On integral zeros of Krawtchouk polynomials

Journal of Combinatorial Theory. Series A

Volumn 74, Issue 1, 1996, Pages 71-99

  Krasikov, Ilia ^a,b   Litsyn, Simon ^a  

^a TEL AVIV UNIVERSITY   (Israel)

^b BEIT BERL COLLEGE   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030120244     PISSN: 00973165     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcta.1996.0038     Document Type: Article

References (44)

1. 1. A. BAKER, Bounds for the solution of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (1969), 439-444.
2. 2. E. BANNAI, On perfect codes in the Hamming schemes H(n, q) with q arbitrary, J. Combin. Theory Ser. A 23 (1977), 52-67.
3. 3. E. BANNAI, Orthogonal polynomials in coding theory and algebraic combinatorics, in "Orthogonal Polynomials" (P. Nevai, Ed.), pp. 25-53, Kluwer, Dordrecht, 1990.
4. 4. E. BANNAI AND T. ITO, "Algebraic Combinatorics. I. Association Schemes," Benjamin/ Cummings, Menlo Park, CA, 1984.
5. 5. M. BEST, "A Contribution to the Nonexistence of Perfect Codes," Ph.D. thesis, University of Amsterdam, 1982.
6. 6. F. BEUKERS, The multiplicity of binary recurrences, Compositio Math. 40 (1980), 251-267.
7. 7. F. BEUKERS, The zero-multiplicity of ternary recurrences, Compositio Math. 77 (1991), 165-177.
8. 8. G. COHEN, I. HONKALA, S. LITSYN, AND H. F. MATTSON, JR., Weighted coverings and packings, IEEE Trans. Info. Theory 41 (1995), 1601-1612.
9. 9. L. CHIHARA AND D. STANTON, Zeros of generalized Krawtchouk polynomials, J. Approx. Theory 60, No. 1 (1990), 43-57.
10. 10. R. CLAYTON, "Multiple Packings and Coverings in Algebraic Coding Theory," Ph.D. Thesis, UCLA, 1987.
11. 11. G. COHEN AND P. FRANKL, On tilings of the binary vector space, Discrete Math. 31 (1980), 271-277.
12. 12. G. COHEN, S. LITSYN, AND H. F. MATTSON, JR., "On Perfect Weighted Coverings with Small Radius," Springer Lecture Notes in Computer Science, Vol. 573, pp. 32-41, Springer-Verlag, New York/Berlin, 1992.
13. 13. K. S. DAVIS AND W. A. WEBB, Lucas' theorem for prime powers, Euopean J. Combin. 11 (1990), 229-233.
14. 2, Pacific J. Math. 118 (1985), 323-345.
15. 15. N. J. FINE, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589-592.
16. 16. L. HABSIEGER AND D. STANTON, More zeros of Krawtchouk polynomials, Graphs and Combin. 2 (1993), 163-172.
17. 17. H. O. HÄMÄLAINEN, I. S. HONKALA, M. K. KAIKKONEN, AND S. N. LITSYN, Bounds for multiple covering codes, Designs, Codes and Cryptogr. 3 (1993), 251-275.
18. 18. Y. HONG, 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 Combin. 2 (1986), 145-164.
19. 19. D. L. HILLIKER, An algorithm for solving a certain class of diophantine equations, I, Math. Comput. 38, No. 158 (1982), 611-626.
20. 20. D. L. HILLIKER AND E. G. STRAUS, Determination of bounds for the solutions to those binary diophantine equations that satisfy the hypotheses of Runge's theorem, Trans. Amer. Math. Soc. 280, No. 2 (1983), 637-657.
21. 21. M. KARPOVSKY, Weight distribution of translates, covering radius and perfect codes correcting errors of the given multiplicities, IEEE Trans. Inform. Theory 27 (1981), 462-472.
22. 22. I. KRASIKOV AND Y. RODITTY, Switching reconstruction and Diophantine equations, J. Combin. Theory Ser. B 54 (1992), 189-195.
23. 24. I. KRASIKOV AND Y. RODDITY, More on vertex switching reconstruction, J. Combin. Theory Ser. B 60 (1994), 40-55.
24. 25. K. K. KUBOTA, On a conjecture of Morgan Ward, I, II, Acta Arithmetica (1977), 11-28, 29-48.
25. 26. V. LEVENSHTEIN, Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Trans. Info. Theory 41 (1995), 1303-1321.
26. 27. V. LEVENSHTEIN, Bounds for packings of metric spaces and some their applications, in "Problemy Kibernetiki," Vol. 40, pp. 43-110, Nauka, Moscow, 1983. [in Russian]
27. 28. J. H. VAN LINT, A survey of perfect codes, Rocky Mountain J. Math. 5 (1975), 199-224.
28. 29. J. H. VAN LINT, "Introduction to Coding Theory," Springer-Verlag, New York/Berlin, 1992.
29. 30. E. LUCAS, Sur les cogruences des nombres eulériens et des coefficients différentiels des fonctions trigonometriques, suivant un module premier, Bull. Soc. Math. France 6 (1878), 49-54.
30. 31. F. J. MACWILLIAMS AND N. J. A. SLOANE, "The Theory of Error-Correcting Codes," North-Holland, New York, 1977.
31. 32. L. J. MORDELL, "Diophantine Equations," Academic Press, London/New York, 1969.
32. 33. A. J. VAN DER POORTEN AND H. P. SCHLICKEWEI, Zeros of recurrence sequences, Bull. Austral. Math. Soc. 44, (1991), 215-223.
33. 34. C. RUNGE, Ueber ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen, J. Reine Angew. Math. 100 (1887), 425-435.
34. 35. A. SHINZEL, An improvement of Runge's theorem on diophantine equations, Comment. Pontific. Acad. Sci. 2, No. 20 (1969), 1-9.
35. 36. T. N. SHOREY AND R. TIJDEMAN, "Exponential Diophantine Equations," Cambridge Univ. Press, London/New York, 1986.
36. 37. T. N. SHOREY AND R. TIJDEMAN, Perfect powers in products of terms in an arithmetical progression, III, Acta Arithmetica 61.4 (1992), 391-398.
37. n - 1 + ··· + k, J. London Math. Soc. 1 (1920), 66-68.
38. 39. R. P. STANLEY, Reconstruction from vertex switching, J. Combin. Theory Ser. B 38 (1985), 132-138.
39. 40. G. SZÉGÖ, "Orthogonal Polynomials," Amer. Math. Soc. Colloq. Publ., Vol. 23, Providence, RI, 1975.
40. 41. A. TIETÄVÄINEN, On the nonexistence of perfect codes over finite fields, SIAM J. Appl. Math. 24 (1973), 88-96.
41. 42. A. TIETÄVÄINEN AND A. PERKO, There are no unknown perfect binary codes, Ann. Univ. Turku, Ser. A, I 148 (1971), 3-10.
42. 43. H. C. A. VAN TILBORG, "Uniformly Packed Codes," Ph.D. thesis, Eindhoven, 1976.
43. 44. G. J. M. VAN WEE, G. D. COHEN, AND S. N. LITSYN, A note on perfect multiple covering of the Hamming space, IEEE Trans. Inform. Theory 37 (1991), 678-682.
44. 45. V. ZINOVIEV AND V. LEONTJEV, The nonexistence of perfect codes over Galois fields, Problemy upravleniya i teorii informatsii 2 (1973), 123-132. [in Russian]