Let $ p\equiv 1\pmod 4 $ be a prime, $ m $ a positive integer, $ \frac{\phi(p^m)}2 $ the multiplicative order of $ 2 $ modulo $ p^m $, and let $ q = 2^{ \frac{\phi(p^m)}2} $, where $ \phi(\cdot) $ is the Euler's function. In this paper, we construct two classes of linear codes over $ \Bbb F_q $ and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained.
Citation: Jianying Rong, Fengwei Li, Ting Li. Two classes of two-weight linear codes over finite fields[J]. AIMS Mathematics, 2023, 8(7): 15317-15331. doi: 10.3934/math.2023783
Let $ p\equiv 1\pmod 4 $ be a prime, $ m $ a positive integer, $ \frac{\phi(p^m)}2 $ the multiplicative order of $ 2 $ modulo $ p^m $, and let $ q = 2^{ \frac{\phi(p^m)}2} $, where $ \phi(\cdot) $ is the Euler's function. In this paper, we construct two classes of linear codes over $ \Bbb F_q $ and investigate their weight distributions. By calculating two classes of special exponential sums, the desired results are obtained.
| [1] |
K. T. Arasu, C. Ding, T. Helleseth, P. V. Kumar, H. M. Martinsen, Almost difference sets and their sequences with optimal autocorrelation, IEEE Trans. Inf. Theory, 47 (2001), 2934–2943. http://dx.doi.org/10.1109/18.959271 doi: 10.1109/18.959271
|
| [2] |
C. Carlet, C. Ding, J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089–2102. http://dx.doi.org/10.1109/TIT.2005.847722 doi: 10.1109/TIT.2005.847722
|
| [3] | L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math., 57 (1935), 391–424. https://doi.org/10.2307/2371217 |
| [4] |
K. Ding, C. Ding, Binary linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879–1882. http://dx.doi.org/10.1109/LCOMM.2014.2361516 doi: 10.1109/LCOMM.2014.2361516
|
| [5] |
K. Ding, C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835–5842. http://dx.doi.org/10.1109/TIT.2015.2473861 doi: 10.1109/TIT.2015.2473861
|
| [6] | M. Grassl, Bounds on the minumum distance of linear codes, 2022. Avaiable from: http://www.codetables.de. |
| [7] |
Z. Heng, C. Ding, Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176–196. https://doi.org/10.1016/j.ffa.2018.08.010 doi: 10.1016/j.ffa.2018.08.010
|
| [8] |
Z. Heng, W. Wang, Y. Wang, Projective binary linear codes from special Boolean functions, AAECC, 32 (2021), 521–552. https://doi.org/10.1007/s00200-019-00412-z doi: 10.1007/s00200-019-00412-z
|
| [9] |
Z. Heng, Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488–1491. http://dx.doi.org/10.1109/LCOMM.2015.2455032 doi: 10.1109/LCOMM.2015.2455032
|
| [10] |
Z. Heng, Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72–92. https://doi.org/10.1016/j.ffa.2015.12.002 doi: 10.1016/j.ffa.2015.12.002
|
| [11] |
Z. Heng, Q. Yue, Evaluation of the Hamming weights of a class of linear codes based on Gauss sums, Des. Codes Cryptogr., 83 (2017), 307–326. http://dx.doi.org/10.1007/s10623-016-0222-7 doi: 10.1007/s10623-016-0222-7
|
| [12] |
Z. Heng, Q. Yue, C. Li, Three classes of linear codes with two or three weights, Discrete Math., 339 (2016), 2832–2847. https://doi.org/10.1016/j.disc.2016.05.033 doi: 10.1016/j.disc.2016.05.033
|
| [13] |
C. Li, Q. Yue, F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94–114. http://dx.doi.org/10.1016/j.ffa.2014.01.009 doi: 10.1016/j.ffa.2014.01.009
|
| [14] |
F. Li, Several classes of exponential sums and three-valued Walsh spectrums over finite fields, Finite Fields Appl., 87 (2023), 102142. https://doi.org/10.1016/j.ffa.2022.102142 doi: 10.1016/j.ffa.2022.102142
|
| [15] |
M. Moisio, A note on evaluations of some exponential sums, Acta Arith., 93 (2000), 117–119. https://doi.org/10.4064/aa-93-2-117-119 doi: 10.4064/aa-93-2-117-119
|
| [16] |
M. Moisio, Explicit evaluation of some exponential sums, Finite Fields Appl., 15 (2009), 644–651. https://doi.org/10.1016/j.ffa.2009.05.005 doi: 10.1016/j.ffa.2009.05.005
|
| [17] | T. Storer, Cyclotomic and difference sets, Markham, Chicago, 1967. |
| [18] |
Q. Wang, K. Ding, D. D. Lin, R. Xue, A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315–322. http://dx.doi.org/10.1007/s12095-015-0180-3 doi: 10.1007/s12095-015-0180-3
|
| [19] |
Q. Wang, K. Ding, R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097–1100. https://doi.org/10.1109/LCOMM.2015.2431253 doi: 10.1109/LCOMM.2015.2431253
|
| [20] |
Y. Wu, Q. Yue, X. Shi, At most three-weight binary linear codes from generalized Moisio's exponential sums, Des. Codes Cryptogr., 87 (2019), 1927–1943. https://doi.org/10.1007/s10623-018-00595-5 doi: 10.1007/s10623-018-00595-5
|
| [21] |
Z. Zhou, C. Ding, J. Luo, A. Zhang, A family of five-weight cyclic codes and their weight enumerators, IEEE Trans. Inf. Theory, 59 (2013), 6674–6682. http://dx.doi.org/10.1109/TIT.2013.2267722 doi: 10.1109/TIT.2013.2267722
|
| [22] |
Z. Zhou, A. Zhang, C. Ding, M. Xiong, The weight enumerator of three families of cyclic codes, IEEE Trans. Inf. Theory, 59 (2013), 6002–6009. http://dx.doi.org/10.1109/TIT.2013.2262095 doi: 10.1109/TIT.2013.2262095
|
Jianying Rong, Fengwei Li, Ting Li. Two classes of two-weight linear codes over finite fields[J]. AIMS Mathematics, 2023, 8(7): 15317-15331. doi: 10.3934/math.2023783
DownLoad: