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Volumn 41, Issue 5, 1995, Pages 1303-1321

Krawtchouk Polynomials and Universal Bounds for Codes and Designs in Hamming Spaces

Author keywords

crosscorrelation; duality in bounding codes and orthogonal arrays; extreme roots; Linear programming; optimal polynomials; self dual codes

Indexed keywords

APPROXIMATION THEORY; CORRELATION THEORY; CRYPTOGRAPHY; LINEAR PROGRAMMING; MATHEMATICAL TRANSFORMATIONS; OPTIMIZATION; POLYNOMIALS; VECTORS;

EID: 0029378543     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.412678     Document Type: Article
Times cited : (108)

References (43)
  • 1
    • 0018320553 scopus 로고
    • Linear programming bounds for tree codes
    • M. J. Aaltonen, “Linear programming bounds for tree codes,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 85–90, 1977.
    • (1977) IEEE Trans. Inform. Theory , vol.IT-25 , pp. 85-90
    • Aaltonen, M.J.1
  • 2
    • 38249016502 scopus 로고
    • A new bound on nonbinary block codes
    • —, “A new bound on nonbinary block codes,” Discr. Math., vol. 83, pp. 139–160, 1990.
    • (1990) Discr. Math. , vol.83 , pp. 139-160
    • Aaltonen, M.J.1
  • 3
    • 84990678058 scopus 로고
    • Simple construction of almost k-wise independent random variables
    • N. Alon, O. Goldreich, J. Hastad, and R. Peralta, “Simple construction of almost k-wise independent random variables,” Random Structures and Algorithms, vol. 3, no. 3, pp. 289–304, 1992.
    • (1992) Random Structures and Algorithms , vol.3 , Issue.3 , pp. 289-304
    • Alon, N.1    Goldreich, O.2    Hastad, J.3    Peralta, R.4
  • 6
    • 0000434687 scopus 로고
    • Mathematical theory of the symmetrical factorial design
    • R. C. Bose, “Mathematical theory of the symmetrical factorial design,” Sankhya, vol. 8, pp. 107–166, 1947.
    • (1947) Sankhya , vol.8 , pp. 107-166
    • Bose, R.C.1
  • 8
    • 0010906160 scopus 로고
    • On uniformly packed [n, n — k — 4] codes over GF(q) and a class of caps in PG(k − 1, q)
    • R. Calderbank, “On uniformly packed [n, n — k — 4] codes over GF(q) and a class of caps in PG(k − 1, q),” J. London Math. Soc., vol. 26, pp. 365–384, 1982.
    • (1982) J. London Math. Soc. , vol.26 , pp. 365-384
    • Calderbank, R.1
  • 9
  • 10
    • 0025522511 scopus 로고
    • A new upper bound on the minimal distance of self-dual codes
    • J. H. Conway and N. J. A. Sloane, “A new upper bound on the minimal distance of self-dual codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 1319–1333, 1990.
    • (1990) IEEE Trans. Inform. Theory , vol.36 , pp. 1319-1333
    • Conway, J.H.1    Sloane, N.J.A.2
  • 11
    • 0015735548 scopus 로고
    • Four fundamental parameters of a code and their combinatorial significance
    • Ph. Delsarte, “Four fundamental parameters of a code and their combinatorial significance,” Inform. Contr., vol. 23, pp. 407–438, 1973.
    • (1973) Inform. Contr. , vol.23 , pp. 407-438
    • Delsarte, Ph.1
  • 12
    • 85041979556 scopus 로고
    • An algebraic approach to the association schemes of coding theory
    • —, “An algebraic approach to the association schemes of coding theory,” Philips Res. Rep., Suppl., vol. 10, 1973.
    • (1973) Philips Res. Rep., Suppl. , vol.10
    • Delsarte, Ph.1
  • 13
    • 0000641964 scopus 로고
    • Some maximal arcs in finite projective planes
    • R. H. F. Denniston, “Some maximal arcs in finite projective planes,” J. Combin. Theory, vol. 6, pp. 317–319, 1969.
    • (1969) J. Combin. Theory , vol.6 , pp. 317-319
    • Denniston, R.H.F.1
  • 14
    • 0040442927 scopus 로고
    • Discrete quadrature and bounds on t — design
    • C. F. Dunkl, “Discrete quadrature and bounds on t — design,” Mich. Math. J., vol. 26, pp. 81–102, 1979.
    • (1979) Mich. Math. J. , vol.26 , pp. 81-102
    • Dunkl, C.F.1
  • 16
    • 0000499681 scopus 로고
    • Notes on digital coding
    • M. J. E. Golay, “Notes on digital coding,” Proc. IRE, vol. 37, p. 657, 1949.
    • (1949) Proc. IRE , vol.37 , pp. 657
    • Golay, M.J.E.1
  • 17
    • 84943817322 scopus 로고
    • Error detecting and error correcting codes
    • R. W. Hamming, “Error detecting and error correcting codes,” Bell Syst. Tech. J., vol. 29, pp. 147–160, 1950.
    • (1950) Bell Syst. Tech. J. , vol.29 , pp. 147-160
    • Hamming, R.W.1
  • 18
    • 0001631552 scopus 로고
    • On the largest size cap in S5.3
    • R. Hill, “On the largest size cap in S5.3,” Rend. Acad. Naz. Lincei, vol. 54, no. 8, pp. 378–384, 1973.
    • (1973) Rend. Acad. Naz. Lincei , vol.54 , Issue.8 , pp. 378-384
    • Hill, R.1
  • 20
    • 0002468745 scopus 로고
    • Bounds for packings on a sphere and in space
    • G. A. Katabiansky and V. I. Levenshtein, “Bounds for packings on a sphere and in space,” Prob. Inform. Transmission, vol. 14, no. 1, pp. 1–17, 1978.
    • (1978) Prob. Inform. Transmission , vol.14 , Issue.1 , pp. 1-17
    • Katabiansky, G.A.1    Levenshtein, V.I.2
  • 21
    • 0000856269 scopus 로고
    • Sur une généralisation des polynômes d'Hermite
    • M. Krawtchouk, “Sur une généralisation des polynômes d'Hermite,” Compt. Rend., vol. 189, pp. 620–622, 1929.
    • (1929) Compt. Rend. , vol.189 , pp. 620-622
    • Krawtchouk, M.1
  • 22
    • 0025483055 scopus 로고
    • There is no (24, 12, 10) self-dual quaternary codes
    • C. W. M. Lam and V. Pless, “There is no (24, 12, 10) self-dual quaternary codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 1153–1156, 1990.
    • (1990) IEEE Trans. Inform. Theory , vol.36 , pp. 1153-1156
    • Lam, C.W.M.1    Pless, V.2
  • 24
    • 3643078812 scopus 로고
    • Bounds on the maximal cardinality of a code with bounded modulus of the inner product
    • —, “Bounds on the maximal cardinality of a code with bounded modulus of the inner product,” Soviet Math—Dokl., vol. 25, no. 2, pp. 526–531, 1982.
    • (1982) Soviet Math—Dokl , vol.25 , Issue.2 , pp. 526-531
    • Levenshtein, V.I.1
  • 25
    • 0002613871 scopus 로고
    • Bounds for packings of metric spaces and some their applications
    • Moscow USSR: Nauka, (in Russian)
    • —, “Bounds for packings of metric spaces and some their applications,” in Problemy Cybernetiki, vol. 40. Moscow, USSR: Nauka, 1983, pp. 43–110 (in Russian).
    • (1983) Problemy Cybernetiki , vol.40 , pp. 43-110
    • Levenshtein, V.I.1
  • 26
    • 34249842879 scopus 로고
    • Designs as maximum codes in polynomial metric spaces
    • —, “Designs as maximum codes in polynomial metric spaces,” Acta Applicandae Mathematicae, vol. 29, pp. 1–82, 1992.
    • (1992) Acta Applicandae Mathematicae , vol.29 , pp. 1-82
    • Levenshtein, V.I.1
  • 27
    • 0040031370 scopus 로고
    • Bounds for self-complementary codes and their applications
    • Wien—New York: Springer-Verlag
    • —, “Bounds for self-complementary codes and their applications,” in Eurocode-92, CISM Courses and Lectures, vol. 339. Wien—New York: Springer-Verlag, 1993, pp. 159–171.
    • (1993) Eurocode-92, CISM Courses and Lectures , vol.339 , pp. 159-171
    • Levenshtein, V.I.1
  • 29
    • 0017468074 scopus 로고
    • New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities
    • R. J. McEliece, E. R. Rodemich, H. Rumsey Jr., and L. R. Welch, “New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 157–166, 1977.
    • (1977) IEEE Trans. Inform. Theory , vol.IT-23 , pp. 157-166
    • McEliece, R.J.1    Rodemich, E.R.2    Rumsey, H.3    Welch, L.R.4
  • 30
    • 0015602468 scopus 로고
    • An upper bound for self-dual codes
    • C. L. Mallows and N. J. A. Sloane, “An upper bound for self-dual codes,” Inform. Contr., vol. 22, pp. 188–200, 1973.
    • (1973) Inform. Contr. , vol.22 , pp. 188-200
    • Mallows, C.L.1    Sloane, N.J.A.2
  • 31
    • 0038925519 scopus 로고
    • Combinatorial configuratons in terms of distances
    • Eindhoven Univ. Technol., Eindhoven, The Netherlands, 81-09 (Wiskunde)
    • A. Neumaier, “Combinatorial configuratons in terms of distances,” Eindhoven Univ. Technol., Eindhoven, The Netherlands, Memo. 81-09 (Wiskunde), 1981.
    • (1981) Memo
    • Neumaier, A.1
  • 33
    • 0003204609 scopus 로고
    • Some remarks concerning curves of the second degree in a finite plane
    • B. Quist, “Some remarks concerning curves of the second degree in a finite plane,” Ann. Acad. Fenn. Sci. Ser. A, vol. 134, 1952.
    • (1952) Ann. Acad. Fenn. Sci. Ser. A , vol.134
    • Quist, B.1
  • 34
    • 0000580915 scopus 로고
    • Factorial experiments derivable from combinatorial arrangements of arrays
    • C. R. Rao, “Factorial experiments derivable from combinatorial arrangements of arrays,” J. Roy. Stat. Soc., vol. 89, pp. 128–139, 1947.
    • (1947) J. Roy. Stat. Soc. , vol.89 , pp. 128-139
    • Rao, C.R.1
  • 35
    • 0040665950 scopus 로고
    • An extremum problem for polynomials
    • I. Schoenberg and G. Szegö, “An extremum problem for polynomials,” Composito Math., vol. 14, pp. 260–268, 1960.
    • (1960) Composito Math. , vol.14 , pp. 260-268
    • Schoenberg, I.1    Szegö, G.2
  • 36
    • 0002073534 scopus 로고
    • Equidistant q-ary codes and resolved balanced incomplete block designs
    • N. V. Semakov and V. A. Zinoviev, “Equidistant q-ary codes and resolved balanced incomplete block designs,” Prob. Inform. Transmission, vol. 4, no. 2, pp. 1–7, 1968.
    • (1968) Prob. Inform. Transmission , vol.4 , Issue.2 , pp. 1-7
    • Semakov, N.V.1    Zinoviev, V.A.2
  • 38
    • 0000977083 scopus 로고
    • On mutual correlation of sequences
    • V. M. Sidelnikov, “On mutual correlation of sequences,” Soviet Math.—Dokl., vol. 12, no. 1, pp. 197–201, 1971.
    • (1971) Soviet Math.—Dokl , vol.12 , Issue.1 , pp. 197-201
    • Sidelnikov, V.M.1
  • 39
    • 0019032458 scopus 로고
    • On extremal polynomials used to estimate the size of codes
    • —, “On extremal polynomials used to estimate the size of codes,” Probl. Inform. Transmission, vol. 16, no. 3, pp. 174–186, 1980.
    • (1980) Probl. Inform. Transmission , vol.16 , Issue.3 , pp. 174-186
    • Sidelnikov, V.M.1
  • 40
    • 33746357208 scopus 로고
    • Maximum distance q—ary codes
    • R. C. Singleton, “Maximum distance q—ary codes,” IEEE Trans. Inform. Theory, vol. IT-10, pp. 116–118, 1964.
    • (1964) IEEE Trans. Inform. Theory , vol.IT-10 , pp. 116-118
    • Singleton, R.C.1
  • 41
    • 0004073954 scopus 로고
    • Providence, RI: AMS Col. Publ.
    • G. Szegö, Orthogonal Polynomials, vol. 23. Providence, RI: AMS Col. Publ., 1939.
    • (1939) Orthogonal Polynomials , vol.23
    • Szegö, G.1
  • 42
    • 0026623869 scopus 로고
    • A bound for divisible codes
    • H. N. Ward, “A bound for divisible codes,” IEEE Trans. Inform. Theory, vol. 38, pp. 191–194, 1992.
    • (1992) IEEE Trans. Inform. Theory , vol.38 , pp. 191-194
    • Ward, H.N.1
  • 43
    • 0016060859 scopus 로고
    • Lower bounds on the maximum correlation of signals
    • L. R. Welch, “Lower bounds on the maximum correlation of signals,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 397–399, 1974.
    • (1974) IEEE Trans. Inform. Theory , vol.IT-20 , pp. 397-399
    • Welch, L.R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.