# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021022
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## Five-weight codes from three-valued correlation of M-sequences

 1 Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, China 3 The Selmer Center, Department of Informatics, University of Bergen, Bergen, Norway 4 I2M, CNRS, Centrale Marseille, University of Aix-Marseille, Marseilles, France

* Corresponding author: Minjia Shi

Received  October 2020 Revised  April 2021 Early access July 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (12071001), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20) and by The Research Council of Norway (247742/O70)

In this paper, for each of six families of three-valued $m$-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring $R = \mathbb{F}_2+u\mathbb{F}_2$, where $u^2 = 0.$ The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are $[2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)]$ for some explicit $r.$

Citation: Minjia Shi, Liqin Qian, Tor Helleseth, Patrick Solé. Five-weight codes from three-valued correlation of M-sequences. Advances in Mathematics of Communications, doi: 10.3934/amc.2021022
##### References:
 [1] A. Ashikmin and A. Barg, Minimal vectors in linear codes, IEEE Transactions on Information Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584. [2] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer New York, 2012. doi: 10.1007/978-1-4614-1939-6. [3] A. Canteaut, P. Charpin and H. Dobbertin, Binary $m$-sequences with three valued crosscorrelation: A proof of Welch's conjecture, IEEE Transactions on Information Theory, 46 (2000), 4-8.  doi: 10.1109/18.817504. [4] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Transactions on Information Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722. [5] T. W. Cusick and H. Dobbertin, Some new three-valued crosscorrelation functions for binary $m$-sequences, IEEE Transactions on Information Theory, 42 (1996), 1238-1240.  doi: 10.1109/18.508848. [6] C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, Coding and Cryptology, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, NJ, 4 (2008), 119–124. doi: 10.1142/9789812832245_0009. [7] C. Ding and J. Yuan, Covering and secret sharing with linear codes, Lecture Notes in Computer Science, 2731 (2003), 11-25.  doi: 10.1007/3-540-45066-1_2. [8] H. Dobbertin, Almost perfect nonlinear power functions on $GF(2^n)$: The Welch case, IEEE Transactions on Information Theory, 45 (1999), 1271-1275.  doi: 10.1109/18.761283. [9] R. Gold, Maximal recursive sequences with 3-valued cross-correlation functions, IEEE Transactions on Information Theory, 14 (1968), 154-156. [10] S. W. Golomb and G. Guang, Signal Design for Good Correlation for Wireless Communication, Cryptography, and Radar, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511546907. [11] J. H. Griesmer, A bound for error-Correcting codes, IBM Journal of Research & Development, 4 (1960), 532-542.  doi: 10.1147/rd.45.0532. [12] T. Helleseth, Some results about the cross-correlation between two maximal linear sequences, Discrete Mathematics, 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X. [13] T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, North-Holland, Amsterdam, 1 (1998), 1765–1853. [14] Y. Liu, M. J. Shi and P. Solé, Two-weight and three-weight codes from trace codes over $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p$, Discrete Math, 341 (2018), 350-357.  doi: 10.1016/j.disc.2017.09.003. [15] M. Pursley and D. Sarwate, Crosscorrelation properties of pseudorandom and related sequences, Proceedings of the IEEE, 68 (1980), 593-619. [16] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003. [17] T. Kasami, The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes, Information and Control, 18 (1971), 369-394.  doi: 10.1016/S0019-9958(71)90473-6. [18] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [19] Y. Niho, Multivalued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph. D. dissertation, Univ. Southern Calif, Los Angeles, 1972. [20] M. J. Shi, Y. Liu and P. Solé, Optimal binary codes from trace codes over a non-chain ring, Discrete Applied Mathematics, 219 (2017), 176-181.  doi: 10.1016/j.dam.2016.09.050. [21] M. J. Shi, Y. Liu and P. Solé, Optimal two weight codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Communications Letters, 20 (2016), 2346-2349. [22] M. J. Shi, Y. Guan and P. Solé, Two new families of two-weight codes, IEEE Trans. Inform. Theory, 63 (2017), 6240-6246.  doi: 10.1109/TIT.2017.2742499. [23] M. J. Shi, Y. Guan and P. Solé, Few-weight codes from trace codes over $R_k$, Bulletin of the Australian Mathematical Society, 98 (2018), 167-174.  doi: 10.1017/S0004972718000291. [24] M. J. Shi, D. T. Huang and P. Solé, Optimal ternary cubic two-weight codes, Chinese Journal of Electronic, (2018), 734–738. [25] M. J. Shi, L. Q. Qian and P. Solé, Few-weight codes from trace codes over a local ring, Applicable Algebra in Engineering Communication and computing, 29 (2018), 335-350.  doi: 10.1007/s00200-017-0345-8. [26] M. J. Shi, R. S. Wu, Y. Liu and P. Solé, Two and three weight codes over $\mathbb{F}_2+u\mathbb{F}_2$, Cryptography and Communications, 9 (2017), 637-646.  doi: 10.1007/s12095-016-0206-5. [27] M. J. Shi, R. S. Wu, L. Q. Qian, L. Sok and P. Solé, New classes of $p$-ary few weight codes, Bull. Malays. Math. Sci. Soc, 42 (2019), 1393-1412.  doi: 10.1007/s40840-017-0553-1. [28] M. J. Shi, S. X. Zhu and S. L. Yang, A class of optimal $p$-ary codes from one-weight codes over $\mathbb{F}_p[u]/(u^m)$, Journal of the Franklin Institute, 350 (2013), 929-937.  doi: 10.1016/j.jfranklin.2012.05.014. [29] M. J. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Appl. Comput. Math, 87 (2019), 2394-2404.  doi: 10.1007/s10623-019-00628-7. [30] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

show all references

##### References:
 [1] A. Ashikmin and A. Barg, Minimal vectors in linear codes, IEEE Transactions on Information Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584. [2] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer New York, 2012. doi: 10.1007/978-1-4614-1939-6. [3] A. Canteaut, P. Charpin and H. Dobbertin, Binary $m$-sequences with three valued crosscorrelation: A proof of Welch's conjecture, IEEE Transactions on Information Theory, 46 (2000), 4-8.  doi: 10.1109/18.817504. [4] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Transactions on Information Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722. [5] T. W. Cusick and H. Dobbertin, Some new three-valued crosscorrelation functions for binary $m$-sequences, IEEE Transactions on Information Theory, 42 (1996), 1238-1240.  doi: 10.1109/18.508848. [6] C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, Coding and Cryptology, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, NJ, 4 (2008), 119–124. doi: 10.1142/9789812832245_0009. [7] C. Ding and J. Yuan, Covering and secret sharing with linear codes, Lecture Notes in Computer Science, 2731 (2003), 11-25.  doi: 10.1007/3-540-45066-1_2. [8] H. Dobbertin, Almost perfect nonlinear power functions on $GF(2^n)$: The Welch case, IEEE Transactions on Information Theory, 45 (1999), 1271-1275.  doi: 10.1109/18.761283. [9] R. Gold, Maximal recursive sequences with 3-valued cross-correlation functions, IEEE Transactions on Information Theory, 14 (1968), 154-156. [10] S. W. Golomb and G. Guang, Signal Design for Good Correlation for Wireless Communication, Cryptography, and Radar, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511546907. [11] J. H. Griesmer, A bound for error-Correcting codes, IBM Journal of Research & Development, 4 (1960), 532-542.  doi: 10.1147/rd.45.0532. [12] T. Helleseth, Some results about the cross-correlation between two maximal linear sequences, Discrete Mathematics, 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X. [13] T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory, North-Holland, Amsterdam, 1 (1998), 1765–1853. [14] Y. Liu, M. J. Shi and P. Solé, Two-weight and three-weight codes from trace codes over $\mathbb{F}_p+u\mathbb{F}_p+v\mathbb{F}_p+uv\mathbb{F}_p$, Discrete Math, 341 (2018), 350-357.  doi: 10.1016/j.disc.2017.09.003. [15] M. Pursley and D. Sarwate, Crosscorrelation properties of pseudorandom and related sequences, Proceedings of the IEEE, 68 (1980), 593-619. [16] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003. [17] T. Kasami, The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes, Information and Control, 18 (1971), 369-394.  doi: 10.1016/S0019-9958(71)90473-6. [18] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. [19] Y. Niho, Multivalued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences, Ph. D. dissertation, Univ. Southern Calif, Los Angeles, 1972. [20] M. J. Shi, Y. Liu and P. Solé, Optimal binary codes from trace codes over a non-chain ring, Discrete Applied Mathematics, 219 (2017), 176-181.  doi: 10.1016/j.dam.2016.09.050. [21] M. J. Shi, Y. Liu and P. Solé, Optimal two weight codes over $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Communications Letters, 20 (2016), 2346-2349. [22] M. J. Shi, Y. Guan and P. Solé, Two new families of two-weight codes, IEEE Trans. Inform. Theory, 63 (2017), 6240-6246.  doi: 10.1109/TIT.2017.2742499. [23] M. J. Shi, Y. Guan and P. Solé, Few-weight codes from trace codes over $R_k$, Bulletin of the Australian Mathematical Society, 98 (2018), 167-174.  doi: 10.1017/S0004972718000291. [24] M. J. Shi, D. T. Huang and P. Solé, Optimal ternary cubic two-weight codes, Chinese Journal of Electronic, (2018), 734–738. [25] M. J. Shi, L. Q. Qian and P. Solé, Few-weight codes from trace codes over a local ring, Applicable Algebra in Engineering Communication and computing, 29 (2018), 335-350.  doi: 10.1007/s00200-017-0345-8. [26] M. J. Shi, R. S. Wu, Y. Liu and P. Solé, Two and three weight codes over $\mathbb{F}_2+u\mathbb{F}_2$, Cryptography and Communications, 9 (2017), 637-646.  doi: 10.1007/s12095-016-0206-5. [27] M. J. Shi, R. S. Wu, L. Q. Qian, L. Sok and P. Solé, New classes of $p$-ary few weight codes, Bull. Malays. Math. Sci. Soc, 42 (2019), 1393-1412.  doi: 10.1007/s40840-017-0553-1. [28] M. J. Shi, S. X. Zhu and S. L. Yang, A class of optimal $p$-ary codes from one-weight codes over $\mathbb{F}_p[u]/(u^m)$, Journal of the Franklin Institute, 350 (2013), 929-937.  doi: 10.1016/j.jfranklin.2012.05.014. [29] M. J. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Appl. Comput. Math, 87 (2019), 2394-2404.  doi: 10.1007/s10623-019-00628-7. [30] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.
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