m-Sequences of Lengths 2-1 and 2-1 with at Most Four-Valued Cross Correlation - Sequences and Their Applications-SETA 2008 - page 120

n − 1
2
i =0

n − 1
2
i =0

n− 1

B 2 ( j +1) l
n−

B 2 ( j − 2 i ) l
n

B 2 l
n−

B 2 (2 i +1) l
n

( a ) ( a )

1 ( a )

( a )

1

=Tr ne
e

=

B 2 l +1 B 2 ( j +1) l +2 jl

( a ) ( a )

n n

j =0

2 2 il

n − 1
2
i =0

B 2 l B n +1 ( a )+ B 2 l

n− 1 ( a ) n +1 ( a )

( 4 ) =Tr ne
e

B 2 l +1 ( a )

n

B 2 l

n− 1 ( a )(Tr ne

( B n +1 ( a )) + B n +1 ( a ))
B 2 l +1

=Tr ne
e

e

( a ) n

B 2 l
n−

B 2 l
n− 1 ( a )

B 2 l +1

1 ( a ) B n +1 ( a )
B 2 l +1

=Tr ne
e

( B n +1 ( a ))Tr ne +Tr ne

e

=0 ,

e

( a ) ( a )

n n

where the latest identity follows by Corollary 2 (note that Tr ne
e

( v )=Tr nl
l

( v )if

v ∈ GF(2 ne )).

B Remaining Proof of Corollary 3

We can compute

ac − 2 v 2 l +1
µ

Tr k

2 l +1 + ac − 1 μ 2 l B n ( a )+ ac − 1 μ

B 2 l

n ( a )+ ac − 2 μ 2 B 2 l +1

=Tr k a V V V ( a )

n

( 11 =Tr k

2 l +1 + c − 1 μV 2 l

B 2 l

n ( a )+ ac − 2 μ 2 B 2 l +1

aV ( a )

n

2

v 0
v 0 + v 1

2 l +1 + c − 1 μ

B n ( a )+ c − 2 μ 2 N ne Tr ne

e

( v 0 ) 2 v − 2 =Tr k a V V

e 1

v 0
v 0 + v 1

2 l +1

c − 1 μ N ne
e

1+Tr ne
e

( v 0 )Tr ne ( v − 0 ) =Tr k a V +Tr e

e

v 0
v 0 + v 1

+ c − 1 μ N ne
e

Tr ne
e

Tr ne
e

( v 0 ) v − 1

1

v 0
v 0 + v 1

2 l +1

c − 1 μ N ne
e

=Tr k a V +Tr e .

Thus,

ac − 2 v 2 l + μ + v μ

1) Tr k ( −

µ∈ GF(2 e )

2 l +1

1) Tr k

aV

c − 1 µ N ne
e

v 0
v 0 + v 1

1) Tr e =( − ( − =0 .

µ∈ GF(2 e )