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On the dimension of the subfield subcodes of 1-point Hermitian codes
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 |
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.
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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