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On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $
A new class of $ p $-ary regular bent functions
1. | School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China |
2. | School of Mathematical Sciences, Peking University, Beijing, 100871, China |
3. | School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China |
4. | Center of Applied Mathematics, School of Mathematics, Tianjin University, Tianjin, 300072, China |
$ p $ |
$ f(x) = \sum\limits_{i = 0}^{q-1}(Tr^n_1(a_1x^{(r i+s)(q-1)})+Tr^n_1(a_2x^{(r i+s)(q-1)+\frac{q^2-1}{2}}))+bx^{\frac{q^2-1}{2}}, $ |
$ n = 2m $ |
$ q = p^m $ |
$ p $ |
$ a_1,a_2\in \mathbb{F}_{p^n} $ |
$ b\in \mathbb{F}_p $ |
$ p $ |
$ gcd(s-r,\frac{q+1}{2}) = 1 $ |
$ gcd(r,q+1) = 1 $ |
$ 2 $ |
References:
[1] |
A. Canteaut, P. Charpin and G. M. Kyureghyan,
A new class of monomial bent functions, Finite Fields Appl., 14 (2008), 221-241.
doi: 10.1016/j.ffa.2007.02.004. |
[2] |
P. Charpin and G. M. Kyureghyan,
Cubic monomial bent functions: A subclass of $\mathcal{M}$, SIAM J. Discr. Math., 22 (2008), 650-665.
doi: 10.1137/060677768. |
[3] |
P. Charpin, E. Pasalic and C. Tavernier,
On bent and semi-bent quadratic boolean functions, IEEE Trans. Inf. Theory, 51 (2005), 4286-4298.
doi: 10.1109/TIT.2005.858929. |
[4] |
J. F. Dillon, Elementary Hadamard difference sets, Thesis (Ph.D.)-University of Maryland, College Park, (1974), 126 pp. |
[5] |
H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit,
Construction of bent functions via Niho power functions, J. Comb. Theory Ser. A, 113 (2006), 779-798.
doi: 10.1016/j.jcta.2005.07.009. |
[6] |
T. Helleseth and A. Kholosha, On generalized bent functions, Proc. IEEE Inf. Theory Appl. Workshop, (2010), 1–6.
doi: 10.1109/ITA.2010.5454124. |
[7] |
T. Helleseth and A. Kholosha,
Sequences, bent functions and Jacob-sthal sums, Sequences and Their Applications-SETA 2010, Lecture Notes Comput. Sci. Springer, Berlin, 6338 (2010), 416-429.
doi: 10.1007/978-3-642-15874-2_35. |
[8] |
T. Helleseth and A. Kholosha,
Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032.
doi: 10.1109/TIT.2006.872854. |
[9] |
W. J. Jia, X. Y. Zeng, T. Helleseth and C. L. Li,
A class of binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 58 (2012), 6054-6063.
doi: 10.1109/TIT.2012.2199736. |
[10] |
P. V. Kumar, R. A. Scholtz and L. R. Welch,
Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107.
doi: 10.1016/0097-3165(85)90049-4. |
[11] |
N. G. Leander,
Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743.
doi: 10.1109/TIT.2005.862121. |
[12] |
N. Li, T. Helleseth and X. H. Tang andd A. Kholosha,
Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818-1831.
doi: 10.1109/TIT.2012.2229782. |
[13] |
R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, 1994.
doi: 10.1017/CBO9781139172769.![]() ![]() |
[14] |
S. Mesnager,
Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials, IEEE Trans. Inf. Theory, 57 (2011), 5996-6009.
doi: 10.1109/TIT.2011.2124439. |
[15] |
S. Mesnager and J.-P. Flori,
Hyperbent functions via Dillon-like exponents, IEEE Trans. Inf. Theory, 59 (2013), 3215-3232.
doi: 10.1109/TIT.2013.2238580. |
[16] |
C. M. Tang, Y. F. Qi, M. Z. Xu, B. C. Wang and Y. X. Yang, A new class of hyper-bent Boolean functions in binomial forms [Online], Available: https://arXiv.org/abs/1112.0062v2. Google Scholar |
[17] |
N. Y. Yu and G. Gong,
Constructions of quadratic bent functions in polynomial forms, IEEE Trans. Inf. Theory, 52 (2006), 3291-3299.
doi: 10.1109/TIT.2006.876251. |
[18] |
D. B. Zheng, L. Yu and L. Hu,
On a class of binomial bent functions over the finite fields of odd characteristic, Applicable Algebra in Engineering, Communication and Computing, 24 (2013), 461-475.
doi: 10.1007/s00200-013-0202-3. |
show all references
References:
[1] |
A. Canteaut, P. Charpin and G. M. Kyureghyan,
A new class of monomial bent functions, Finite Fields Appl., 14 (2008), 221-241.
doi: 10.1016/j.ffa.2007.02.004. |
[2] |
P. Charpin and G. M. Kyureghyan,
Cubic monomial bent functions: A subclass of $\mathcal{M}$, SIAM J. Discr. Math., 22 (2008), 650-665.
doi: 10.1137/060677768. |
[3] |
P. Charpin, E. Pasalic and C. Tavernier,
On bent and semi-bent quadratic boolean functions, IEEE Trans. Inf. Theory, 51 (2005), 4286-4298.
doi: 10.1109/TIT.2005.858929. |
[4] |
J. F. Dillon, Elementary Hadamard difference sets, Thesis (Ph.D.)-University of Maryland, College Park, (1974), 126 pp. |
[5] |
H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit,
Construction of bent functions via Niho power functions, J. Comb. Theory Ser. A, 113 (2006), 779-798.
doi: 10.1016/j.jcta.2005.07.009. |
[6] |
T. Helleseth and A. Kholosha, On generalized bent functions, Proc. IEEE Inf. Theory Appl. Workshop, (2010), 1–6.
doi: 10.1109/ITA.2010.5454124. |
[7] |
T. Helleseth and A. Kholosha,
Sequences, bent functions and Jacob-sthal sums, Sequences and Their Applications-SETA 2010, Lecture Notes Comput. Sci. Springer, Berlin, 6338 (2010), 416-429.
doi: 10.1007/978-3-642-15874-2_35. |
[8] |
T. Helleseth and A. Kholosha,
Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032.
doi: 10.1109/TIT.2006.872854. |
[9] |
W. J. Jia, X. Y. Zeng, T. Helleseth and C. L. Li,
A class of binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 58 (2012), 6054-6063.
doi: 10.1109/TIT.2012.2199736. |
[10] |
P. V. Kumar, R. A. Scholtz and L. R. Welch,
Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107.
doi: 10.1016/0097-3165(85)90049-4. |
[11] |
N. G. Leander,
Monomial bent functions, IEEE Trans. Inf. Theory, 52 (2006), 738-743.
doi: 10.1109/TIT.2005.862121. |
[12] |
N. Li, T. Helleseth and X. H. Tang andd A. Kholosha,
Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818-1831.
doi: 10.1109/TIT.2012.2229782. |
[13] |
R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, 1994.
doi: 10.1017/CBO9781139172769.![]() ![]() |
[14] |
S. Mesnager,
Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials, IEEE Trans. Inf. Theory, 57 (2011), 5996-6009.
doi: 10.1109/TIT.2011.2124439. |
[15] |
S. Mesnager and J.-P. Flori,
Hyperbent functions via Dillon-like exponents, IEEE Trans. Inf. Theory, 59 (2013), 3215-3232.
doi: 10.1109/TIT.2013.2238580. |
[16] |
C. M. Tang, Y. F. Qi, M. Z. Xu, B. C. Wang and Y. X. Yang, A new class of hyper-bent Boolean functions in binomial forms [Online], Available: https://arXiv.org/abs/1112.0062v2. Google Scholar |
[17] |
N. Y. Yu and G. Gong,
Constructions of quadratic bent functions in polynomial forms, IEEE Trans. Inf. Theory, 52 (2006), 3291-3299.
doi: 10.1109/TIT.2006.876251. |
[18] |
D. B. Zheng, L. Yu and L. Hu,
On a class of binomial bent functions over the finite fields of odd characteristic, Applicable Algebra in Engineering, Communication and Computing, 24 (2013), 461-475.
doi: 10.1007/s00200-013-0202-3. |
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