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An Assmus–Mattson theorem for codes over commutative association schemes

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Abstract

We prove an Assmus–Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with s classes). This in particular generalizes the Assmus–Mattson-type theorems for \(\mathbb {Z}_4\)-linear codes due to Tanabe (Des Codes Cryptogr 30:169–185, 2003) and Shin et al. (Des Codes Cryptogr 31:75–92, 2004), as well as the original theorem by Assmus and Mattson (J Comb Theory 6:122–151, 1969). The weights of a code are s-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining t-designs from the code involve concepts from polynomial interpolation in s variables. The Terwilliger algebra is the main tool to establish our results.

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Notes

  1. This term is meant as only provisional; cf. [33].

References

  1. Assmus Jr. E.F., Mattson Jr. H.F.: New \(5\)-designs. J. Comb. Theory 6, 122–151 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachoc C.: On harmonic weight enumerators of binary codes. Des. Codes Cryptogr. 18, 11–28 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bannai E., Bannai E., Suda S., Tanaka H.: On relative \(t\)-designs in polynomial association schemes. Electron. J. Comb. 22, #P4.47 (2015). arXiv:1303.7163.

  4. Bannai E., Ito T.: Algebraic Combinatorics I: Association Schemes. Benjamin/Cummings, Menlo Park (1984).

    MATH  Google Scholar 

  5. Bannai E., Koike M., Shinohara M., Tagami M.: Spherical designs attached to extremal lattices and the modulo \(p\) property of Fourier coefficients of extremal modular forms. Mosc. Math. J. 6, 225–264 (2006).

    MathSciNet  MATH  Google Scholar 

  6. Bonnecaze A., Rains E., Solé P.: \(3\)-Colored \(5\)-designs and \(\mathbf{Z}_4\)-codes. J. Stat. Plan. Inference 86, 349–368 (2000).

    Article  MATH  Google Scholar 

  7. Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989).

    Book  MATH  Google Scholar 

  8. Calderbank A.R., Delsarte P.: On error-correcting codes and invariant linear forms. SIAM J. Discret. Math. 6, 1–23 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  9. Calderbank A.R., Delsarte P., Sloane N.J.A.: A strengthening of the Assmus-Mattson theorem. IEEE Trans. Inf. Theory 37, 1261–1268 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  10. de Boor C., Ron A.: On multivariate polynomial interpolation. Constr. Approx. 6, 287–302 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  11. de Boor C., Ron A.: The least solution for the polynomial interpolation problem. Math. Z. 210, 347–378 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  12. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 1–97 (1973).

    MathSciNet  MATH  Google Scholar 

  13. Delsarte P., Levenshtein V.I.: Association schemes and coding theory. IEEE Trans. Inf. Theory 44, 2477–2504 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  14. Gasca M., Sauer T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12, 377–410 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  15. Gijswijt D., Schrijver A., Tanaka H.: New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Comb. Theory A 113, 1719–1731 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  16. Godsil C.D.: Generalized Hamming schemes, manuscript (2010). arXiv:1011.1044.

  17. Gulliver T.A., Harada M.: Extremal double circulant type II codes over \(\mathbb{Z}_4\) and construction of \(5\)-\((24,10,36)\) designs. Discret. Math. 194, 129–137 (1999).

    Article  MATH  Google Scholar 

  18. Harada M.: New \(5\)-designs constructed from the lifted Golay code over \(\mathbb{Z}_4\). J. Comb. Des. 6, 225–229 (1998).

    Article  MATH  Google Scholar 

  19. Helleseth T., Rong C., Yang K.: On \(t\)-designs from codes over \(\mathbf{Z}_4\). Discret. Math. 238, 67–80 (2001).

    Article  MATH  Google Scholar 

  20. Höhn G.: Self-dual codes over the Kleinian four group. Math. Ann. 327, 227–255 (2003). arXiv:math/0005266.

    Article  MathSciNet  MATH  Google Scholar 

  21. Iliev P.: A Lie-theoretic interpretation of multivariate hypergeometric polynomials. Compos. Math. 148, 991–1002 (2012). arXiv:1101.1683.

    Article  MathSciNet  MATH  Google Scholar 

  22. Iliev P., Terwilliger P.: The Rahman polynomials and the Lie algebra \(\mathfrak{sl}_3(\mathbb{C})\). Trans. Am. Math. Soc. 364, 4225–4238 (2012). arXiv:1006.5062.

    Article  MathSciNet  MATH  Google Scholar 

  23. Kim J.-L., Pless V.: Designs in additive codes over \(GF(4)\). Des. Codes Cryptogr. 30, 187–199 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  24. Lahtonen J., Ranto K., Vehkalahti R.: \(3\)-designs from all \(\mathbf{Z}_4\)-Goethals-like codes with block size \(7\) and \(8\). Finite Fields Appl. 13, 815–827 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  25. Martin W.J., Tanaka H.: Commutative association schemes. Eur. J. Comb. 30, 1497–1525 (2009). arXiv:0811.2475.

    Article  MathSciNet  MATH  Google Scholar 

  26. Miezaki T., Nakasora H.: An upper bound of the value of \(t\) of the support \(t\)-designs of extremal binary doubly even self-dual codes. Des. Codes Cryptogr. 79, 37–46 (2016). arXiv:1311.2122.

    Article  MathSciNet  MATH  Google Scholar 

  27. Mizukawa H., Tanaka H.: \((n+1, m+1)\)-Hypergeometric functions associated to character algebras. Proc. Am. Math. Soc. 132, 2613–2618 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  28. Morales J.V.S.: On Lee association schemes over \(\mathbb{Z}_4\) and their Terwilliger algebra. Linear Algebra Appl. 510, 311–328 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  29. Schrijver A.: New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inf. Theory 51, 2859–2866 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  30. Shin D.-J., Kumar P.V., Helleseth T.: An Assmus-Mattson-type approach for identifying \(3\)-designs from linear codes over \(\mathbf{Z}_4\). Des. Codes Cryptogr. 31, 75–92 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  31. Simonis J.: MacWilliams identities and coordinate partitions. Linear Algebra Appl. 216, 81–91 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  32. Solé P.: The Lee association scheme. In: Cohen G., Godlewski P. (eds.) Coding Theory and Applications. Lecture Notes in Computer Science, vol. 311, pp. 45–55. Springer, Berlin (1988).

    Google Scholar 

  33. Srivastava J.N., Chopra D.V.: Balanced arrays and orthogonal arrays. In: Srivastava J.N. (ed.) A Survey of Combinatorial Theory, pp. 411–428. North-Holland, Amsterdam (1973).

    Chapter  Google Scholar 

  34. Tanabe K.: An Assmus-Mattson theorem for \(\mathbf{Z}_4\)-codes. IEEE Trans. Inf. Theory 46, 48–53 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  35. Tanabe K.: A new proof of the Assmus-Mattson theorem for non-binary codes. Des. Codes Cryptogr. 22, 149–155 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  36. Tanabe K.: A criterion for designs in \(\mathbb{Z}_4\)-codes on the symmetrized weight enumerator. Des. Codes Cryptogr. 30, 169–185 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  37. Tanaka H.: New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra. Eur. J. Comb. 30, 736–746 (2009). arXiv:math/0612740.

    Article  MathSciNet  MATH  Google Scholar 

  38. Tanaka H., Tanaka R., Watanabe Y.: The Terwilliger algebra of a \(Q\)-polynomial distance-regular graph with respect to a set of vertices (in preparation).

  39. Tarnanen H.: On extensions of association schemes. In: Laakso H., Salomaa A. (eds.) The Very Knowledge of Coding, pp. 128–142. University of Turku, Institute for Applied Mathematics, Turku (1987).

    Google Scholar 

  40. Terwilliger P.: The subconstituent algebra of an association scheme I. J. Algebr. Comb. 1, 363–388 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  41. Terwilliger P.: The subconstituent algebra of an association scheme II. J. Algebr. Comb. 2, 73–103 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  42. Terwilliger P.: The subconstituent algebra of an association scheme III. J. Algebr. Comb. 2, 177–210 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  43. Terwilliger P.: The displacement and split decompositions for a \(Q\)-polynomial distance-regular graph. Graphs Comb. 21, 263–276 (2005). arXiv:math.CO/0306142.

  44. Terwilliger P.: Six Lectures on Distance-regular Graphs. Lecture Notes. De La Salle University, Metro Manila. http://www.math.wisc.edu/~terwilli/teaching.html (2010).

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Acknowledgements

The authors thank Masaaki Harada for helpful discussions. HT was supported in part by JSPS KAKENHI Grant No. 25400034.

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Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    John Vincent S. Morales

  2. Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Hajime Tanaka

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  1. John Vincent S. Morales
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  2. Hajime Tanaka
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Correspondence to Hajime Tanaka.

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Communicated by T. Helleseth.

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Morales, J.V.S., Tanaka, H. An Assmus–Mattson theorem for codes over commutative association schemes. Des. Codes Cryptogr. 86, 1039–1062 (2018). https://doi.org/10.1007/s10623-017-0376-y

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  • DOI: https://doi.org/10.1007/s10623-017-0376-y

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