 13.
    R.E. Blahut, Fast Algorithms for Digital Signal Processing (Addison-Wesley, Reading, 1985)Google Scholar
  14. 14.
    D.L. Boley, T.J. Lee, F.T. Luk, The Lanczos algorithm and Hankel matrix factoriztion. Linear Algebr. Appl. 172, 109–133 (1992)MATHCrossRefGoogle Scholar
  15. 15.
    D.L. Boley, F.T. Luk, D. Vandevoorde, A fast method to diagonalize a Hankel matrix. Linear Algebr. Appl. 284, 41–52 (1998)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    M. Bousquet Mélou, L. Habsieger, Sur les matrices à signes alternants. Discret. Math. 139, 57–72 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    D.M. Bressoud, Proofs and Confirmations (Cambridge University Press, Cambridge, 1999)MATHCrossRefGoogle Scholar
  18. 18.
    L. Carlitz, D.P. Rosselle, R.A. Scoville, Some remarks on ballot - type sequences. J. Combin. Theory 11, 258–271 (1971)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    L. Carroll, Alice’s Adventures in Wonderland (1865)Google Scholar
  20. 20.
    P.L. Chebyshev, Sur l’interpolation par la méthode des moindres carrés. Mém. Acad. Impér. Sci. St. Pétersbourg (7) 1 (15), 1–24; also: Oeuvres I, 473–489 (1859)Google Scholar
  21. 21.
    U. Cheng, On the continued fraction and Berlekamp’s algorithm. IEEE Trans. Inf. Theory 30, 541–544 (1984)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    S.H. Choi, D. Gouyou-Beauchamps, Enumeration of generalized Young tableaux with bounded height. Theor. Comput. Sci. 117, 137–51 (1993)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    W. Chu, Binomial convolutions and determinant identities. Discret. Math. 204, 129–153 (1999)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes (Elsevier, Amsterdam, 1997)MATHGoogle Scholar
  25. 25.
    G.D. Cohen, S. Litsyn, A. Vardy, G. Zemor, Tilings of binary spaces. SIAM J. Discret. Math. 9, 393–412 (1996)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    S.D. Constantin, T.R.N. Rao, On the theory of binary asymmetric error correcting codes. Inf. Control 40, 20–26 (1979)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    N.G. de Bruijn, On the factorization of finite abelian groups. Indag. Math. Kon. Ned. Akad. Wet. Amst. 15, 258–264 (1953)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    N.G. de Bruijn, On the factorization of cyclic groups. Indag. Math. Kon. Ned. Akad. Wet. Amst. 15, 370–377 (1953)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    G. de Prony, Essai expérimental et analytique sur les lois de la dilatabilité de fluides élastiques et sur les celles de la force expansive de la vapeur de l’ alcool, à différentes températures. J. de l’École Polytechnique 1, cahier 22, 24–76 (1795)Google Scholar
  30. 30.
    P. Delsarte, Nombres de Bell et polynômes de Charlier. C. R. Acad. Sci. Paris (Ser. A) 287, 271–273 (1978)Google Scholar
  31. 31.
    M. Desainte-Catherine, X.G. Viennot, Enumeration of certain Young tableaux with bounded height. Combinatoire Énumérative (Montreal 1985). Lecture Notes in Mathematics, vol. 1234 (Springer, Berlin, 1986), pp. 58–67Google Scholar
  32. 32.
    C.L. Dodgson, Condensation of determinants. Proc. R. Soc. Lond. 15, 150–155 (1866)CrossRefGoogle Scholar
  33. 33.
    T. Etzion, A. Vardy, Perfect codes: constructions, properties, and enumeration. IEEE Trans. Inf. Theory 40(3), 754–763 (1994)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    T. Etzion, A. Vardy, On perfect codes and tilings: problems and solutions, in Proceedings of 1997 IEEE Symposium on Information Theory, Ulm (1997), p. 450Google Scholar
  35. 35.
    H. Everett, D. Hickerson, Packing and covering by translates of certain starbodies. Proc. Am. Math. Soc. 75(1), 87–91 (1979)MATHCrossRefGoogle Scholar
  36. 36.
    P. Flajolet, Combinatorial aspects of continued fractions. Discret. Math. 32, 125–161 (1980)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    P. Flajolet, On congruences and continued fractions for some classical combinatorial quantities. Discret. Math. 41, 145–153 (1982)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    D. Foata, Combinatoire des identités sur les polynômes orthogonaux, in Proceedings of the International Congress of Mathematicians, Warsaw (1983), pp. 1541–1553Google Scholar
  39. 39.
    L. Fuchs, Abelian Groups (Pergamon Press, New York, 1960)MATHGoogle Scholar
  40. 40.
    S. Galovich, S. Stein, Splittings of Abelian groups by integers. Aequationes Math. 22, 249–267 (1981)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    I. Gessel, A probabilistic method for lattice path enumeration. J. Stat. Plan. Inference 14, 49–58 (1986)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    I. Gessel, R. Stanley, Algebraic enumeration, Handbook of Combinatorics, vol. 2, ed. by R.L. Graham, M. Grötschel, L. Lovasz (Wiley, New York, 1996), pp. 1021–1069Google Scholar
  43. 43.
    I. Gessel, X.G. Viennot, Binomial determinants, paths and hook length formulae. Adv. Math. 58, 300–321 (1985)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    I. Gessel, X G. Viennot, Determinants, Paths, and Plane Partitions, Preprint (1989)Google Scholar
  45. 45.
    S. Golomb, A general formulation of error metrics. IEEE Trans. Inf. Theory 15, 425–426 (1969)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    S. Golomb, Polyominoes, 2nd edn. (Princeton University Press, Princeton, 1994)MATHGoogle Scholar
  47. 47.
    S.W. Golomb, L.R. Welch, Algebraic coding and the Lee metric, in Error Correcting Codes, ed. by H.B. Mann (Wiley, New York, 1968), pp. 175–194Google Scholar
  48. 48.
    S.W. Golomb, L.R. Welch, Perfect codes in the Lee metric and the packing of polyominoes. SIAM J. Appl. Math. 18, 302–317 (1970)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    V.D. Goppa, A new class of linear correcting codes. Probl. Peredachi Informatsii 6(3), 24–30 (1970) (in Russian)Google Scholar
  50. 50.
    V.D. Goppa, Rational representation of codes and (L,g) codes, Probl. Peredachi Informatsii 7(3), 41–49 (1971) (in Russian)Google Scholar
  51. 51.
    V.D. Goppa, Decoding and diophantine approximations. Probl. Control Inf. Theory 5(3), 195–206 (1975)MathSciNetGoogle Scholar
  52. 52.
    B. Gordon, A proof of the Bender - Knuth conjecture. Pac. J. Math. 108, 99–113 (1983)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    D.C. Gorenstein, N. Zierler, A class of error-correcting codes in \(p^m\) symbols. J. Soc. Indus. Appl. Math. 9, 207–214 (1961)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics (Addison Wesley, Reading, 1988)Google Scholar
  55. 55.
    S. Gravier, M. Mollard, On domination numbers of Cartesian products of paths. Discret. Appl. Math. 80, 247–250 (1997)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    A.J. Guttmann, A.L. Owczarek, X.G. Viennot, Vicious walkers and Young tableaux I: without walls. J. Phys. A: Math. General 31, 8123–8135 (1998)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    R.K. Guy, Catwalks, sandsteps and Pascal pyramids. J. Integer Seq. 3, Article 00.1.6 (2000)Google Scholar
  58. 58.
    W. Hamaker, S. Stein, Combinatorial packing of \(R^3\) by certain error spheres. IEEE Trans. Inf. Theory 30(2), 364–368 (1984)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    A.J. Han Vinck, H. Morita, Codes over the ring of integers modulo m. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E81-A(10), 1564–1571 (1998)Google Scholar
  60. 60.
    G. Hajós, Über einfache und mehrfache Bedeckungen des n-dimensionalen Raumes mit einem Würfelgitter. Math. Zeit. 47, 427–467 (1942)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    D. Hickerson, Splittings of finite groups. Pac. J. Math. 107, 141–171 (1983)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    D. Hickerson, S. Stein, Abelian groups and packing by semicrosses. Pac. J. Math. 122(1), 95–109 (1986)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    P. Hilton, J. Pedersen, Catalan numbers, their generalization, and their uses. Math. Intell. 13(2), 64–75 (1991)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    K. Imamura, W. Yoshida, A simple derivation of the Berlekamp - Massey algorithm and some applications. IEEE Trans. Inf. Theory 33, 146–150 (1987)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    E. Jonckheere, C. Ma, A simple Hankel interpretation of the Berlekamp - Massey algorithm. Linear Algebr. Appl. 125, 65–76 (1989)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    S. Klavžar, N. Seifter, Dominating Cartesian products of cycles. Discret. Appl. Math. 59, 129–136 (1995)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    V.I. Levenshtein, Binary codes with correction for deletions and insertions of the symbol \(1\). Probl. Peredachi Informacii 1, 12–25 (1965). (in Russian)MathSciNetMATHGoogle Scholar
  68. 68.
    V.I. Levenshtein, A.J. Han, Vinck, Perfect (d, k)-codes capable of correcting single peak shifts. IEEE Trans. Inf. Theory 39(2), 656–662 (1993)MATHCrossRefGoogle Scholar
  69. 69.
    S. Lin, D.J. Costello, Error - Control Coding (Prentice-Hall, Englewood Cliffs, 1983)MATHGoogle Scholar
  70. 70.
    B. Lindström, On the vector representation of induced matroids. Bull. Lond. Math. Soc. 5, 85–90 (1973)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    S.S. Martirossian, Single – error correcting close packed and perfect codes, in Proceedings of 1st INTAS International Seminar on Coding Theory and Combinatorics, Thahkadzor, Armenia (1996), pp. 90 – 115Google Scholar
  72. 72.
    M.E. Mays, J. Wojciechowski, A determinant property of Catalan numbers. Discret. Math. 211, 125–133 (2000)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    J.L. Massey, Shift register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15, 122–127 (1969)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    W.H. Mills, Continued fractions and linear recurrences. Math. Comput. 29(129), 173–180 (1975)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    W.H. Mills, D.P. Robbins, H. Rumsey Jr., Enumeration of a symmetry class of plane partitions. Discret. Math. 67, 43–55 (1987)MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    H. Minkowski, Diophantische Approximationen (Teubner, Leipzig, 1907)MATHCrossRefGoogle Scholar
  77. 77.
    S.G. Mohanty, Lattice Path Counting and Applications (Academic Press, New York, 1979)MATHGoogle Scholar
  78. 78.
    T. Muir, Theory of Determinants (Dover, New York, 1960)Google Scholar
  79. 79.
    A. Munemasa, On perfect t-shift codes in Abelian groups. Des. Codes Cryptography 5, 253–259 (1995)MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    T.V. Narayana, Lattice Path Combinatorics (University of Toronto Press, Toronto, 1979)MATHGoogle Scholar
  81. 81.
    P. Peart, W.-J. Woan, Generating functions via Hankel and Stieltjes matrices. J. Integer Sequences 3, Article 00.2.1 (2000)Google Scholar
  82. 82.
    O. Perron, Die Lehre von den Kettenbrüchen (Chelsea Publishing Company, New York, 1929)MATHGoogle Scholar
  83. 83.
    W.W. Peterson, Encoding and error-correction procedures for the Bose-Chaudhuri codes. Trans. IRE 6, 459–470 (1960)MathSciNetMATHGoogle Scholar
  84. 84.
    M. Petkovšek, H.S. Wilf, A high-tech proof of the Mills-Robbins-Rumsey determinant formula. Electron. J. Comb. 3(2), 19 (1996)MathSciNetMATHGoogle Scholar
  85. 85.
    J.L. Phillips, The triangular decomposition of Hankel matrices. Math. Comput. 25(115), 599–602 (1971)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    G. Polya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, vol. II, 3rd edn. (Springer, Berlin, 1964)Google Scholar
  87. 87.
    C. Radoux, Déterminants de Hankel et théorème de Sylvester, in Proceedings of the 28th Séminaire Lotharingien (1992), pp. 115 – 122Google Scholar
  88. 88.
    C. Radoux, Addition formulas for polynomials built on classical combinatorial sequences. J. Comput. Appl. Math. 115, 471–477 (2000)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    L. Rédei, Die neue Theorie der endlichen abelschen Gruppen und eine Verallgemeinerung des Hauptsatzes von Hajós. Acta Math. Acad. Sci Hung. 16, 329–373 (1965)MATHCrossRefGoogle Scholar
  90. 90.
    J. Riordan, An Introduction to Combinatorial Analysis (Wiley, New York, 1958)MATHGoogle Scholar
  91. 91.
    H. Rutishauser, Der Quotienten-Differenzen-Algorithmus (Birkhäuser, Basel, 1957)MATHCrossRefGoogle Scholar
  92. 92.
    S. Saidi, Codes for perfectly correcting errors of limited size. Discret. Math. 118, 207–223 (1993)MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    S. Saidi, Semicrosses and quadratic forms. Eur. J. Comb. 16, 191–196 (1995)MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    A.D. Sands, On the factorization of finite abelian groups. Acta Math. 8, 65–86 (1957)MATHGoogle Scholar
  95. 95.
    A.D. Sands, On the factorization of finite abelian groups II. Acta Math. 13, 45–54 (1962)MATHGoogle Scholar
  96. 96.
    L.W. Shapiro, A Catalan triangle. Discret. Math. 14, 83–90 (1976)CrossRefGoogle Scholar
  97. 97.
    V. Sidorenko, Tilings of the plane and codes for translational metrics, in Proceedings of 1994 IEEE Symposium on Information Theory, Trondheim (1994), p. 107Google Scholar
  98. 98.
    F. Solove’eva, Switchings and perfect codes, in Numbers, Information and Complexity, Special Volume in Honour of Rudolf Ahlswede, ed. by I. Althöfer, N. Cai, G. Dueck, L. Khachatrian, M. Pinsker, A. Sárközy, I. Wegener, Z. Zhang (Kluwer Publishers, Boston, 2000), pp. 311–324CrossRefGoogle Scholar
  99. 99.
    R.P. Stanley, Theory and application of plane partitions. Stud. Appl. Math. 50 Part 1, 167–189; Part 2, 259–279 (1971)Google Scholar
  100. 100.
    R.P. Stanley, A baker’s dozen of conjectures concerning plane partitions, in Combinatoire Énumérative (Montreal 1985).Lecture Notes in Mathematics, vol. 1234 (Springer, Berlin, 1986), pp. 285–293Google Scholar
  101. 101.
    R.P. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge University Press, Cambridge, 1999)MATHCrossRefGoogle Scholar
  102. 102.
    S. Stein, Factoring by subsets. Pac. J. Math. 22(3), 523–541 (1967)MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    S. Stein, Algebraic tiling. Am. Math. Mon. 81, 445–462 (1974)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    S. Stein, Packing of \(R^n\) by certain error spheres. IEEE Trans. Inf. Theory 30(2), 356–363 (1984)MathSciNetMATHCrossRefGoogle Scholar
  105. 105.
    S. Stein, Tiling, packing, and covering by clusters. Rocky Mt. J. Math. 16, 277–321 (1986)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    S. Stein, Splitting groups of prime order. Aequationes Math. 33, 62–71 (1987)MathSciNetMATHCrossRefGoogle Scholar
  107. 107.
    S. Stein, Packing tripods. Math. Intell. 17(2), 37–39 (1995)MATHCrossRefGoogle Scholar
  108. 108.
    S. Stein, S. Szabó, Algebra and Tiling. The Carus Mathematical Monographs, vol. 25 (The Mathematical Association of America, Washington, 1994)Google Scholar
  109. 109.
    T.J. Stieltjes, Recherches sur les fractions continue. Ann. Fac. Sci. Toulouse 8, J.1– 22 (1895); A.1–47 (1894)Google Scholar
  110. 110.
    T.J. Stieltjes, Oeuvres Complètes (Springer, Berlin, 1993)MATHGoogle Scholar
  111. 111.
    V. Strehl, Contributions to the combinatorics of some families of orthogonal polynomials, mémoire, Erlangen (1982)Google Scholar
  112. 112.
    Y. Sugiyama, M. Kasahara, S. Hirawawa, T. Namekawa, A method for solving key equation for decoding Goppa code. Inf. Control 27, 87–99 (1975)MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    R.A. Sulanke, A recurrence restricted by a diagonal condition: generalized Catalan arrays. Fibonacci Q. 27, 33–46 (1989)MathSciNetMATHGoogle Scholar
  114. 114.
    S. Szabó, Lattice coverings by semicrosse of arm length 2. Eur. J. Comb. 12, 263–266 (1991)MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    U. Tamm, Communication complexity of sum-type functions, Ph.D. thesis, Bielefeld, 1991, also Preprint 91–016, SFB 343, University of Bielefeld (1991)Google Scholar
  116. 116.
    U. Tamm, Still another rank determination of set intersection matrices with an application in communication complexity. Appl. Math. Lett. 7, 39–44 (1994)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    U. Tamm, Communication complexity of sum - type functions invariant under translation. Inf. Comput. 116(2), 162–173 (1995)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    U. Tamm, Deterministic communication complexity of set intersection. Discret. Appl. Math. 61, 271–283 (1995)MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    U. Tamm, On perfect \(3\)–shift \(N\)–designs, in Proceedings of 1997 IEEE Symposium on Information Theory, Ulm (1997), p. 454Google Scholar
  120. 120.
    U. Tamm, Splittings of cyclic groups, tilings of Euclidean space, and perfect shift codes, Proceedings of 1998 IEEE Symposium on Information Theory (MIT, Cambridge, 1998), p. 245Google Scholar
  121. 121.
    U. Tamm, Splittings of cyclic groups and perfect shift codes. IEEE Trans. Inf. Theory 44(5), 2003–2009 (1998)MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    U. Tamm, Communication complexity of functions on direct sums, in Numbers, Information and Complexity, Special Volume in Honour of Rudolf Ahlswede, ed. by I. Althöfer, N. Cai, G. Dueck, L. Khachatrian, M. Pinsker, A. Sárközy, I. Wegener, Z. Zhang (Kluwer Publishers, Boston, 2000), pp. 589–602CrossRefGoogle Scholar
  123. 123.
    U. Tamm, Communication complexity and orthogonal polynomials, in Proceedings of the Workshop Codes and Association Schemes. DIMACS Series, Discrete Mathematics and Computer Science, vol. 56 (2001), pp. 277–285Google Scholar
  124. 124.
    U. Tamm, Some aspects of Hankel matrices in coding theory and combinatorics. Electron. J. Comb. 8(A1), 31 (2001)MathSciNetMATHGoogle Scholar
  125. 125.
    U. Tamm, Lattice paths not touching a given boundary. J. Stat. Plan. Interf. 2(2), 433–448 (2002)MathSciNetMATHCrossRefGoogle Scholar
  126. 126.
    W. Ulrich, Non-binary error correction codes. Bell Syst. Tech. J. 36(6), 1341–1388 (1957)MATHCrossRefGoogle Scholar
  127. 127.
    R.R. Varshamov, G.M. Tenengolts, One asymmetric error correcting codes (in Russian). Avtomatika i Telemechanika 26(2), 288–292 (1965)Google Scholar
  128. 128.
    X.G. Viennot, A combinatorial theory for general orthogonal polynomials with extensions and applications, in Polynômes Orthogonaux et Applications, Proceedings, Bar-le-Duc (Springer, Berlin, 1984), pp. 139–157Google Scholar
  129. 129.
    X.G. Viennot, A combinatorial interpretation of the quotient – difference algorithm, Preprint (1986)Google Scholar
  130. 130.
    H.S. Wall, Analytic Theory of Continued Fractions (Chelsea Publishing Company, New York, 1948)MATHGoogle Scholar
  131. 131.
    H. Weber, Beweis des Satzes, daß jede eigentlich primitive quadratische Form unendlich viele prime Zahlen darzustellen fähig ist. Math. Ann. 20, 301–329 (1882)MathSciNetMATHCrossRefGoogle Scholar
  132. 132.
    L.R. Welch, R.A. Scholtz, Continued fractions and Berlekamp’s algorithm. IEEE Trans. Inf. Theory 25, 19–27 (1979)MathSciNetMATHCrossRefGoogle Scholar
  133. 133.
    D. Zeilberger, Proof of the alternating sign matrix conjecture. Electronic J. Comb. 3(2), R13, 1–84 (1996)Google Scholar
  134. 134.
    D. Zeilberger, Proof of the refined alternating sign matrix conjecture. N. Y. J. Math. 2, 59–68 (1996)MathSciNetMATHGoogle Scholar
  135. 135.
    D. Zeilberger, Dodgson’s determinant-evaluation rule proved by TWO-TIMING MEN and WOMEN. Electron. J. Comb. 4(2), 22 (1997)MathSciNetMATHGoogle Scholar

Further Readings

  1. 136.
    R. Ahlswede, N. Cai, U. Tamm, Communication complexity in lattices. Appl. Math. Lett. 6, 53–58 (1993)MathSciNetMATHCrossRefGoogle Scholar
  2. 137.
    M. Aigner, Motzkin numbers. Eur. J. Comb. 19, 663–675 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 138.
    R. Askey, M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials. Mem. Am. Math. Soc. 49(300), 108 (1984)MathSciNetMATHGoogle Scholar
  4. 139.
    C. Brezinski, Padé-Type Approximation and General Orthogonal Polynomials (Birkhäuser, Basel, 1980)MATHCrossRefGoogle Scholar
  5. 140.
    D.C. Gorenstein, W.W. Peterson, N. Zierler, Two-error correcting Bose-Chaudhuri codes are quasi-perfect. Inf. Control 3, 291–294 (1960)MathSciNetMATHCrossRefGoogle Scholar
  6. 141.
    V.I. Levenshtein, On perfect codes in the metric of deletions and insertions (in Russian), Diskret. Mat. 3(1), 3–20; English translation. Discret. Math. Appl. 2(3), 1992 (1991)Google Scholar
  7. 142.
    H. Morita, A. van Wijngaarden, A.J. Han Vinck, Prefix synchronized codes capable of correcting single insertion/deletion errors, in Proceedings of 1997 IEEE Symposium on Information Theory, Ulm (1997), p. 409Google Scholar
  8. 143.
    J. Riordan, Combinatorial Identities (Wiley, New York, 1968)MATHGoogle Scholar
  9. 144.
    S. Szabó, Some problems on splittings of groups. Aequationes Math. 30, 70–79 (1986)MathSciNetMATHCrossRefGoogle Scholar
  10. 145.
    S. Szabó, Some problems on splittings of groups II. Proc. Am. Math. Soc. 101(4), 585–591 (1987)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BielefeldBielefeldGermany

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