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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)
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