Some results on cross-correlation distribution between a p-ary m-sequence and its decimated sequences

Transcription

1 Some results on cross-correlation distribution between a p-ary m-sequence and its decimated sequences A joint work with Chunlei Li, Xiangyong Zeng, and Tor Helleseth Selmer Center, University of Bergen Sept. 5, 014

2 Outline 1 Background and preliminaries

3 Outline 1 Background and preliminaries The exponential sum S d (u, v)

4 Outline 1 Background and preliminaries The exponential sum S d (u, v) 3 Cross-correlation distribution

5 Outline 1 Background and preliminaries The exponential sum S d (u, v) 3 Cross-correlation distribution 4 Connection between our decimations and some known ones

6 Notation p: an odd prime. m: a positive integer. F p m: the finite field with p m elements. α: a primitive element of F p m. {s(t)} pm t=0 : a p-ary m-sequence of period p m 1.

7 The d-decimations of {s(t)} Trace representation (after suitable cyclic shift): s(t) = Tr m 1 (α t ). The decimation exponent d. The l-th d-decimated sequence {s(dt + l)} of {s(t)}: s(dt + l) = Tr m 1 (α dt+l ), 0 l < gcd(d, p m 1). {s(dt + l)} has period p m 1 gcd(d, p m 1).

8 Cross-correlation function C d,l (τ) The cross-correlation function of {s(t)} and {s(dt + l)}: C d,l (τ) = p m t=0 ω Trm 1 (αt ) Tr m 1 (αd(t+τ)+l ) p. To determine C d,l (τ), it suffices to investigate C d (γ) = x F p m ω Trm 1 (x+γxd ) p 1, γ F pm. (1.1)

9 Cross-correlation distribution Two important problems in sequence design. Find new decimation exponents d such that max γ F p m C d (γ) is low. Determine the cross-correlation distribution, i.e., the multiset { Cd (γ) γ F p m }.

10 Exponential sums related to C d (γ) Define Then, S d (u, v) = x F p m S d (1, γ) = C d (γ) + 1. ω Trm 1 (ux+vx d ) p. (1.)

11 Some known results (1/4) odd prime p, e = gcd(k, m), m e 3 odd, d = pk +1 or p3k +1 p k +1, 3-valued, p m+e + 1. p m (mod 3), m even, d = p m 1, 4-valued, p m 1. T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discr. Math., 16: 09-3 (1976)

12 Some known results (/4) p = 3, m odd, d = 3 m 1 + 1, 3-valued, 3 m p = 3, m = 3r (r ), d = 3 r + or 3 r +, 4 or 6-valued, 3 r 1. H. Dobbertin, T. Helleseth, P. V. Kumar, and H. Martinsen, Ternary m-sequences with three-valued cross-correlation function: new decimations of Welch and Niho type, IEEE Trans. Inf. Theory, 47(4): (001) T. Zhang, S. Li, T. Feng and G. Ge, Some new results on the cross correlation of m-sequences, IEEE Trans. Inf. Theory, 60(5): (014). Y. Xia, T. Helleseth and G. Wu, A note on cross-correlation distribution between a ternary m-sequence and its decimated sequence, to appear in SETA014.

13 Some known results (3/4) odd prime p, e = gcd(k, m), m e, d = pk +1, k e odd, 9-valued, pe 1 p m + 1. odd prime p, m = 4k, d = ( pk +1 ), 4-valued, p m 1. J. Luo and K. Feng, Cyclic codes and sequence from generalized Coulter-Matthews functions, IEEE Trans. Inf. Theory, 54(1): (008) E. Y. Seo, Y. S. Kim, J. S. No and D. J. Shin, Cross-correlation distribution of p-ary m-sequence of period p 4k 1 and its decimated sequences by ( pk +1 ), IEEE Trans. Inf. Theory, 54(7): (008)

14 Some known results (4/4) p 3 (mod 4), m odd, e m, m e 3, d = pm +1 p e +1 ± pm 1, gcd(d, p m 1) =, 9-valued, pe +1 p m + 1. E. N. Müller, On the crosscorrelation of sequences over GF(p) with short periods, IEEE Trans. Inf. Theory, 45(1): (1999) Z. Hu, X. Li, D. Mills, E. N. Müller, W. Sun, W. Willems, Y. Yang and Z. Zhang, On the crosscorrelation of sequences with the decimation factor d = pn +1 p+1 pn 1, Appl. Algebra Eng. Commun. Comput., 1(3): (001) Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor d = pn +1 p+1 pn 1, Appl. Algebra Eng. Commun. Comput., 1(5): (010) S. T. Choi, J. Y. Kim, and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by d = pn +1 p k +1 + pn 1, IEICE Trans. Commun., vol. E96-B(9): (013)

15 The topic of this talk An odd prime p and two positive integers m, k: a decimation d: m gcd(k, m) is odd and m > 1; (1.3) gcd(k, m) d ( ) p k + 1 (mod p m 1). (1.4) The purpose is to determine the cross-correlation distribution for every decimation d satisfying Eq. (1.4).

16 d(p k +1) (mod p m -1) Cross-correlation Function C d (γ) An exponential sum S d (u,v) C d (γ)= S d (1, γ) 1 Correlation distribution d p k + 1 (mod p m -1) Relationship Some known decimations

17 Some auxiliary results (1/4) Lemma 1 For p, m and k satisfying (1.3), there are two distinct integers d 1, d satisfying d ( p k + 1 ) (mod p m 1) in Z p m 1. Then (i) d 1 1 (mod p e 1), and d 1 + pe 1 (mod p e 1); (ii) gcd(d 1, p m 1) = 1, and { gcd(d, p m 1, if p 1) = e 1 (mod 4),, if p e 3 (mod 4).

18 Some auxiliary results (/4) Let m, k be two positive integers satisfying (1.3). Define ( Q u,v (x) = Tr m e ux pk +1 + vx ), u, v F p m. (1.5) J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54(1): (008) J. Luo and K. Feng, Cyclic codes and sequence from generalized Coulter-Matthews functions, IEEE Trans. Inf. Theory, 54(1): (008) S. T. Choi, J. Y. Kim, and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by d = pn +1 p k +1 + pn 1, IEICE Trans. Commun., vol. E96-B(9): (013) Z. Zhou and C. Ding, A class of three-weight cyclic codes, Finite Fields Appl., 5: (014)

19 Some auxiliary results (3/4) Lemma (Luo and Feng, 008; Choi et. al., 013; Zhou and Ding, 014) Let Q u,v (x) be the quadratic form defined by (1.5), (u, v) F p m \ {(0, 0)} and s = m e. (i) The rank of Q u,v (x) is s, s 1 or s. Especially, both Q u,0 (x) with u F p m and Q 0,v(x) with v F pm have rank s. (ii) For any given (u, v) F p m \ {(0, 0)}, at least one of Q u,v(x) and Q u, v (x) has rank s.

20 Some auxiliary results (4/4) Lemma 3 (Luo and Feng, 008) T (u, v) = x F p m ω Tre 1 (Qu,v(x)) p (1.6) Table 1: Value distribution for T (u, v) Value Frequency (each) p m 1 ±ɛp m p m+e p m+e ±ɛp m+e (p m 1)p e (p m p m e p m e +1) (p e 1) (p m 1)(p m e +p m e ) (p m 1)(p m e p m e ) (p m 1)(p m e 1) (p e 1)

21 The Exponential sum S d (u, v) Then d: d ( p k + 1 ) (mod p m 1). θ: a fixed nonsquare in F p e. } S = {x pk +1 : x F p m = { x : x F } p m. Then, F p m = S θs. = S d (u, v) ω Trm 1 (ux+vx d ) p x F p m ( ) ω Trm 1 (ux pk +1 +vx p + ω Trm 1 p = 1 x F p m (uθx pk +1 +vθ d x ) ).

22 A relation between S d (u, v) and T (u, v) If d satisfies d 1 (mod p e 1), i.e., θ d = θ, = S d (u, v) ( ) ω Tre 1 p (Qu,v(x)) + ω Tre 1 p (θqu,v(x)) x F p m (.1) = 1 (T (u, v) + T (uθ, vθ)). If d satisfies d 1 + pe 1 (mod p e 1), i.e., θ d = θ, = S d (u, v) ( ) ω Tre 1 p (Qu,v(x)) + ω Tre 1 (θq u, v(x)) p x F p m (.) = 1 (T (u, v) + T (uθ, vθ)).

23 The definition of T (u, v) Define Denote c i = { T (u, v) = (T (u, v), T (uθ, vθ)) (.3) ɛp m+ie, i = 0,, p m+ie, i = 1, where ɛ = η e ( 1). T (u, v), T (uθ, vθ) {εc i ε = ±1, i = 0, 1, }. T (u, v) {(ε 1 c i1, ε c i ) ε 1, ε {±1}, i 1, i {0, 1, }}. (36 possible values!)

24 A characterization of T (u, v) Define two sets N ε,i = { (u, v) F p m T (u, v) = εc i}, M ε,i = { (u, v) F p m T (uθ, vθ) = εc } (.4) i, where ε {±1} and i {0, 1, }. Then, T (u, v) = (ε 1 c i1, ε c i ) (u, v) N ε1,i 1 M ε,i, where ε 1, ε {±1} and i 1, i {0, 1, }.

25 Some properties of N ε,i and M ε,i Let A be a set of F pm, and define (θ, θ)a = {(θ, θ)(a, b) (a, b) A} = {(aθ, bθ) (a, b) A} and θa = {(aθ, bθ) (a, b) A}. Lemma 4 Let N ε,i and M ε,i be the sets defined in (.4). Then, (i) for any ε {±1} and any i {0, 1, }, (θ, θ)n ε,i = M ε,i, N ε,i = (θ, θ)m ε,i ; (ii) for any ε {±1} and any i {0, }, θn ε,i = N ε,i, θm ε,i = M ε,i ; (iii) for any ε {±1}, θn ε,1 = N ε,1, θm ε,1 = M ε,1.

26 Some properties of T (u, v) Lemma 5 Let T (u, v) be the exponential sum defined in (1.6) and N be the number given in Lemma 6. Then (i) T (u, v)t (uθ, vθ) = p m ; (u,v) F p m (ii) T 3 (u, v)t (uθ, vθ) = p m N. (u,v) F p m

27 The number of solutions to a system of equations Lemma 6 With the notation above, let N denote the number of solutions of { x + y + z θw = 0, x pk +1 + y pk +1 + z pk +1 + θw pk +1 = 0, where (x, y, z, w) F 4 p m and θ is a fixed nonsquare in F pe. Then N = { p m+e + p m p e, if p e 1 (mod 4), p m p m+e p m + p e, if p e 3 (mod 4).

28 Proof sketch of Lemma 6 N 1 (a, b): the number of solutions to { x + y = a, x pk+1 + y pk+1 = b. N (a, b): the number of solutions to { z θw = a, z pk+1 + θw pk+1 = b. N : the number of solutions to the system in Lemma 5 N = N 1 (a, b)n (a, b). (a,b) F p m

29 Value distribution of T (u, v) Theorem 1 Let T (u, v) be the function defined by (.3). Then, the value distribution of T (u, v) as (u, v) runs through F p m is given in Table if p e 1 (mod 4) and in Table 3 if p e 3 (mod 4), where c i, i = 0, 1,, are defined by (0).

30 Table : Value distribution of T (u, v) if p e 1 (mod 4) Value Frequency (each) (p m, p m ) 1 (c 0, c 0 ) ( c 0, c 0 ) ( c 0, c 0 ), (c 0, c 0 ) (c 0, c 1 ), (c 1, c 0 ) ( c 0, c 1 ), (c 1, c 0 ) ( c 0, c 1 ), ( c 1, c 0 ) (c 0, c 1 ), ( c 1, c 0 ) (c 0, c ), (c, c 0 ) ( c 0, c ), ( c, c 0 ) ( c 0, c ), (c, c 0 ) (c 0, c ), ( c, c 0 ) (p m 1)(p e 1) 4(p e +1) (p m 1)[(p m +1)(p e 3)+4] 4(p e 1) (p m 1)(p m e +p m e ) 4 (p m 1)(p m e p m e ) 4 0 (p m 1)(p m e 1) (p e 1)

31 Table 3: Value distribution of T (u, v) if p e 3 (mod 4) Value Frequency (each) (p m, p m ) 1 (c 0, c 0 ) ( c 0, c 0 ) ( c 0, c 0 ), (c 0, c 0 ) (c 0, c 1 ), (c 1, c 0 ) ( c 0, c 1 ), (c 1, c 0 ) ( c 0, c 1 ), ( c 1, c 0 ) (c 0, c 1 ), ( c 1, c 0 ) (c 0, c ), (c, c 0 ) ( c 0, c ), ( c, c 0 ) ( c 0, c ), (c, c 0 ) (c 0, c ), ( c, c 0 ) (p m 1)[(p m +1)(p e 3)+4] 4(p e 1) (p m 1)(p e 1) 4(p e +1) (p m 1)(p m e +p m e ) 4 (p m 1)(p m e p m e ) 4 (p m 1)(p m e 1) (p e 1) 0

32 Value distribution of S d (u, v) Theorem Let S d (u, v) be the exponential sum defined by (1.). (i) When d 1 (mod p e 1), the value distribution of S d (u, v) is given in Table 4; (ii) When d 1 + pe 1 (mod p e 1), the value distribution of S d (u, v) is given in Table 5 if p e 1 (mod 4) and in Table 6 if p e 3 (mod 4).

33 Table 4: Value distribution of S d (u, v) when d 1 (mod p e 1) Value Frequency (each) p m 1 0 (p m 1)(p m p m e + 1) p m+e p m+e (p m 1)(p m e +p m e ) (p m 1)(p m e p m e )

34 Table 5: Value distribution of S d (u, v) when d 1 + pe 1 (mod p e 1) and p e 1 (mod 4) Value Frequency (each) p m 1 ±p m 0 ±1+ p e p m ±1 p e p m ± pe 1 p m (p m 1)(p e 1) 4(p e +1) (p m 1)[(p m +1)(p e 3)+4] (p e 1) (p m 1)(p m e +p m e ) (p m 1)(p m e p m e ) (p m 1)(p m e 1) (p e 1)

35 Table 6: Value distribution of S d (u, v) when d 1 + pe 1 (mod p e 1) and p e 3 (mod 4) ±p m Value Frequency (each) p m ± 1+ p e p m ± 1 p e p m ± pe +1 p m (p m 1)[(p m +1)(p e 3)+4] 4(p e 1) (p m 1)(p e 1) (p e +1) (p m 1)(p m e +p m e ) (p m 1)(p m e p m e ) 1 (p m 1)(p m e 1) (p e 1)

36 Recall some facts d ( p k + 1 ) (mod p m 1); S d (u, v) = C d (γ) = x F p m x F p m ω Trm 1 (ux+vx d ) p ; ω Trm 1 (x+γxd ) p 1 = S d (1, γ) 1.

37 Some properties of S d (u, v) Lemma 7 Let S d (u, v) be the exponential sum given in (1.). (i) S d (u, 0) = 0 for any u F p m; (ii) When d 1 (mod p e 1), or d 1 + pe 1 (mod p e 1) and p e 1 (mod 4), S d (0, v) = 0 for any v F p m; (iii) When d 1 + pe 1 { (mod p e 1) and p e 3 (mod 4), S d (0, v) ±p m } 1 for any v F p m and each value occurs p m 1 times as v runs through F p m; (iv) For any given u F p m, as v runs through F p m, S d(u, v) and S d (1, v) have the same value distribution.

38 Cross-correlation distribution for d Theorem 3 Let p, m and k satisfy Eq. (1.3), and d satisfy Eq. (1.4). (i) When gcd(d, p m 1) = 1, the value distribution of C d (γ) is given in Table 7 if d 1 (mod p e 1), and in Table 8 if d 1 + pe 1 (mod p e 1) and p e 1 (mod 4). (ii) When gcd(d, p m 1) =, i.e., d 1 + pe 1 (mod p e 1) and p e 3 (mod 4), the value distribution of C d (γ) is given in Table 9.

39 Table 7: Cross-correlation distribution for d 1 (mod p e 1) Value Frequency (each) 1 (p m p m e 1) p m+e 1 p m+e 1 (p m e +p m e ) (p m e p m e )

40 Table 8: Cross-correlation distribution for d 1 + pe 1 (mod p e 1) and p e 1 (mod 4) Value 1 ±p m 1 ±1+ p e p m 1 ±1 p e p m 1 ± pe 1 p m 1 Frequency (each) (p m+e 3p m 3p e +5) (p e 1) (p m +1)(p e 1) 4(p e +1) (p m e +p m e ) (p m e p m e ) p m e 1 p e 1

41 Table 9: Cross-correlation distribution for d 1 + pe 1 (mod p e 1) and p e 3 (mod 4) ±p m Value Frequency (each) p m+e p m p e 3 (p e +1) p m+e 3p m p e +3 4(p e 1) ± 1+ p e p m 1 ± 1 p e p m 1 ± pe +1 p m 1 1 (p m e +p m e ) (p m e p m e ) p m e 1 p e 1

42 Type 1 Type 1: odd prime p, m e 3 odd, e = gcd(k, m), d(p k + 1) (mod p m 1), d 1(mod p e 1), 3-valued, p m+e + 1. (Helleseth, 1976): odd prime p, m e 3 odd, e = gcd(k, m), d = pk +1, k m+e e even, 3-valued, p + 1. (Inverse is covered by Type 1.) T. Helleseth, Some results about the cross-correlation function between two maximal linear sequences, Discr. Math., 16: 09-3 (1976)

43 Recently, Ding et al. reported three new decimations for ternary m-sequences that give a three-valued cross-correlation function: 3 m m 1 3 h +1, m 3 odd, m+1 h even ( ) ( ) ( ) 3 m m m m 1, m( 7 (mod ) ( 8) ) 3 m m These decimations are of Type m 1, m 3 (mod 4) C. Ding, Y. Gao and Z. Zhou, Five families of three-weight ternary cyclic codes and their duals, IEEE Trans. Inf. Theory, 59(1): (013)

44 Type Type : odd prime p, p e 1 (mod 4), m e 3 odd, e = gcd(k, m), d(p k + 1) (mod p m 1), d 1 + pe 1 (mod p e 1), 9-valued, pe 1 p m + 1. (Luo and Feng, 008): odd prime p, p e 1 (mod 4), e = gcd(k, m), m e 3 odd, d = pk +1, k e odd, 9-valued, p e 1 p m + 1. (Inverse is covered by Type.) J. Luo and K. Feng, Cyclic codes and sequence from generalized Coulter-Matthews functions, IEEE Trans. Inf. Theory, 54(1): (008)

45 Type 3 Type 3: odd prime p, p e 3 (mod 4), m e 3 odd, e = gcd(k, m), d(p k + 1) (mod p m 1), d 1 + pe 1 (mod p e 1), gcd(d, p m 1) =, 9-valued, p e +1 p m + 1. ( Xia et. al, 010, and Choi et. al, 013 ): p 3 (mod 4), m odd, e m, m e 3, d = pm +1 p e +1 ± pm 1, gcd(d, p m 1) =, 9-valued, pe +1 p m + 1. (Special cases of Type 3.) Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor d = pn +1 p+1 pn 1, Appl. Algebra Eng. Commun. Comput., 1(5): (010) S. T. Choi, J. Y. Kim, and J. S. No, On the cross-correlation of a p-ary m-sequence and its decimated sequences by d = pn +1 p k +1 + pn 1, IEICE Trans. Commun., vol. E96-B(9): (013)

46 Thank you!