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2
k
+2
e
−
2
k
+
e
−
2
k
+1
distribution of
S
0
(
a
)
2
proven in Proposition 5 we get
r
+
s
=
2
2
e
−
1
and
t
+
z
=
2
k
−
e
−
1
2
2
e
. Taking the simple sum and the sum-of-cubes identities from
−
1
Lemma1weget
2
k
(
r
s
)+2
k
+
e
(
t
z
)=2
k
−
−
2
3
k
(
r
s
)+2
3(
k
+
e
)
(
t
2
4
k
+(2
e
+1)2
3
k
−
−
z
)=
−
.
Solving this system of four equations for four unknowns gives the claimed dis-
tribution.
As noted above in the proof, from the conditions of Theorem 2 it follows that
k/e
is odd. Thus, the condition for
k
to be odd can safely be removed from
Theorem 2 in [4], where particular case of
e
= 1 is considered. Also note that if
e
= 1 then the cross correlation is three-valued since the value
2
k
is not taken
on. If
l
= 0 then the cross correlation is two-valued since the values 0 and
−
2
k
+
e
are not taken on, which gives the Kasami case where
d
= 1. Also note that the
value of
C
d
(
τ
)and
S
(
a
) can be found directly using Corollary 3 which is usually
a much more dicult task than finding the value distribution.
−
Conjecture 1.
Except for the case when
m
=8and
d
= 7, all decimations
leading to at most four-valued cross correlation between two
m
-sequences of
different lengths 2
m
1and2
k
−
−
1, where
m
=2
k
, are described in Theorem 2.
4
Conclusions
We have identified pairs of
m
-sequences having different lengths 2
2
k
1and2
k
1
with at most four-valued cross correlation and we have completely determined
the cross-correlation distribution. In these pairs, decimation
d
is taken such that
d
(2
l
+1)
−
−
2
i
(mod 2
k
0.
Our results cover the two-valued Kasami case where
d
= 1 and all three-valued
decimations found in [4]. Conjectured is that we have a characterization of all
decimations leading to at most four-valued cross correlation of
m
-sequences with
the described parameters except for a single, seemingly degenerate, case.
≡
−
1) for some integer
l
with 0
≤
l<k
and
i
≥
References
1. Ness, G.J., Helleseth, T.: Cross correlation of
m
-sequences of different lengths. IEEE
Trans. Inf. Theory 52(4), 1637-1648 (2006)
2. Kasami, T.: Weight distribution formula for some classes of cyclic codes. Technical
Report R-285 (AD 637524), Coordinated Science Laboratory, University of Illinois,
Urbana (April 1966)
3. Ness, G.J., Helleseth, T.: A new three-valued cross correlation between
m
-sequences
of different lengths. IEEE Trans. Inf. Theory 52(10), 4695-4701 (2006)
4. Helleseth, T., Kholosha, A., Ness, G.J.: Characterization of
m
-sequences of lengths
2
2
k
−
1and2
k
−
1 with three-valued crosscorrelation. IEEE Trans. Inf. Theory 53(6),
2236-2245 (2007)