A Note on Cross Correlation Distribution of Ternary m-Sequences
aa r X i v : . [ c s . I T ] S e p A Note on Cross Correlation Distribution of Ternary m -Sequences Hai Xiong and Longjiang Qu
College of Science, National University of Defense Technology, Changsha 410073, China [email protected],[email protected]
Abstract.
In this note, we prove a conjecture proposed by Tao Zhang, Shuxing Li, Tao Fengand Gennian Ge, IEEE Transaction on Information Theory, vol. 60, no. 5, May 2014. Thisconjecture is about the cross correlation distribution of ternary m -sequences. Index Terms— cross correlation; decimation; ternary m -sequences Let ω be a 3-rd complex root of unity. Let { a t } and { b t } be two ternary sequences of period N . The cross correlation function of { a t } and { b t } is defined by C a,b ( τ ) = N − X t =0 ω a t + τ − b t , τ ∈ Z / ( N ) . Let n be a positive integer. Let χ be an additive character of GF (3 n ) which is definedby χ ( x ) = ω Tr n ( x ) , where Tr n ( x ) = x + x + · · · + x n − is the trace function from GF (3 n )to GF (3). Generally, for a positive integer r | n , we denote by Tr nr ( x ) the trace function from GF (3 n ) to GF (3 r ), which is defined by Tr nr ( x ) = x + x r + · · · + x r ( nr − .Let { a t } be a ternary m -sequence of period 3 n − { b t } be its d -decimation wheregcd( d, n −
1) = 1. Let α be a primitive element of GF (3 n ). Then we denote C a,b ( τ ) by C d ( z ), where z = α τ . Clearly, we have C d ( z ) + 1 = P x ∈ GF (3 n ) χ ( zx − x d ). And we define S d ( z ) := P x ∈ GF (3 n ) χ ( zx − x d ). Hence computing the cross correlation distribution of m -sequences is equivalent to compute the values distribution of Weil sum S d ( z ).The cross correlation distribution of m -sequences is an important topic in sequences,coding theory and communications. It essentially arises in many contexts with various names,please refer to the appendix of [1] for more details. Many results on this topic have beenreported. Please see [2] for an exhaustive survey. In [2], Zhang et al. also determined thedistribution of cross correlation of some ternary m -sequences and got some interesting resultsabout the cross correlation of some binary m -sequences. One of their main results can bepresented as follows. Theorem 1. [2, Theorem II.5] Let r be a positive integer such that gcd( r,
3) = 1 . Let n = 3 r , d = 3 r + 2 or d = 3 r + 2 . Let s be a ternary m -sequence of period n − . Then thecross correlation values between s and its d -decimation are showed as in Tables 1 and 2. Zhang et al. conjectured that Theorem 1 is also right if gcd( r,
3) = 3. In this paper weprove their conjecture for any positive integer r . And our technique is generalized from theirs.In the rest of this paper, we always assume that r is a positive integer, d = 3 r +2 or 3 r +2,and n = 3 r . Let E = GF (3 n ) and F = GF (3 r ). It is easy to verify that gcd( d, n −
1) = 1.
Xiong Hai, Longjiang Qu
Table 1.
The distribution of the case r evenCross Correlation Value Occurs Times − r +3 r − r r − r r − r − +3 r − − r − r − +3 r − · r − r − +3 r − − · r − r − +3 r − Table 2.
The distribution of the case r oddCross Correlation Value Occurs Times − · r − + 3 r − − r r − r r +12 − r − +3 r − − r +12 − r − +3 r − Before proving the conjecture, let us review the sketch of the proof of Theorem 1 in [2]. Atfirst, a suitable element was chosen to construct field extension from F to E . Hence everyelement in E can be expressed by some elements in the subfield F with the aforementionedelement. And then S d ( z ) can be expressed by some exponential sums over F . Finally, S d ( z )was computed by using some characterizations of quadratic Weil sum [2, Lemma II.2] andquadratic Gauss sum [2, Lemma II.1].In the following, we will prove the conjecture. The main difference between our proof andthe one in [2] is that we choose different elements to construct field extension. Then we canremove the restriction gcd( r,
3) = 1 and the discussion of cases r ≡ , d . Case d = 3 r + 2 : Let u be an element of F such that Tr r ( u −
1) = 1. Hence x − x − ( u − is an irreduciblepolynomial over F . And let α be a root of x − x − ( u − = 0. Then we can get E = F ( α ),which means that for any x ∈ E , it can be uniquely expressed as x = x + x α + x α , where x , x , x ∈ F . Lemma 1.
Let x = x + x α + x α and z = z + z α + z α be two elements of E . Thenwe have Tr nr ( x d ) = (( u − + 1) x + x x + x x + 2 x x + 2 x and Tr nr ( zx ) = 2( z + z ) x + 2 z x + 2 z x . Proof.
Noting that α = α + ( u − , we can deduce that α r = α + Tr r (( u − ) = α + 1.Thus x r = x + x ( α + 1) + x ( α + 1) and x r = x + x ( α + 2) + x ( α + 2) . Hence we Note on Cross Correlation Distribution of Ternary m -Sequences 3 have Tr nr ( x d ) = x d + ( x d ) r + ( x d ) r = x r x + x r ( x r ) + x ( x r ) = ( x + x ( α + 1) + x ( α + 1) )( x + x α + x α ) + ( x + x ( α + 2) + x ( α + 2) )( x + x ( α + 1) + x ( α + 1) ) + ( x + x α + x α )( x + x ( α + 2) + x ( α + 2) ) = (( u − + 1) x + x x + x x + 2 x x + 2 x . The last step is got from a complicated but not difficult computation. Comparing with theabove equation, it is easier to verify thatTr nr ( zx ) = zx + ( zx ) r + ( zx ) r = ( z + z α + z α )( x + x α + x α )+ ( z + z ( α + 1) + z ( α + 1) )( x + x ( α + 1) + x ( α + 1) )+ ( z + z ( α + 2) + z ( α + 2) )( x + x ( α + 2) + x ( α + 2) )= 2( z + z ) x + 2 z x + 2 z x . (cid:3) According to Lemma 1, we can get thatTr n ( x d ) = Tr r (Tr nr ( x d )) = Tr r ( x x + x x + 2 x x + ux + 2 x )and Tr n ( zx ) = Tr r (Tr nr ( zx )) = Tr r (2( z + z ) x + 2 z x + 2 z x ) . Define χ F ( x ) = ω Tr r ( x ) , for x ∈ F . Then we can get S d ( z )= X x ∈ E ω Tr n ( zx − x d ) = X x ,x ,x ∈ F χ F ( x x + (2 z − x + 1) x + (2 z + 2 z − u ) x + (2 z − x ) x )= X x ,x ∈ F χ F ( x x + (2 z − x + 1) x + (2 z + 2 z − u ) x ) X x ∈ F χ F ((2 z − x ) x )=3 r · X x ∈ F,x ∈ M χ F ( x x + (2 z − x + 1) x + (2 z + 2 z − u ) x ) , (1)where M = { x ∈ F | x = − z } . Following from similar arguments in [2], we can deduce the following results by Eq.(1), [2, Lemma II.1] and [2, Lemma II.2]. The details are omitted here. – If r is even, then S d ( z ) takes six values, namely 0, 3 r , 3 r , − r , 2 · r and − · r . Andthe number of occurrences of the first two values are r +3 r − r and 3 r respectively. Xiong Hai, Longjiang Qu – If r is odd, then S d ( z ) takes four values, namely 0, 3 r , 3 r +12 and − r +12 . And the numberof occurrence of the second value is 3 r .Then according to [2, Lemma II.3] and [2, Lemma II.4], we can solve the number of oc-currences of all the values. The result is the same as Zhang et al. conjectured in [2]. Herea remark is as follows. In the original form of [2, Lemma II.4], the authors assumed thatgcd( r,
3) = 1. However, this result can be easily generalized to any positive integer r . Case d = 3 r + 2 : Let u be an element of F such that Tr r (1 − u ) = 1 and let α be a root of x − x = (1 − u ) .Then we also can getTr n ( x d ) = Tr r (Tr nr ( x d )) = Tr r (2 x x + x x + 2 x x + ux + x )and Tr n ( zx ) = Tr r (Tr nr ( zx )) = Tr r (2( z + z ) x + 2 z x + 2 z x ) . Then similarly as the first case, we can confirm the conjecture in this case.
In this note, we completely determine the distribution of cross correlation values of a ternary m -sequence with period 3 r − d -decimation, where d = 3 r + 2 or d = 3 r + 2. Hencewe confirm the conjecture presented in [2]. Acknowledgments
The work of H. Xiong was supported by Hunan Provincial Innovation Foundation for Post-graduate (No. CX2013B007) and the Innovation Foundation of NUDT under Grant (No.B130201). The work of L. Qu was supported in part by the Research Project of NationalUniversity of Defense Technology under Grant CJ 13-02-01 and the Program for New Cen-tury Excellent Talents in University (NCET). The first author also gratefully acknowledgefinancial support from China Scholarship Council.
References
1. Katz D. J, “Weil Sums of Binomials, Three-Level Cross-Correlation, and a Conjecture of Helleseth,”