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doi: 10.3934/amc.2022043
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Infinite families of $ t $-designs and strongly regular graphs from punctured codes

Ziling Heng , , Dexiang Li , Fenjin Liu and  Weiqiong Wang

School of Science, Chang'an University, Xi'an 710064, China

*Corresponding author: Ziling Heng

Received  February 2022 Revised  May 2022 Early access June 2022

Fund Project: The first author was supported in part by the National Natural Science Foundation of China under Grant 11901049, in part by the Young Talent Fund of University Association for Science and Technology in Shaanxi, China, under Grant 20200505 and in part by the Fundamental Research Funds for the Central Universities, CHD, under Grant 300102122202. The third author was supported by the Natural Science Basic Research Program of Shaanxi, China, under Grant 2021JM-149

The puncturing technique is sometimes efficient in constructing projective codes from original codes which are not projective. In this paper, several families of projective linear codes punctured from reducible cyclic codes, special linear codes or irreducible cyclic codes are investigated. The parameters and weight enumerators of the punctured codes and their duals are explicitly determined. Some of the codes are optimal and some of the codes are self-orthogonal which can be used to construct quantum codes. Several infinite families of combinatorial $ 2 $-designs and $ 3 $-designs including some families of Steiner systems are constructed from the punctured codes and their duals. Besides, infinite families of strongly regular graphs are also derived from some families of two-weight projective codes.

Keywords: Combinatorial design, linear code, strongly regular graph.
Mathematics Subject Classification: Primary: 94B05; Secondary: 05B05.
Citation: Ziling Heng, Dexiang Li, Fenjin Liu, Weiqiong Wang. Infinite families of $ t $-designs and strongly regular graphs from punctured codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022043
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E. F. Assmus and H. F. Mattson, Coding and combinatorics, SIAM Review, 16 (1974), 349-388.  doi: 10.1137/1016056.

[2]

I. Bouyukliev, V. Fack, W. Willems and J. Winne, Projective two-weight codes with small parameters and their corresponding graphs, Des. Codes Cryptogr., 41 (2006), 59-78.  doi: 10.1007/s10623-006-0019-1.

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C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

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D. Crnkovi$\acute{c}$, A. $\check{S}$vob and V. D. Tonchev, Cyclotomic trace codes, Algorithms, 12 (2019), Paper No. 168, 10 pp. doi: 10.3390/a12080168.

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

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C. Ding, Designs from Linear Codes, orld Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.

[8]

C. Ding, An infinite family of Steiner systems $S(2, 4, 2^m)$ from cyclic codes, J. Combin. Des., 26 (2018), 127-144.  doi: 10.1002/jcd.21565.

[9]

C. Ding, Infinite families of 3-designs from a type of five-weight codes, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.

[10]

C. Ding and C. Li, Infinite families of 2-designs and 3-designs from linear codes, Disc. Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.

[11]

C. Ding, C. Li and Y. Xia, Another generalisation of the binary Reed–Muller codes and its applications, Finite Fields Appl., 53 (2018), 144-174.  doi: 10.1016/j.ffa.2018.06.006.

[12]

C. Ding, A. Munemasa and V. Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inf. Theory, 65 (2019), 7533-7541.  doi: 10.1109/TIT.2019.2922401.

[13]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.

[14]

C. Ding and C. Tang, Combinatorial $t$-designs from special functions, Cryptogr. Commun., 12 (2020), 1011-1033.  doi: 10.1007/s12095-020-00442-2.

[15]

C. Ding and C. Tang, Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inf. Theory, 66 (2020), 5419-5428.  doi: 10.1109/TIT.2020.2990396.

[16]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.

[17]

X. Du, R. Wang and C. Fan, Infinite families of 2-designs from a class of cyclic codes, J. Combin. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[18]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of 2-designs from linear codes, Appl. Algebra Engrg. Comm. Comput., 33 (2022), 193-211.  doi: 10.1007/s00200-020-00438-8.

[19]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros, Adv. Math. Commun., 16 (2022), 157-168.  doi: 10.3934/amc.2020106.

[20]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409.  doi: 10.1016/j.ffa.2007.03.003.

[21]

J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.

[22]

Z. Heng, C. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.

[23]

Z. Heng and Q. Yue, Complete weight distributions of two classes of cyclic codes, Cryptogr. Commun., 9 (2017), 323-343.  doi: 10.1007/s12095-015-0177-y.

[24]

Z. Heng and Q. Yue, A construction of $q$-ary linear codes with two weights, Finite Fields Appli., 48 (2017), 20-42.  doi: 10.1016/j.ffa.2017.07.006.

[25] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[26]

A. Ketkar, A. Klappenecker and S. Kumar, Nonbinary stablizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.

[27]

C. Li, S. Bae, J. Ahn, S. Yang and Z. Yao, Complete eweight enumerators of some linear codes and their applications, Des. Codes Cryptogr., 81 (2016), 153-168.  doi: 10.1007/s10623-015-0136-9.

[28]

C. Li, Q. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[29]

H. Liang, W. Chen and Y. Tang, A class of three-weight cyclic codes, J. of Math. (PRC), 36 (2016), 474-480. 

[30]

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Boston, 1983.

[31]

G. McGuire, Quasi-symmetric designs and codes meeting the Grey-Rankin bound, J. Combin. Theory Ser. A, 78 (1997), 280-291.  doi: 10.1006/jcta.1997.2765.

[32]

S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptogr. Commun., 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.

[33]

C. Tang and C. Ding, An infinite family of linear codes supporting 4-designs, IEEE Trans. Inf. Theory, 67 (2021), 244-254.  doi: 10.1109/TIT.2020.3032600.

[34]

C. Tang, C. Ding and M. Xiong, Codes, differentially $\delta-$uniform functions, and $t$-designs, IEEE Trans. Inf. Theory, 66 (2020), 3691-3703.  doi: 10.1109/TIT.2019.2959764.

[35]

C. Tang, C. Ding and M. Xiong, Steiner systems $S(2, 4, \frac{3^m-1}{2})$ and $2$-designs from ternary linear codes of length $\frac{3^m-1}{2}$, Designs Codes Cryptogr., 87 (2019), 2793-2811.  doi: 10.1007/s10623-019-00651-8.

[36]

C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.

[37]

C. Tang, Y. Qi and M. Huang, Two-weight and three-weight linear codes from square functions, IEEE Commun. Lett., 20 (2016), 29-32. 

[38]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.  doi: 10.1016/j.ffa.2013.08.005.

[39]

Z. Zhou, N. Li, C. Fan and T. Helleseth, Linear codes with two or three weights from quadratic bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.

show all references

References:
[1]

E. F. Assmus and H. F. Mattson, Coding and combinatorics, SIAM Review, 16 (1974), 349-388.  doi: 10.1137/1016056.

[2]

I. Bouyukliev, V. Fack, W. Willems and J. Winne, Projective two-weight codes with small parameters and their corresponding graphs, Des. Codes Cryptogr., 41 (2006), 59-78.  doi: 10.1007/s10623-006-0019-1.

[3]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.

[4]

C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[5]

D. Crnkovi$\acute{c}$, A. $\check{S}$vob and V. D. Tonchev, Cyclotomic trace codes, Algorithms, 12 (2019), Paper No. 168, 10 pp. doi: 10.3390/a12080168.

[6]

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

[7]

C. Ding, Designs from Linear Codes, orld Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.

[8]

C. Ding, An infinite family of Steiner systems $S(2, 4, 2^m)$ from cyclic codes, J. Combin. Des., 26 (2018), 127-144.  doi: 10.1002/jcd.21565.

[9]

C. Ding, Infinite families of 3-designs from a type of five-weight codes, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.

[10]

C. Ding and C. Li, Infinite families of 2-designs and 3-designs from linear codes, Disc. Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.

[11]

C. Ding, C. Li and Y. Xia, Another generalisation of the binary Reed–Muller codes and its applications, Finite Fields Appl., 53 (2018), 144-174.  doi: 10.1016/j.ffa.2018.06.006.

[12]

C. Ding, A. Munemasa and V. Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inf. Theory, 65 (2019), 7533-7541.  doi: 10.1109/TIT.2019.2922401.

[13]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.

[14]

C. Ding and C. Tang, Combinatorial $t$-designs from special functions, Cryptogr. Commun., 12 (2020), 1011-1033.  doi: 10.1007/s12095-020-00442-2.

[15]

C. Ding and C. Tang, Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inf. Theory, 66 (2020), 5419-5428.  doi: 10.1109/TIT.2020.2990396.

[16]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.

[17]

X. Du, R. Wang and C. Fan, Infinite families of 2-designs from a class of cyclic codes, J. Combin. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[18]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of 2-designs from linear codes, Appl. Algebra Engrg. Comm. Comput., 33 (2022), 193-211.  doi: 10.1007/s00200-020-00438-8.

[19]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros, Adv. Math. Commun., 16 (2022), 157-168.  doi: 10.3934/amc.2020106.

[20]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409.  doi: 10.1016/j.ffa.2007.03.003.

[21]

J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542.  doi: 10.1147/rd.45.0532.

[22]

Z. Heng, C. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.

[23]

Z. Heng and Q. Yue, Complete weight distributions of two classes of cyclic codes, Cryptogr. Commun., 9 (2017), 323-343.  doi: 10.1007/s12095-015-0177-y.

[24]

Z. Heng and Q. Yue, A construction of $q$-ary linear codes with two weights, Finite Fields Appli., 48 (2017), 20-42.  doi: 10.1016/j.ffa.2017.07.006.

[25] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[26]

A. Ketkar, A. Klappenecker and S. Kumar, Nonbinary stablizer codes over finite fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.

[27]

C. Li, S. Bae, J. Ahn, S. Yang and Z. Yao, Complete eweight enumerators of some linear codes and their applications, Des. Codes Cryptogr., 81 (2016), 153-168.  doi: 10.1007/s10623-015-0136-9.

[28]

C. Li, Q. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[29]

H. Liang, W. Chen and Y. Tang, A class of three-weight cyclic codes, J. of Math. (PRC), 36 (2016), 474-480. 

[30]

R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Boston, 1983.

[31]

G. McGuire, Quasi-symmetric designs and codes meeting the Grey-Rankin bound, J. Combin. Theory Ser. A, 78 (1997), 280-291.  doi: 10.1006/jcta.1997.2765.

[32]

S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptogr. Commun., 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.

[33]

C. Tang and C. Ding, An infinite family of linear codes supporting 4-designs, IEEE Trans. Inf. Theory, 67 (2021), 244-254.  doi: 10.1109/TIT.2020.3032600.

[34]

C. Tang, C. Ding and M. Xiong, Codes, differentially $\delta-$uniform functions, and $t$-designs, IEEE Trans. Inf. Theory, 66 (2020), 3691-3703.  doi: 10.1109/TIT.2019.2959764.

[35]

C. Tang, C. Ding and M. Xiong, Steiner systems $S(2, 4, \frac{3^m-1}{2})$ and $2$-designs from ternary linear codes of length $\frac{3^m-1}{2}$, Designs Codes Cryptogr., 87 (2019), 2793-2811.  doi: 10.1007/s10623-019-00651-8.

[36]

C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.

[37]

C. Tang, Y. Qi and M. Huang, Two-weight and three-weight linear codes from square functions, IEEE Commun. Lett., 20 (2016), 29-32. 

[38]

Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.  doi: 10.1016/j.ffa.2013.08.005.

[39]

Z. Zhou, N. Li, C. Fan and T. Helleseth, Linear codes with two or three weights from quadratic bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.

Table 1.  The weight distribution of $ \widehat{ {\mathcal{C}}} $ in Lemma 3.1
Weight Frequency
$ 0 $ $ 1 $
$ p^{m-1}-p^{(m+e-2)/2} $ $ (p^m-1)(p^{m-e}+p^{(m-e)/2})/2 $
$ p^{m-1} $ $ (p^m-1)(p^m-p^{m-e}+1) $
$ p^{m-1}+p^{(m+e-2)/2} $ $ (p^m-1)(p^{m-e}-p^{(m-e)/2})/2 $
Table Options

Download as excel

Weight Frequency
$ 0 $ $ 1 $
$ p^{m-1}-p^{(m+e-2)/2} $ $ (p^m-1)(p^{m-e}+p^{(m-e)/2})/2 $
$ p^{m-1} $ $ (p^m-1)(p^m-p^{m-e}+1) $
$ p^{m-1}+p^{(m+e-2)/2} $ $ (p^m-1)(p^{m-e}-p^{(m-e)/2})/2 $
Table 2.  Examples of $ \widehat{ {\mathcal{C}}}^\bot $ for $ e = 1 $ and even $ l $
$ m $ $ p $ Parameters of $ \widehat{ {\mathcal{C}}}^\bot $ Optimality
$ 3 $ $ 3 $ $ [13,7,4] $ Almost Optimal
$ 3 $ $ 5 $ $ [31,25,4] $ Best Known
$ 3 $ $ 7 $ $ [57,51,4] $ Best Known
$ 5 $ $ 3 $ $ [121,111,4] $ Almost Optimal
Table Options

Download as excel

$ m $ $ p $ Parameters of $ \widehat{ {\mathcal{C}}}^\bot $ Optimality
$ 3 $ $ 3 $ $ [13,7,4] $ Almost Optimal
$ 3 $ $ 5 $ $ [31,25,4] $ Best Known
$ 3 $ $ 7 $ $ [57,51,4] $ Best Known
$ 5 $ $ 3 $ $ [121,111,4] $ Almost Optimal
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