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For any 1
1, we have from number theory [14]
H ( e ( i ) ,e ( i ) ; τ )= N ((1 + τ
τ
p
n 1 ) k =1)=gcd( p
·
1 ,k )
1 ,
hence (3) follows. Then (4) follows immediately from (3).
We prove now part 3). For any two k th power residue FH sequences e ( i ) , e ( j )
C ( p , k ), their Hamming crosscorrelation functions can be written as
n =0 h ( c ( i n ,e ( j )
n =0 h ( e ( j )
p− 1
p− 1
e ( i n , 0)
H ( e ( i ) ,e ( j ) ; τ )=
n + τ )=
n + τ
= N ( j ( n + τ ) k
n k =0)= N ( j ( n + τ ) k = i
n k ) .
i
·
·
For τ =0, since i
= j ,weobtain
H ( e ( i ) ,e ( j ) ;0)= N ( j
n k = i
n k )=1 .
·
·
For any 1
τ
p
1, we have from number theory
H ( e ( i ) ,e ( j ) ; τ )= N ((1 + τ
n 1 ) k = i
j 1 )
·
·
= gcd( p
j 1 is a k th power residue modulo p
1 ,k ) ,i
·
j 1 is a k th power nonresidue modulo p
0 ,i
·
Hence (5) follows.
For any fixed 1
j 1
j
p
1, it is noted that i
·
runs independently over
all elements in
{
1,2,. . . , p
1
}
as i takes independently all elements in
{
1,2,. . . ,
j 1
p
1
}
. Therefore, i
·
takes k th power residues modulo p ( p
1)/gcd( p
1,
k )times, k th power nonresidues modulo pp
1-( p
1)/gcd( p
1, k )timesfrom
number theory. Therefore
1
2
H ( e ( i ) ,e ( j ) ; τ )
S c =
1 ≤i,j≤p− 1 , 0 ≤τ≤p− 1 ,i = j
( p
1)( p
2)
+ 2
=
H ( e ( i ) ,e ( j ) ; τ )
2
1
i,j
p
1 , 1
τ
p
1 ,i
p− 1
gcd( p
= j
1
( p− 1)( p− 2)
2
+ 2
=
1 ,k )
·
gcd( p
1 ,k )
p 2
1 ≤j≤p− 1 , 1 ≤τ≤p− 1
1 ,k ) .
p− 1
2
=
p
1
( p
1)gcd( p
The average Hamming crosscorrelation of C ( p , k )is
2 S c
ML ( M
A c =
1)
p 2
1 ,k )
( p− 1) p ( p− 2) . p− 1
2
=
p
1
( p
1)gcd( p
p ( p− 2) p 2
2
1 ,k ) .
1
=
p
1
( p
1)gcd( p
Thus (6) is true. By applying 1), (4) and (5) to (2), it follows that
A a
L ( M
A c
( L
1) +
1)
p ( p− 2) p 2
1 ,k )
1
1
( p− 1)
1
=
p ( p− 2) [gcd( p
1 ,k )
1] +
p
1
( p
1)gcd( p
LM−q
p ( p− 1) −p
p ( p
1
1
=
=
=
1) .
( p
1)
q ( L
1)( M
1)
1)( p
2)
( p
Thus, the FH sequence set C ( p , k ) is an optimal average Hamming correlation
set . This completes the proof.
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