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Infinite families of 2-designs from a class of non-binary Kasami cyclic codes
1. | College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China |
2. | Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China |
3. | School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610000, China |
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.
References:
[1] |
E. F. Assmus Jr. and J. D. Key, Designs 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. Google Scholar |
[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. Google Scholar |
[12] |
X. Du, R. 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. Pless, Fundamentals 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. Kasami, S. 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]. Google Scholar |
[18] |
R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |
show all references
References:
[1] |
E. F. Assmus Jr. and J. D. Key, Designs 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. Google Scholar |
[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. Google Scholar |
[12] |
X. Du, R. 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. Pless, Fundamentals 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. Kasami, S. 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]. Google Scholar |
[18] |
R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |
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