Information Technology Reference
In-Depth Information
×{
F
p
n
\{
S
(
γ,δ,
)
−
1 can be determined when (
γ,δ,
) runs through
F
p
m
×
F
p
n
1
}}
.
Together with the distribution of
S
(
γ,δ,
0)
−
1as(
γ,δ
) runs through
F
p
m
×
F
p
n
,
the correlation distribution is obtained.
By the definition of
{
s
a,b
(
α
t
)
}
0
≤t≤p
n
−
2
,itsperiodis
p
n
−
1. From the corre-
lation distribution given in Theorem 14, one easily knows there are exactly
p
3
2
sequences in the family
and the maximum magnitude is
p
2
+
d
+1.
k
F
Acknowledgment.
The authors thank anonymous reviewers for their useful
comments.
References
1. Bluher, A.W.: On
x
q
+1
+
ax
+
b
. Finite Fields and Their Applications 10, 285-305
(2004)
2. Carlet, C., Ding, C.: Highly Nonlinear Functions. J. Complexity 20, 205-244 (2004)
3. Carlet, C., Ding, C., Niederreiter, H.: Authentication Schemes from Highly Non-
linear Functions. Des. Codes Cryptography 40, 71-79 (2006)
4. Carlet, C., Ding, C., Yuan, J.: Linear Codes from Perfect Nonlinear Mappings and
Their Secret Sharing Schemes. IEEE Trans. Inform. Theory 51, 2089-2102 (2005)
5. Ding, C., Niederreiter, H.: Systematic Authentication Codes from Highly Nonlinear
Functions. IEEE Trans. Inform. Theory 50, 2421-2428 (2004)
6. Ding, C., Yuan, J.: A Family of Skew Hadamard Difference Sets. J. Combin. The-
ory, series A 113, 1526-1535 (2006)
7. Golomb, S.W., Gong, G.: Signal Design for Good Correlation
−
For Wireless Com-
munication, Cryptography and Radar. Cambridge Univ. Press, New York (2005)
8. Hu, L., Zeng, X., Li, N., Jiang, W.: Period-different m-sequences with at Most a
Four-valued Cross Correlation,
http://arxiv.org/abs/0801.0857
9. Kasami, T.: Weight Distribution of Bose-Chaudhuri-Hocquenghem Codes. In:
Bose, R.C., Dowling, T.A. (eds.) Combinatorial Mathematics and Its Applications,
pp. 335-357. University of North Carolina Press, Chapel Hill (1969)
10. Kumar, P.V., Moreno, O.: Prime-phase Sequences with Periodic Correlation Prop-
erties better than Binary Sequences. IEEE Trans. Inform. Theory 37, 603-616
(1991)
11. Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized Bent Functions and Their
Properties. J. Combin. Theory, series A 40, 90-107 (1985)
12. Lahtonen, J.: Two Remarks on a Paper by Moreno and Kumar. IEEE Trans.
Inform. Theory 41, 859-861 (1995)
13. Liu, S.-C., Komo, J.J.: Nonbinary Kasami Sequences over GF(
p
). IEEE Trans.
Inform. Theory 38, 1409-1412 (1992)
14. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Ap-
plications. Addison-Wesley, Reading (1983)
15. Moriuchi, T., Imamura, K.: Balanced Nonbinary Sequences With Good Periodic
Correlation Properties Obtained From Modified Kumar-Moreno Sequences. IEEE
Trans. Inform. Theory 41, 572-576 (1995)
16. Moreno, O., Kumar, P.V.: Minimum Distance Bounds for Cyclic Codes and
Deligne's Theorem. IEEE Trans. Inform. Theory 39, 1524-1534 (1993)
17. MacWilliams, F.J., Sloane, N.J.: The Theory of Error-Correcting Codes. North-
Holland, Amsterdam (1977)
