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Table 1. The generating polynomials
n The generating polynomial for
F 2 n
x 5 + x 3 +1
5
x 6 + x 5 +1
6
x 7 + x +1
7
x 8 + x 4 + x 3 + x +1
8
x 9 + x 4 +1
9
x 10 + x 3 +1
10
x 11 + x 2 +1
11
x 12 + x 3 +1
12
x 13 + x 4 + x 3 + x +1
13
x 14 + x 5 + x 3 + x +1
14
x 15 + x +1
15
x 16 + x 5 + x 3 + x 2 +1
16
x 17 + x 3 +1
17
x 18 +
x 5 +
x 2 +
18
x
+1
Table 2. The case of 1 , ..., α n } = { 1 ,x,x 2 , ..., x n 1 }
n period max τ =0 |C S ( τ ) | of max τ =0 |C S ( τ ) | of
A ( x )= Tr ( x 2 n 2 ) A ( x )= Tr ( x 2 n 4 )
5
32
16
8
6
64
24
20
7
128
28
36
8
256
40
40
9
512
64
72
10 1024
104
88
11 2048
152
144
12 4096
208
264
13 8192
336
348
14 16384
452
468
15 32768
700
688
16 65536
1000
1088
17 131072
1592
1684
18 262144
2184
2224
Proof. Let f ( x )= x p n
2 . Then by Lemma 2, we have
ψ
ξ τ
ξ ( ξ + ξ τ )
p n/ 2 +3 ,
ψ ( f ( ξ )
f ( ξ + ξ τ ))
2+
2
·
ξ∈ F p n
ξ =0 ,−ξ τ
and for any η
=0,wehave
ηξ
ψ
ξ τ
ξ ( ξ + ξ τ )
p n/ 2 +2 .
ψ ( f ( ξ )
f ( ξ + ξ τ )
ηξ )
2+
4
·
ξ∈ F 2 n
ξ =0 ,−ξ τ
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