Skip to main content
SpringerLink
Log in

Designs as maximum codes in polynomial metric spaces

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

Finite and infinite metric spaces % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] that are polynomial with respect to a monotone substitution of variable t(d) are considered. A finite subset (code) W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is characterized by the minimal distance d(W) between its distinct elements, by the number l(W) of distances between its distinct elements and by the maximal strength τ(W) of the design generated by the code W. A code W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is called a maximum one if it has the greatest cardinality among subsets of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with minimal distance at least d(W), and diametrical if the diameter of W is equal to the diameter of the whole space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. Delsarte codes are codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with τ(W)≥2l(W)−1 or τ(W)=2l(W)−2>0 and W is a diametrical code. It is shown that all parameters of Delsarte codes W) % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are uniquely determined by their cardinality |W| or minimal distance d(W) and that the minimal polynomials of the Delsarte codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are expansible with positive coefficients in an orthogonal system of polynomials {Q i(t)} corresponding to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. The main results of the present paper consist in a proof of maximality of all Delsarte codes provided that the system {Q i)} satisfies some condition and of a new proof confirming in this case the validity of all the results on the upper bounds for the maximum cardinality of codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \]% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with a given minimal distance, announced by the author in 1978. Moreover, it appeared that this condition is satisfied for all infinite polynomial metric spaces as well as for distance-regular graphs, decomposable in a sense defined below. It is also proved that with one exception all classical distance-regular graphs are decomposable. In addition for decomposable distance-regular graphs an improvement of the absolute Delsarte bound for diametrical codes is obtained. For the Hamming and Johnson spaces, Euclidean sphere, real and complex projective spaces, tables containing parameters of known Delsarte codes are presented. Moreover, for each of the above-mentioned infinite spaces infinite sequences (of maximum) Delsarte codes not belonging to tight designs are indicated.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bannai E., Ito T., Algebraic Combinatorics. I. Association Schemes, Benjamin/Cummings, London, 1984.

    Google Scholar 

  2. Bose R. C., Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13 (1963) 389–419.

    Google Scholar 

  3. Brouwer A. E., van Lint J. H., Strongly regular graphs and partial geometries, In: Enumeration and Design, Academic Press, Toronto, 1984, pp. 85–122.

    Google Scholar 

  4. Brouwer A. E., Cohen A. M., Neumaier A., Distance-Regular graphs, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  5. Calderbank R., 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. 26 (1982), 365–384.

    Google Scholar 

  6. Calderbank R., Kantor W.M., The geometry of two-weight codes, Bull. London Math. Soc. 18 (1986), 97–122.

    Google Scholar 

  7. Cameron P. J., Goethals J.-M., Seidel J. J., Strongly regular graphs having strongly regular subconstituents, J. Algebra 43 (1978), 257–280.

    Google Scholar 

  8. Delsarte Ph., An algebraic approach to the association schemes of coding theory, Philips Res. Reports, Suppl. 10 (1973).

  9. Delsarte Ph., Goethals J.-M., Seidel J. J., Bounds for systems of lines, and Jacobi polynomials, Philips Res. Reports 30 (1975), 91∓105.

    Google Scholar 

  10. Delsarte Ph., Goethals J.-M., Alternating bilinear forms over GF(q), J. Combin. Th. (A) 19 (1975), 26–50.

    Google Scholar 

  11. Delsarte Ph., Associations schemes and t-design in regular semilattices, J. Combin. Th. (A) 20 (1976), 230–243.

    Google Scholar 

  12. Delsarte Ph., Goethals J.-M., Seidel J. J., Spherical codes and designs, Geometriae Dedicata 6 (1977), 363–388.

    Google Scholar 

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

    Google Scholar 

  14. Delsarte Ph., Bilinear forms over a finite field with application to coding theory J. Combin. Th. (A) 25 (1978), 226–241.

    Google Scholar 

  15. Denniston R. H. F., Some maximal arcs in finite projective planes. I, Combin. Th. 6 (1969), 317–319.

    Google Scholar 

  16. Dunkl C. F., Discrete quadrature and bounds on t-design, Mich. Math. J. 26 (1979), 81–102.

    Google Scholar 

  17. Fazekas G., Levenshtein V. I., On upper bounds for code distance and covering radius of designs in polynomial metric spaces In: Fifth Joint Soviet-Swedish Intern. Workshop Inform. Theory, Moscow, 1990, pp. 65–68.

  18. Egawa Y., Association schemes of quadratic forms, J. Combin. Th. (A) 38 (1985), 1–14.

    Google Scholar 

  19. Gabidulin E. M., Theory of codes with maximal rank distance, Problems of Information Transmission 21 (1985).

  20. Gasper G., Linearization of the product of Jacobi polynomials. I, Canad. J. Math. 22: (1) (1970), 171–175.

    Google Scholar 

  21. Gelfand I. M., Spherical functions on symmetric Riemannian spaces, Amer. Math. Soc. Transl. 37: (2) (1964), 39–43.

    Google Scholar 

  22. Godsil C. D., Polynomial spaces, Discrete Math. 73 (1988/89), 71–88.

    Google Scholar 

  23. Helgason S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.

    Google Scholar 

  24. Hill R., On the largest size cap in S 5,3Rend. Accad. Naz. Lincei 54: (8) (1973), 378–384.

    Google Scholar 

  25. Hill R., Caps and groups, In: Atti dei Covegni Lincei, Colloquio Intern. sulle Theorie Combinatorie (Roma, 1973), No. 17 (Accad. Naz. Licei), 1976, 384–394.

  26. Hoggar S. G., Bounds for quaternionic line systems and reflection groups, Math. Scand. 43 (1978), 241–249.

    Google Scholar 

  27. Hoggar S.G., I-designs in projective spaces Europ. J. Comb. 3 (1982), 233–254.

    Google Scholar 

  28. Hoggar S. G., Tight 4 and 5-designs in projective spaces, Graphs and Comb. 5 (1989), 87–94.

    Google Scholar 

  29. Kabatiansky G. A., Levenshtein V. I., Bounds for packings on a sphere and in space, Problems of Information Transmission 14: (1) (1978), 1–17.

    Google Scholar 

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

    Google Scholar 

  31. Krein M. G., Nudelman A. A., Problems of Markov's Moments and Extremal Problems, Nauka, Moscow, 1973 (in Russian).

    Google Scholar 

  32. Lemmens P. W. H., Seidel J. J., Equiangular lines, J. Algebra 24 (1973), 494–512.

    Google Scholar 

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

    Google Scholar 

  34. Levenshtein V. I., On choosing polynomials to obtain bounds in packing problems, In: Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius, 1978, pp. 103–108 (in Russian).

  35. Levenshtein V. I., On bounds for packings in n-dimensional Euclidean space, Soviet Math. Dokl. 20: (2) (1979), 417–421.

    Google Scholar 

  36. Levenshtein V. I., Bounds on the maximal cardinality of a code with bounded modulus of the inner product, Soviet Math. Dokl. 25: (2) (1982), 526–531.

    Google Scholar 

  37. Levenshtein V.I., Bounds for packings of metric spaces and some their applications In: Probl. Cybern. 40, Nauka, Moscow, 1983, pp. 43–110 (in Russian).

    Google Scholar 

  38. Levenshtein V. I., Packing of polynomial metric spaces, In: Proc. Third International Workshop on Information Theory, Sochi, 1987, pp. 271–274.

  39. MacWilliams F. J., Sloane N. J. A., The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977.

    Google Scholar 

  40. McEliece R. J., Rodemich E. R., Rumsey H., Jr., Welch L.R., New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities, IEEE Trans. Inform. Theory IT-23 (1977), 157–166.

    Google Scholar 

  41. Neumaier A., Combinatorial Configurations in Terms of Distances, Memorandum 81-09 (Wiskunde), Eindhoven Univ. Technol., 1981.

  42. Milnor J., Husemoller D., Symmetric Bilinear Forms, Springer-Verlag, New York, 1973.

    Google Scholar 

  43. Odlyzko A. M., Sloane N. J. A., New upper bounds on the number of units spheres that can touch a unit sphere in n dimensions, J. Combin. Th. (A) 26 (1979), 210–214.

    Google Scholar 

  44. Schoenberg I. J., Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–107.

    Google Scholar 

  45. Semakov N. V., Zinov'ev V. A., Equidistant q-ary codes and resolved balanced imcomplete block designs, Problems of Information Transmission, 4: (2) (1968).

  46. Semakov N. V., Zinov'ev V. A., Zaitsev G. V., Class of maximal equidistant codes, Problems of Information Transmission 5: (2) (1969).

  47. Semakov N. V., Zinov'ev V. A., Equally-weighted codes and tactical configurations, Problems of Information Transmission 5 (3) (1969) 22–28.

    Google Scholar 

  48. Sidelnikov V. M., On extremal polynomials used to estimate the size of codes, Problems of Information Transmission 16: (3) (1980), 174–186.

    Google Scholar 

  49. Sloane N. J. A., Recent bounds for codes, sphere packing and related problems obtained by linear programming and others methods, Contemporary Mathematics 9 (1982), 153–185.

    Google Scholar 

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

    Google Scholar 

  51. Stanton D., A partially ordered set and q-Krawtchouk polynomials, J. Combin. Th.(A) 30 (1981), 276–284.

    Google Scholar 

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

    Google Scholar 

  53. Szego G., Orthogonal Polynomials, AMS Col. Publ., vol. 23, Providence, RI, 1939.

  54. Terwilliger P., A characterization of P-and Q-polynomial schemes, J. Combin. Th. (A) 45 (1987), 8–26.

    Google Scholar 

  55. Thas J. A., On 4-gonal configuration with parameters r=q 2 +1 and k=q+1. Part I, Geom. Dedicata 3 (1974), 365–375.

    Google Scholar 

  56. Tietavainen A., Covering radius problems and character sums, In: Proc. Fourth Joint Swedish-Soviet International Workshop on Information Theory, Gotland, Sweden, 1989, pp. 196–198.

  57. Vilenkin N. J., Special Functions and the Theory of Group Representations, Amer. Math. Soc., Providence, RI, 1968.

    Google Scholar 

  58. Wang H.-C., Two-point homogeneous spaces, Ann. Math. 55 (1952), 177–191.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Keldysh Institute for Applied Mathematics, Miusskaya sq. 4, 125047, Moscow, Russia

    V. I. Levenshtein

Authors
  1. V. I. Levenshtein
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levenshtein, V.I. Designs as maximum codes in polynomial metric spaces. Acta Appl Math 29, 1–82 (1992). https://doi.org/10.1007/BF00053379

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00053379

Mathematics Subject Classifications (1991)

Search

Navigation