Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes


In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in Ding and Li (Discret Math 340(10):2415–2431, 2017) are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed–Muller codes.

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  1. 1.

    More generally, a code \({\mathsf {C}}\) holds (or supports) a t-\((v,k,\lambda )\) design \({{\mathbb {D}}}\) if every block of \({{\mathbb {D}}}\) is the support of some codeword of \({\mathsf {C}}\) [23].


  1. 1.

    Assmus Jr. E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).

    Book  Google Scholar 

  2. 2.

    Assmus Jr. E.F., Key J.D.: Polynomial codes and finite geometries. In: Pless V.S., Huffman W.C. (eds.) The Handbook of Coding Theory, vol. II, pp. 1269–1343. Elsevier, Amsterdam (1998).

    Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1999).

    Book  Google Scholar 

  5. 5.

    Ceccherini P.V., Hirschfeld J.W.P.: The dimension of projective geometry codes. Discret. Math. 107, 117–126 (1992).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Charpin P.: Codes cycliques étendus affines-invariants et antichaînes d’un ensemble partiellement ordonné. Discret. Math. 80, 229–247 (1990).

    Article  Google Scholar 

  7. 7.

    Delsarte P.: On subfield subcodes of modified Reed–Solomon codes. IEEE Trans. Inf. Theory 21(5), 575–576 (1975).

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ding C.: Codes from Difference Sets. World Scientific, Singapore (2015).

    Google Scholar 

  9. 9.

    Ding C.: Designs from Linear Codes. World Scientific, Singapore (2018).

    Book  Google Scholar 

  10. 10.

    Ding C., Li C.: Infinite families of 2-designs and 3-designs from linear codes. Discret. Math. 340(10), 2415–2431 (2017).

    MathSciNet  Article  Google Scholar 

  11. 11.

    Du X., Wang R., Fan C.: Infinite families of \(2\)-designs from a class of cyclic codes with two non-zeros. arXiv:1904.04242 [math.CO] (2019).

  12. 12.

    Grassl M.: Code Tables.

  13. 13.

    Hamada N.: On the \(p\)-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J. 3, 153–226 (1973).

    MathSciNet  Article  Google Scholar 

  14. 14.

    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  Google Scholar 

  15. 15.

    Jungnickel D., Tonchev V.D.: Counting Steiner triple systems with classical parameters and prescribed rank. J. Comb. Theory Ser. A 162, 10–33 (2019).

    MathSciNet  Article  Google Scholar 

  16. 16.

    Jungnickel D., Magliveras S.S., Tonchev V.D., Wassermann A.: The classification of Steiner triple systems on 27 points with 3-rank 24. Des. Codes Cryptogr. 87, 831–839 (2019).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kasami T., Lin S., Peterson W.: Some results on cyclic codes which are invariant under the affine group and their applications. Inf. Control 11, 475–496 (1968).

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kennedy G.T., Pless V.: A coding-theoretic approach to extending designs. Discret. Math. 142, 155–168 (1995).

    MathSciNet  Article  Google Scholar 

  19. 19.

    Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics and Its Application, vol. 20. Cambridge University Press, Cambridge (1997).

    Google Scholar 

  20. 20.

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

    MATH  Google Scholar 

  21. 21.

    Osuna O.P.: There are 1239 Steiner triple systems \(STS(31)\) of 2-rank 27. Des. Codes Cryptogr. 40(2), 187–190 (2006).

    MathSciNet  Article  Google Scholar 

  22. 22.

    Tonchev V.D.: Quasi-symmetric designs, codes, quadrics, and hyperplane sections. Geom. Dedic. 48, 295–308 (1993).

    MathSciNet  Article  Google Scholar 

  23. 23.

    Tonchev V.D.: Codes and designs. In: Pless V.S., Huffman W.C. (eds.) The Handbook of Coding Theory, vol. II, pp. 1229–1268. Elsevier, Amsterdam (1998).

    Google Scholar 

  24. 24.

    Tonchev V.D.: A mass formula for Steiner triple systems \(STS(2^n-1)\) of 2-rank \(2^n-n\). J. Comb. Theory Ser. A 95, 197–208 (2001).

    Article  Google Scholar 

  25. 25.

    Tonchev V.D.: A formula for the number of Steiner quadruple systems on \(2^n\) points of 2-rank \(2^n-n\). J. Comb. Des. 11, 260–274 (2003).

    Article  Google Scholar 

  26. 26.

    Tonchev V.D.: Codes. In: Colbourn C.J., Dinitz J.H. (eds.) The Handbook of Combinatorial Designs, 2nd edn, pp. 677–701. CRC Press, New York (2007).

    Google Scholar 

  27. 27.

    Zinoviev D.V.: The number of Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+2\) over \({\mathbb{F}}_2\). Discret. Math. 339, 2727–2736 (2016).

    Article  Google Scholar 

  28. 28.

    Zinoviev V.A., Zinoviev D.V.: Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+1\) over \({\mathbb{F}}_2\). Probl. Inf. Transm. 48, 102–126 (2012).

    MathSciNet  Article  Google Scholar 

  29. 29.

    Zinoviev V.A., Zinoviev D.V.: Structure of Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+2\) over \({\mathbb{F}}_2\). Probl. Inf. Transm. 49, 232–248 (2013).

    MathSciNet  Article  Google Scholar 

  30. 30.

    Zinoviev V.A., Zinoviev D.V.: Remark on “Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+1\) over \({\mathbb{F}}_2\) published in Probl. Peredachi Inf., 2012, no. 2. Probl. Inf. Transm. 49, 107–111 (2013).

    Google Scholar 

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The authors are grateful to the reviewers and the Editors for their comments and suggestions that improved the presentation of this paper.

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Correspondence to Chunming Tang.

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C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16300418. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds).

Communicated by J. D. Key.

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Ding, C., Tang, C. & Tonchev, V.D. Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes. Des. Codes Cryptogr. 88, 625–641 (2020).

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  • Cyclic code
  • Linear code
  • Reed–Muller code
  • t-design

Mathematics Subject Classification

  • 94B05
  • 94B15
  • 05B05