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May  2021, 15(2): 227-240. 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  May 2021 Early access  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, 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.

[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.

[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.

[4]

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186. 

[5]

Y. Borissov and N. Manev, Minimal codewords in linear codes, Serdica Math. J., 30 (2004), 303-324. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

J. L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish-Russian Int. Workshop on Info. Theory, (1993), 276–279.

[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.

[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.

[19]

A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.

[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.

[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.

[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.

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.

[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.

[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.

[4]

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin., 18 (1984), 181-186. 

[5]

Y. Borissov and N. Manev, Minimal codewords in linear codes, Serdica Math. J., 30 (2004), 303-324. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

J. L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish-Russian Int. Workshop on Info. Theory, (1993), 276–279.

[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.

[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.

[19]

A. Shamir, How to share a secret, Communications of the ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.

[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.

[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.

[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.

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