\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Infinite families of 2-designs from a class of non-binary Kasami cyclic codes

  • * Corresponding author: Xiaoni Du

    * Corresponding author: Xiaoni Du 
The second author is supported by NSFC grant No. 61772022. The third author is supported by NSFC grant No. 11971395
Abstract Full Text(HTML) Figure(0) / Table(10) Related Papers Cited by
  • Combinatorial $ t $-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $ t $-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $ t $-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $ t $-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of $ 2 $-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.

    Mathematics Subject Classification: Primary: 05B05, 94B05, 11T23, 11T71.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is odd

    Weight Multiplicity
    $ 0 $ $ 1 $
    $ (p-1)(p^{m-1}-p^{s-1}) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
    $ p^{m-1}(p-1)+p^{s-1} $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
    $ (p-1)(p^{m-1}+p^{s-1}) $ $ \frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
    $ p^{m-1}(p-1)-p^{s-1} $ $ \frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
    $ p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
    $ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
    $ p^{m-1}(p-1)-p^{s+d-1} $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
    $ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d} $
    $ -p^{m-2d}+1)(p^m-1) $
    $ p^m $ $ p-1 $
     | Show Table
    DownLoad: CSV

    Table 2.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is even

    Weight Multiplicity
    $ 0 $ $ 1 $
    $ p^{s-1}(p-1)(p^s-1) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
    $ p^{s-1}(p^{s+1}-p^s+1) $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
    $ p^{s-1}(p-1)(p^s+1) $ $ p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
    $ p^{s-1}(p^{s+1}-p^s-1) $ $ p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
    $ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1) $ $ \frac{1}{2}p^{3s-2d}(p^m-1) $
    $ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1) $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
    $ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
    $ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
    $ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d} $
    $ +1)(p^m-1) $
    $ p^m $ $ p-1 $
     | Show Table
    DownLoad: CSV

    Table 3.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = 2d $

    Weight Multiplicity
    $ 0 $ $ 1 $
    $ p^{s-1}(p-1)(p^s+1) $ $ \frac{p^{m+3d}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
    $ p^{s-1}(p^{s+1}-p^s-1) $ $ \frac{p^{m+3d}(p-1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
    $ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ \frac{p^{m-d}(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
    $ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ \frac{p^{m-d}(p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
    $ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ \frac{p^{m-4d}(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
    $ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ \frac{p^{m-4d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
    $ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d} $
    $ +p^{3s-5d}+p^{m-d}-2p^{m-2d}+p^{m-3d} $
    $ -p^{m-4d}+1)(p^m-1) $
    $ p^m $ $ p-1 $
     | Show Table
    DownLoad: CSV

    Table 4.  The value of $ T(a,b,c,h) $ when $ d' = d $ is odd ($ j\in \mathbb{F}_p $)

    Value Corresponding Condition
    $ p^{m-1}+\varepsilon p^{s-1}(p-1) $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0 $
    $ p^{m-1}-\varepsilon p^{s-1} $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0 $
    $ 0 $ $ S(a,b,c)= p^m\; \mathrm{and}\; h\neq0 $
    $ p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon $
    $ p^{m-1}-p^{s+d-1}(p-1) $ $ S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0 $
    $ p^{m-1}+p^{s+d-1} $ $ S(a,b,c) $ $ =-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0 $
    $ p^{m-1} $ $ S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and} $
    $ h+j = 0 $
    $ p^m $ $ S(a,b,c)=p^m\; \mathrm{and}\; h = 0 $
     | Show Table
    DownLoad: CSV

    Table 5.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is odd

    Value Multiplicity
    $ (p-1)(p^{m-1}-p^{s-1}) $ $ M_1+(p-1)M_3 $
    $ p^{m-1}(p-1)+p^{s-1} $ $ (p-1)M_1+(p-1)^2M_3 $
    $ (p-1)(p^{m-1}+p^{s-1}) $ $ M_2+(p-1)M_4 $
    $ p^{m-1}(p-1)-p^{s-1} $ $ (p-1)M_2+(p-1)^2M_4 $
    $ p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ (p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1}) $
    $ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
    $ p^{m-1}(p-1)-p^{s+d-1} $ $ (p-1)M_7+(p-1)^2M_8 $
    $ p^{m-1}(p-1) $ $ pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1}) $
    $ p^m $ $ p-1 $
     | Show Table
    DownLoad: CSV

    Table 6.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is even

    Value Multiplicity
    $ p^{s-1}(p-1)(p^s-1) $ $ M_1+(p-1)M_3 $
    $ p^{s-1}(p^{s+1}-p^s+1) $ $ (p-1)M_1+(p-1)^2M_3 $
    $ p^{s-1}(p-1)(p^s+1) $ $ M_2+(p-1)M_4 $
    $ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_2+(p-1)^2M_4 $
    $ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1) $ $ (p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1} $
    $ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1) $ $ M_{5,\mp1}+(p-1)M_{6,\mp1} $
    $ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
    $ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ (p-1)M_7+(p-1)^2M_8 $
    $ p^{m-1}(p-1) $ $ pM_9 $
    $ p^m $ $ p-1 $
     | Show Table
    DownLoad: CSV

    Table 7.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = 2d $

    Value Multiplicity
    $ p^{s-1}(p-1)(p^s+1) $ $ M_1+(p-1)M_2 $
    $ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_1+(p-1)^2M_2 $
    $ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ M_3+(p-1)M_4 $
    $ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ (p-1)M_3+(p-1)^2M_4 $
    $ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ M_5+(p-1)M_6 $
    $ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ (p-1)M_5+(p-1)^2M_6 $
    $ p^{m-1}(p-1) $ $ pM_7 $
    $ p^m $ $ p-1 $
     | Show Table
    DownLoad: CSV

    Table 8.  The value distribution of $ S(a,b,c) $ when $ d' = d $ is odd

    Value Multiplicity
    $ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
    $ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
    $ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
    $ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
    $ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}} $ $ M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1) $
    $ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1) $
    $ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
    $ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1) $
    $ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
    $ p^m $ $ 1 $
     | Show Table
    DownLoad: CSV

    Table 9.  The value distribution of $ S(a,b,c) $ when $ d' = d $ is even

    Value Multiplicity
    $ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
    $ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
    $ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
    $ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
    $ \varepsilon p^{s+\frac{d}{2}} $ $ M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1) $
    $ \varepsilon p^{s+\frac{d}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1) $
    $ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
    $ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1) $
    $ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
    $ p^m $ $ 1 $
     | Show Table
    DownLoad: CSV

    Table 10.  The value distribution of $ S(a,b,c) $ when $ d' = 2d $

    Value Multiplicity
    $ -p^s $ $ M_1=\frac{p^{s+3d-1}(p^s-1)(p^s-p+1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)}{(p^d+1)(p^{2d}-1)} $
    $ -\zeta^j_pp^s $ $ M_2=\frac{p^{s+3d-1}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
    $ p^{s+d} $ $ M_3=\frac{p^{s-1}(p^{s-d}+p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
    $ p^{s+d}\zeta^j_p $ $ M_4=\frac{p^{s-1}(p^{s-d}-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
    $ -p^{s+2d} $ $ M_5=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}-p+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
    $ -p^{s+2d}\zeta^j_p $ $ M_6=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
    $ 0 $ $ M_7=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}+p^{3s-5d}+p^{m-d}- $
    $ 2p^{m-2d}+p^{m-3d}-p^{m-4d}+1)(p^m-1) $
    $ p^m $ $ 1 $
     | Show Table
    DownLoad: CSV
  • [1] E. F. Assmus Jr. and  J. D. KeyDesigns and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
    [2] E. F. Assmus Jr. and H. F. Mattson Jr, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-152.  doi: 10.1016/S0021-9800(69)80115-8.
    [3] E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.
    [4] M. Antweiler and L. Bömer, Complex sequences over GF$ {(p^M)} $ with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory, 38 (1992), 120-130.  doi: 10.1109/18.108256.
    [5] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. II. Encyclopedia of Mathematics and its Applications, Vol. 78, 2nd edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139507660.003.
    [6] C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.
    [7] C. Ding, Infinite families of $3$-designs from a type of five-weight code, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.
    [8] C. Ding and C. Li, Infinite families of $2$-designs and $3$-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.
    [9] K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.
    [10] X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of linear codes, preprint, arXiv: 1903.07459.
    [11] X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros, preprint, arXiv: 1903.08153.
    [12] X. DuR. Wang and C. Fan, Infinite families of $2$-designs from a class of cyclic codes, J. Comb. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.
    [13] R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110.  doi: 10.1016/j.ffa.2004.06.002.
    [14] W. C. Huffman and  V. PlessFundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
    [15] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, $2^nd$ edition, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.
    [16] T. KasamiS. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control, 11 (1967), 475-496.  doi: 10.1016/S0019-9958(67)90691-2.
    [17] J. Luo, Y. Tang, and H. Wang, Exponential sums, cycle codees and sequences: the odd characteristic Kasami case, preprint, arXiv: 0902.4508v1 [cs.IT].
    [18] R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997.
    [19] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, I, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
    [20] C. Reid and A. Rosa, Steiner systems $ {S} (2, 4, v) $-a survey, The Electronic Journal of Combinatorics, 18 (2010), 1–34. https://www.researchgate.net/publication/266996333. doi: 10.37236/39.
    [21] J. Serrin, C. J. Colbourn and R. Mathon, Steiner systems, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, (2006), 128–135. https://www.researchgate.net/publication/329786723.
    [22] V. D. Tonchev, Codes and designs, in Handbook of Coding Theory, Vol. I, II North-Holland, Amsterdam, (1998), 1229–1267. https://www.researchgate.net/publication/268549395.
    [23] V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.
    [24] M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.
  • 加载中

Tables(10)

SHARE

Article Metrics

HTML views(1417) PDF downloads(266) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return