# American Institute of Mathematical Sciences

May  2021, 15(2): 227-240. doi: 10.3934/amc.2020055

## Construction of minimal linear codes from multi-variable functions

 1 Glocal Campus, Konkuk University, Chungju, 27478, South Korea 2 Department of Mathematics, Sungkyunkwan University, Suwon, 16419, South Korea 3 Innovation Center for Industrial Mathematics, National Institute for Mathematical Sciences, Suwon, 16229, South Korea

* Corresponding author: Minwon Na

Received  July 2019 Revised  October 2019 Published  January 2020

Fund Project: The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (NRF-2017R1A2B2004574). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2019R1I1A1A01060467). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2018R1D1A1B07046315)

In this paper, we define a linear code by using multi-variable functions, and construct three classes of minimal linear codes with few-weight from multi-variable functions.

Citation: Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055
##### References:
 [1] A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584.  Google Scholar [2] D. Bartoli and M. Bonini, Minimal linear codes in odd characteristic, IEEE Transactions on Information Theory, 65 (2019), 4152-4155.  doi: 10.1109/TIT.2019.2891992.  Google Scholar [3] G. R. Blakley, Safeguarding cryptographic keys, Proceedings of AFIPS National Computer Conference. New York, USA, AFIPS Press, 48 (1979), 313-317.  doi: 10.1109/MARK.1979.8817296.  Google Scholar [4] A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.   Google Scholar [5] Y. Borissov and N. Manev, Minimal codewords in linear codes, Serdica Math. J., 30 (2004), 303-324.   Google Scholar [6] Y. Borissov, N. Manev and S. Nikova, On the non-minimal codewords in binary Reed-Muller codes, Discrete Appl. Math., 128 (2003), 65-74.  doi: 10.1016/S0166-218X(02)00436-5.  Google Scholar [7] S. Chang and J. Y. Hyun, Linear codes from simplicial complexes, Des. Codes Cryptogr., 86 (2018), 2167-2181.  doi: 10.1007/s10623-017-0442-5.  Google Scholar [8] G. D. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes, IMACC 2013, LNCS, Springer, Heidelberg, 8308 (2013), 85-98.   Google Scholar [9] C. S. Ding, A construction of binary linear codes from boolean functions, Discrete mathematics, 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029.  Google Scholar [10] K. L. Ding and C. S. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Transactions on Information Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar [11] C. S. Ding, Z. L. Heng and Z. C. Zhou, Minimal binary linear codes, IEEE Trans. Inf. Theory, 64 (2018), 6536-6545.  doi: 10.1109/TIT.2018.2819196.  Google Scholar [12] Z. L. Heng, C. S. Ding and Z. C. Zhou, Minimal linear codes over finite fields, Finite Fields and Their Applications, 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.  Google Scholar [13] Z. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Communications Letters, 19 (2015), 1488-1491.  doi: 10.1109/LCOMM.2015.2455032.  Google Scholar [14] Z. L. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields and Their Applications, 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002.  Google Scholar [15] T. Y. Hwang, Decoding linear block codes for minimizing word error rate, IEEE Trans. Inform. Theory, 25 (1979), 733-737.  doi: 10.1109/TIT.1979.1056120.  Google Scholar [16] J. L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish-Russian Int. Workshop on Info. Theory, (1993), 276–279. Google Scholar [17] R. J. McEliece and D. V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM, 24 (1981), 583-584.  doi: 10.1145/358746.358762.  Google Scholar [18] J. Schillewaert, L. Storme and J. A. Thas, A Minimal codewords in Reed-Muller codes, Des. Codes Cryptogr., 54 (2010), 273-286.  doi: 10.1007/s10623-009-9323-x.  Google Scholar [19] A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [20] C. M. Tang, N. Li, Y. F. Qi, Z. C. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Transactions on Information Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.  Google Scholar [21] J. Yuan and C. S. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.  Google Scholar [22] W. Q. Zhang, H. D. Yan and H. L. Wei, Four families of minimal binary linear codes with $\frac{w_{\min}}{w_{\max}} \leq\frac{1}{2}$, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 175-184.  doi: 10.1007/s00200-018-0367-x.  Google Scholar

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##### References:
 [1] A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584.  Google Scholar [2] D. Bartoli and M. Bonini, Minimal linear codes in odd characteristic, IEEE Transactions on Information Theory, 65 (2019), 4152-4155.  doi: 10.1109/TIT.2019.2891992.  Google Scholar [3] G. R. Blakley, Safeguarding cryptographic keys, Proceedings of AFIPS National Computer Conference. New York, USA, AFIPS Press, 48 (1979), 313-317.  doi: 10.1109/MARK.1979.8817296.  Google Scholar [4] A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186.   Google Scholar [5] Y. Borissov and N. Manev, Minimal codewords in linear codes, Serdica Math. J., 30 (2004), 303-324.   Google Scholar [6] Y. Borissov, N. Manev and S. Nikova, On the non-minimal codewords in binary Reed-Muller codes, Discrete Appl. Math., 128 (2003), 65-74.  doi: 10.1016/S0166-218X(02)00436-5.  Google Scholar [7] S. Chang and J. Y. Hyun, Linear codes from simplicial complexes, Des. Codes Cryptogr., 86 (2018), 2167-2181.  doi: 10.1007/s10623-017-0442-5.  Google Scholar [8] G. D. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes, IMACC 2013, LNCS, Springer, Heidelberg, 8308 (2013), 85-98.   Google Scholar [9] C. S. Ding, A construction of binary linear codes from boolean functions, Discrete mathematics, 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029.  Google Scholar [10] K. L. Ding and C. S. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Transactions on Information Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar [11] C. S. Ding, Z. L. Heng and Z. C. Zhou, Minimal binary linear codes, IEEE Trans. Inf. Theory, 64 (2018), 6536-6545.  doi: 10.1109/TIT.2018.2819196.  Google Scholar [12] Z. L. Heng, C. S. Ding and Z. C. Zhou, Minimal linear codes over finite fields, Finite Fields and Their Applications, 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.  Google Scholar [13] Z. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Communications Letters, 19 (2015), 1488-1491.  doi: 10.1109/LCOMM.2015.2455032.  Google Scholar [14] Z. L. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields and Their Applications, 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002.  Google Scholar [15] T. Y. Hwang, Decoding linear block codes for minimizing word error rate, IEEE Trans. Inform. Theory, 25 (1979), 733-737.  doi: 10.1109/TIT.1979.1056120.  Google Scholar [16] J. L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish-Russian Int. Workshop on Info. Theory, (1993), 276–279. Google Scholar [17] R. J. McEliece and D. V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM, 24 (1981), 583-584.  doi: 10.1145/358746.358762.  Google Scholar [18] J. Schillewaert, L. Storme and J. A. Thas, A Minimal codewords in Reed-Muller codes, Des. Codes Cryptogr., 54 (2010), 273-286.  doi: 10.1007/s10623-009-9323-x.  Google Scholar [19] A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [20] C. M. Tang, N. Li, Y. F. Qi, Z. C. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Transactions on Information Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.  Google Scholar [21] J. Yuan and C. S. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.  Google Scholar [22] W. Q. Zhang, H. D. Yan and H. L. Wei, Four families of minimal binary linear codes with $\frac{w_{\min}}{w_{\max}} \leq\frac{1}{2}$, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 175-184.  doi: 10.1007/s00200-018-0367-x.  Google Scholar
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