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doi: 10.3934/amc.2020055

Construction of minimal linear codes from multi-variable functions

Jong Yoon Hyun 1, , Boran Kim 2, and  Minwon Na 3,,

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.

Keywords: Minimal linear code, multi-variable function, AB-condition, HDZ-condition.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94A60.
Citation: Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 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

show all references

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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