On the Correlation Distribution of Delsarte Goethals Sequences

Transcription

1 On the Correlation Distribution of Delsarte Goethals Seuences Kai-Uwe Schmidt 9 July 009 (revised 01 December 009) Abstract For odd integer m 3 and t = 0, 1,..., m 1, we define Family V (t) to be a set of size m(t+1) containing binary seuences of period m+1. The nontrivial correlations between seuences in Family V (t) are bounded in magnitude by + (m+1)/+t. Families V (0) and V (1) compare favourably to the small and large Kasami sets, respectively. So far, the correlation distribution of Family V (t) is only known for t = 0. A general framework for computing the correlation distribution of Family V (t) is established. The correlation distribution of V (1) is derived, and a way to obtain the correlation distribution of V () is described. Keywords Galois ring, Low correlation, Quadratic Form, Seuence Set 1 Introduction We consider families of binary seuences for use in code-division multiple access (CDMA) systems (for background see [HK98], for example). The size and the maximum nontrivial correlation are key parameters of such designs. Large family size is reuired to support a large number of simultaneous users. Small nontrivial correlation is reuired to ensure message synchronisation and to minimise interference among different users. Knowledge of the distribution of the possible correlation values allows to evaluate the system performance without doing extensive simulations. For odd integer m 3 and t = 0, 1,..., m 1, Family V (t), to be defined in Section 4, is a set of binary seuences of period m+1, size m(t+1), and maximum nontrivial correlation + m+1 +t (see Theorem 9). The respective Families V (0), V (1), and V () coincide with Families Q(), Q(3), and Q(5), as defined by Helleseth and Kumar [HK98, p. 183], and are the largest known designs among all binary seuence families with asymptotically the same period and maximum nontrivial correlation. Table 1 shows that Families V (0) and V (1) compare favourably to the small and the large Kasami set, respectively. In general, Family V (t) is a subset of Family Q( t + 1). Following Nechaev s treatment [Nec91] of the Kerdock code, it can be shown that the seuences in V (t) are codewords of the Delsarte Goethals code DG (m + 1, m+1 t) [MS77, Ch. 15], punctured in two coordinates. Kai-Uwe Schmidt is with Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 1S6, Canada, kuschmidt@sfu.ca. He is supported by Deutsche Forschungsgemeinschaft (German Research Foundation) under Research Fellowship SCHM 609/1-1. 1

2 Table 1: Comparison of the Kasami Sets with Families V (0) and V (1) Family Period Family Size Max. Correlation m Kasami (Small set) m+1 1 m m+1 odd V (0) m+1 m + m+1 odd Kasami (Large set) m+1 1 3(m+1) + m m+3 1 (mod 4) Kasami (Large set) m+1 1 3(m+1) + m m+3 3 (mod 4) V (1) m+1 m + m+3 odd The distribution of the correlation values of Family V (0) was recently established by Tang, Helleseth, and Johansen [THJ08]. In the present paper, we pursue a rather different approach and use the theory of Z 4 -valued uadratic forms [Sch09]. After giving some background on Galois rings and fields in Section, we review Z 4 -valued uadratic forms in Section 3. In Section 4, we relate the correlation distribution of Family V (t) to the distribution of certain exponential sums and prove an upper bound on the magnitude of the nontrivial correlations of Family V (t). These exponential sums are then analysed in Section 5 for the specific cases of Families V (0) and V (1) using the theory of Z 4 -valued uadratic forms. In Section 6 we describe how to obtain the correlation distribution of Family V () and comment on the difficulty of extending our techniue to t >. Background on Galois Rings and Fields In this section, we briefly recall some facts about Galois rings and fields. Let R be a Galois extension of Z 4 of degree m. Then (R, +, ) is a Galois ring of characteristic 4 and cardinality 4 m. For details on Galois rings we refer to Nechaev [Nec91] and Helleseth and Kumar [HK98]. Define L := z R : z m = z} to be the set of Teichmuller representatives in R. Each z R can be uniuely written as We define an operation on L by z = a + b, where a, b L. (1) a b := a + b + ab. () Then (L,, ) is a Galois field of size m [Nec91, Statement ]. Let K be the prime subfield of L. Note that K = z Z 4 : z = z}. Informally, K can be identified with the subset 0, 1} of Z 4. The Frobenius automorphism σ on L is given by σ(x) = x, and the absolute trace function on L is the mapping tr : L K given by tr(x) := m 1 j=0 σ j (x).

3 It is easy to check that tr(σ(x)) = tr(x) and tr(αx βy) = α tr(x) β tr(y) for α, β K. Another useful property is that the mapping (x, y) tr(xy) is an inner product in L, as a vector space over K. This last fact can be used to prove the following elementary lemma. Lemma 1. Let E be a subspace of L, and define E := x L : tr(xy) = 0 for all y E}. Then ( 1) tr(cx) = x E E for c E 0 for c L \ E. The Frobenius automorphism ϱ on R is given by ϱ(a + b) := σ(a) + σ(b), where a, b L, and the absolute trace function on R is defined to be the mapping Tr : R Z 4 given by Tr(x) := m 1 j=0 ϱ j (x). We have Tr(ϱ(x)) = Tr(x) and Tr(αx + βy) = α Tr(x) + β Tr(y) for α, β Z 4. Moreover, the identity Tr(x) = tr(x) holds for each x L. 3 Z 4 -Valued Quadratic Forms In this section, we review some facts about Z 4 -valued uadratic forms. A symmetric bilinear form on L is a mapping B : L L K that satisfies symmetry B(x, y) = B(y, x) and the bilinearity condition B(αx βy, z) = α B(x, z) β B(y, z) for α, β K. (3) Moreover, B is called alternating if B(x, x) = 0 for each x L. Otherwise, B is called nonalternating. The radical rad(b) of B contains all elements x L such that B(x, y) = 0 for each y L. The bilinearity condition (3) implies that this set is a subspace of L. The rank of B is defined as rank(b) := m dim K (rad(b)). A Z 4 -valued uadratic form Q is a mapping Q : L Z 4 that satisfies Q(0) = 0 and Q(x y) = Q(x) + Q(y) + B(x, y), where B : L L K is a symmetric bilinear form. We say that the Z 4 -valued uadratic form Q has rank r and write rank(q) = r if its associated bilinear form has rank r. Moreover, Q is called alternating if its associated bilinear form is alternating. Otherwise, Q is called nonalternating. It is readily verified that Q takes values only in Z (and can therefore be identified with an ordinary 3

4 Z -valued uadratic form) if and only if Q is alternating. It is well-known [HK98, p. 1800] that the rank of an alternating Z 4 -valued uadratic form is always even. Given a Z 4 -valued uadratic form Q : L Z 4, we will be interested in the distribution of the values of the exponential sum χ Q (c) := x L i Q(x) ( 1) tr(cx) for c L (4) (where i := 1). For alternating Z 4 -valued uadratic forms we have the following classical result (see [HK98, Thm. 6.], for example). Theorem ([HK98, Thm. 6.]). Let Q : L Z 4 be an alternating Z 4 -valued uadratic form of (necessarily even) rank r. Then the distribution of χ Q (c) : c L} is given by: 0 occurs m r times ± m r/ occurs r 1 ± r/ 1 times. For nonalternating Z 4 -valued uadratic forms the distribution of the values (4) was established by the author in [Sch09]. As an immediate corollary of [Sch09, Thm. 5] we have the following. Theorem 3 ([Sch09, Thm. 5]). Let Q : L Z 4 be a nonalternating Z 4 -valued uadratic form of rank r, and write s := r/. The distribution of Re(χ Q (c)) : c L} is given by: 0 occurs m s 1 times ± m s/ occurs s ± s/ 1 times. The distribution of Im(χ Q (c)) : c L} is given by: 0 occurs m s 1 times ± m s/ occurs s times (each). In the rest of this section, we consider a particular set of Z 4 -valued uadratic forms, which has been studied by the author in [Sch09] following earlier work in [Sch08]. Let m > 0 be an odd integer, and let t be an integer satisfying 0 t m 1. For a Lt+1 write a = (a 0, a 1,..., a t ), and define Q a : L Z 4 by t Q a (x) := Tr(a 0 x) + Tr(a j x j +1 ). (5) It is straightforward to verify [Sch09] that Q a is a Z 4 -valued uadratic form, and that Q a is alternating if and only if a 0 = 0. The crucial property of Q a is that, if Q a is not identically zero, then the rank of Q a is at least m t. More generally, the rank distributions of the forms Q a and of the forms Q a that are alternating have been established in [Sch09]. In order to state the results, we recall that for real x and nonnegative integer k the 4-ary Gaussian binomial coefficient [ x k] is defined as [ ] x := 1 0 ] [ x k j=1 := (4x 1)(4 x 1 1) (4 x k+1 1) (4 k 1)(4 k 1 1) (4 1) for k > 0. We refer to [MS77, p. 444] for some properties of Gaussian binomial coefficients. 4

5 Theorem 4 ([Sch09]). Let m > 0 be an odd integer, write n := m 1, and let t be an integer satisfying 0 t n. For k = 0, 1,..., t define S k := t [ ] ( 1) j k 4 (j k ) n + 1 k ( m(t j+1) 1). n + 1 j j=k Let A j be the number of elements in Q a : a L t+1 } having rank j, and let B j be the number of elements in Q a : a L t+1 } that are alternating and have rank j. Then A 0 = B 0 = 1, and for j > 0 we have A j = B j = 0 except for [ ] n A m k = S k for k = 0, 1,..., t, k [ ] n A m k+1 = 4 n k+1 S k for k = 1,,..., t, k 1 [ ] n B m k+1 = S k for k = 1,,..., t. k 1 4 Family V (t) In this section, we define Family V (t) and the correlation between two members of this family. We then relate the possible correlation values of Family V (t) to certain exponential sums. The section will be concluded with a bound on the maximum nontrivial correlations between seuences in Family V (t). Let m 3 be an odd integer, let t be an integer satisfying 0 t m 1, and write := m. Let β be a primitive element in L. For γ Z 4 and a = (a 0, a 1,..., a t ) R L t define the uaternary seuence s a,γ (k)} of period 1 by s a,γ (k) := Tr(a 0 β k ) + Let π : Z 4 Z be the mapping defined by t Tr(a j β (j +1)k ) + γ. j=1 π(0) = 0, π(1) = 0, π() = 1, π(3) = 1. Then the binary seuence π(3 k s a,γ (k))} has period ( 1). subspace H of L as Define the (m 1)-dimensional H := a L : tr(a) = 0}, and let Γ(t) := (1 + c 0, c 1,..., c t ) : c 0 H, c 1,..., c t L} be a subset of R L t. Family V (t) is defined to be V (t) := π(3 k s a,γ (k))} : a Γ(t), γ K}. 5

6 The size of V (t) is t+1. For a, b R L t, γ, δ Z 4, and integer u, the correlation at displacement u between the seuences π(3 k s a,γ (k))} and π(3 k s b,δ (k))} is given by C (a,γ),(b,δ) (u) := 3 ( 1) π(3k+u s a,γ(k+u))+π(3 k s b,δ (k)). k=0 If (a, γ) = (b, δ) and u 0 (mod ( 1)), then the correlation value C (a,γ),(b,δ) (u) is called trivial (in which case it euals ( 1)), otherwise it is called nontrivial. The correlation distribution of Family V (t) is the distribution of the values in the multiset C (a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, 0 u < ( 1)}. In what follows, we shall establish a relation between the correlation distribution of Family V (t) and the distribution of certain exponential sums. A key step is the following lemma, which relates the correlation between two binary seuences of the form π(3 k s a,γ (k))} to the correlation of their uaternary counterparts. Lemma 5. For a, b R L t, γ, δ Z 4, and integer u we have C (a,γ),(b,δ) (u) = Re Proof. We readily verify the identity ( i 3u s a,γ(k+u) s b,δ (k) k=0 ( 1) π(α)+π(β) + ( 1) π( α)+π( β) = Re(i α β ) for α, β Z 4. (6) Since s a,γ (k)} has period 1 and one of 3 k+ 1 or 3 k is congruent 1 (mod 4) and the other is congruent 1 (mod 4), we have as reuired. C (a,γ),(b,δ) (u) = [( 1) ] π(3u s a,γ(k+u))+π(s b,δ (k)) + ( 1) π( 3u s a,γ(k+u))+π( s b,δ (k)) k=0 = k=0 ( ) Re i 3u s a,γ(k+u) s b,δ (k), by (6), We also need the following technical lemma on the number of solutions of a certain euation. Henceforth, let R be the following subset of R R := a + b : a L \ K, b L}. ). Lemma 6. Let j be an integer. H H L \ K of Then Given z R, let N(z) be the number of solutions (a, b, c) 3 j (1 + a)c (1 + b) = z. N(z) = 4 for z R 0 otherwise. 6

7 Proof. Using (), we find by direct inspection 3 j (c 1) + (ac b c 1) for even j (1 + a)c (1 + b) = (c 1) + (ac b c c 1) for odd j. When c ranges over L \ K, so does c 1. Therefore, N(z) = 0 for z R. It remains to show that, for each c L \ K, the number of solutions (a, b) H H of ac b = y is eual to /4 for each y L. Now let c L \ K be arbitrary, but fixed. Application of Lemma 1 with E = H (and therefore E = K) shows that the number of solutions a H of tr(ac) = ɛ is eual to /4 for each ɛ K. Hence, as a ranges over H, the set ac b : b H} is either H or L \ H, and each of these cases occurs /4 times. We conclude that, for each y L, the number of solutions (a, b) H H of ac b = y is /4, which completes the proof. We are now in a position to prove the following result, which relates the correlation distribution of Family V (t) to the distribution of certain exponential sums. Theorem 7. For a R L t and γ Z 4 define (a) The distribution of the correlation values is as follows: ( ζ(a, γ) := Re i ). sa,γ(x) C (a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, 0 u < ( 1), u 0, 1}} (b) The distribution of the correlation values is as follows: k=0 ζ(a, 0) (a R L t 3 ) occurs 4 t+1 times, ζ(a, 1) (a R L t 1 ) occurs 4 t+1 times, ζ(a, ) (a R L t 1 ) occurs 4 t+1 times, ζ(a, 3) (a R L t 3 ) occurs 4 t+1 times. C (a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, u 0, 1}} 0 occurs (t+1) times, ζ(a, 0) (a H L t ) occurs t+1 times, ζ(a, 0) (a (L \ H) L t 1 ) occurs t+1 times, ζ(a, ) (a (L \ H) L t 1 ) occurs t+1 times. 7

8 Proof. For a, a R L t and integer u we have s a,γ (k + u)} = s a,γ(k)}, where, writing a = (a 0, a 1,..., a t ) and a = (a 0, a 1,..., a t), a 0 = a 0 β u a j = a j β (j +1)u for j = 1,,..., t. Since β has order 1 and gcd(, 1) = 1, we then have for fixed b R L t and γ, δ Z 4 3 u s a,γ (k + u)} s b,δ (k)} : a Γ(t), 0 u < ( 1), u 0 even} = s a,γ (k)} s b,δ (k)} : a Γ (t)} = s a b,γ δ (k)} : a Γ (t)}, (7) where Γ (t) = ((1 + c 0 )d, c 1,..., c t ) : c 0 H, c 1,..., c t L, d L \ K}. Similarly, since β 1 = 1, 3 u s a,γ (k + u)} s b,δ (k)} : a Γ(t), 0 u < ( 1), u 1 odd} = 3s a,γ (k)} s b,δ (k)} : a Γ (t)} = s 3a b,3γ δ (k)} : a Γ (t)}. (8) Then (7), (8), and application of Lemma 5 give C(a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, 0 u < ( 1), u / 0, 1} } = ζ(3 j a b, 3 j γ δ) : a Γ (t), b Γ(t), γ, δ K, j 0, 1} }. (9) Now, for a, b, z R L t, write a = (a 0, a 1,..., a t ) b = (b 0, b 1,..., b t ) z = (z 0, z 1,..., z t ). Given j 0, 1} and z R L t, let N j (z) be the number of solutions (a, b) Γ (t) Γ(t) of 3 j (a 0, a 1,..., a t ) (b 0, b 1,..., b t ) = (z 0, z 1,..., z t ). (10) Then we have by application of Lemma 6 N j (z) = 1 4 t+1 for z R L t 0 otherwise (11) for either j. For j 0, 1} and y Z 4, let S j (y) be the number of solutions (γ, δ) K K of 3 j γ δ = y. Then S 0 (0) =, S 0 (1) = 1, S 0 () = 0, S 0 (3) = 1, S 1 (0) = 1, S 1 (1) = 0, S 1 () = 1, S 1 (3) =. (1) 8

9 Now (a) follows by combining (9), (11), and (1). To prove (b), observe that we have by Lemma 5 C(a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, u 0, 1} } = ζ(3 j a b, 3 j γ δ) : a, b Γ(t), γ, δ K, j 0, 1} }, (13) where we used that s a,γ (k)} has period 1 and (mod 4). Given j 0, 1} and z R L t, let M j (z) be the number of solutions (a, b) Γ(t) Γ(t) satisfying (10). Since m is odd, we have tr(1) = 1, and therefore, By direct inspection, we then find that c H (1 c) L \ H. M 0 (z) = 1 t+1 for z H L t 0 otherwise (14) and M 1 (z) = 1 t+1 for z (L \ H) L t 0 otherwise. (15) Now (b) follows by combining (13), (14), (15) and (1). In particular, the correlation value 0 occurs ( S0 (1) + S 0 (3) + S 1 (1) + S 1 (3) ) ( 1 t+1) = (t+1) times, since for a L L t and γ 0, the seuence i sa,γ(k) } is imaginary, hence ζ(a, γ) = 0. It will be convenient to rephrase Theorem 7. To this end, let a = (a 0, a 1,..., a t ) L t+1, c L, and γ Z 4, and observe that we have by (1) s (a0 +c,a 1,...,a t),γ(k) = Q a (β k ) + Tr(cβ k ) + γ for each integer k, where Q a is the Z 4 -valued uadratic form defined in (5). Hence ) ζ((a 0 + c, a 1,..., a t ), γ) = Re (i γ i Qa(βk )+ Tr(cβ k ) k=0 ( = Re i γ[ 1 + i Qa(x) ( 1) tr(cx)]). x L We therefore deduce the following. Corollary 8. Let a L t+1, write a = (a 0, a 1,..., a t ), and let Q a : L Z 4 be as defined in (5). For γ Z 4 write ( ξ(a, c, γ) := Re i γ[ 1 + i Qa(x) ( 1) tr(cx)]). x L 9

10 (a) The distribution of the correlation values is as follows: C (a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, 0 u < ( 1), u 0, 1}} ξ(a, c, 0) (a (L \ K) L t 3, c L) occurs 4 t+1 times, ξ(a, c, 1) (a (L \ K) L t 1, c L) occurs 4 t+1 times, ξ(a, c, ) (a (L \ K) L t 1, c L) occurs 4 t+1 times, ξ(a, c, 3) (a (L \ K) L t 3, c L) occurs 4 t+1 times. (b) The distribution of the correlation values is as follows: C (a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, u 0, 1}} 0 occurs (t+1) times, ξ(a, c, 0) (a L t+1, a 0 = 0, c H) occurs t+1 times, ξ(a, c, 0) (a L t+1 1, a 0 = 0, c L \ H) occurs t+1 times, ξ(a, c, ) (a L t+1 1, a 0 = 0, c L \ H) occurs t+1 times. Using Corollary 8, we can easily bound the magnitude of the nontrivial correlations between seuences in Family V (t) as follows. Theorem 9. The nontrivial correlation values of Family V (t) are bounded in magnitude by + t. Proof. Let ξ(a, c, γ) be as defined in Corollary 8. By Corollary 8, in the correlation distribution of Family V (t) the trivial correlation value ξ(0, 0, 0) = ( 1) occurs exactly V (t) = t+1 times, and the value ξ(0, 0, ) never occurs. Therefore, the nontrivial correlation values of Family V (t) are contained in the set ξ(a, c, γ) : a L t+1, c L, γ Z 4, (a, c, γ) nonzero}. (16) We have ξ(0, 0, γ) = 0 for γ 0, and from Lemma 1, applied with E = L, we conclude that ξ(0, c, γ) = Re(i γ ) for c 0. Theorem 4 asserts that the rank of Q a is at least m t for nonzero a. We then conclude from Theorems and 3 that the values in the set (16) are bounded in magnitude by + m+1 +t. Theorem 9 was proved by Helleseth and Kumar [HK98, p. 1833] for t 0, 1}. In general, Family V (t) is a subset of Family Q( t + 1), as defined by Helleseth and Kumar in [HK98]. In [HK98, p. 183] it was proved that all nontrivial correlation values of Family Q( t +1) are at most + t+1 in magnitude, which differs from the bound in Theorem 9 approximately by a factor. 10

11 Table : Correlation Distribution of Family V (0). value freuency ( 1) 0 ± ± ± ( ) (each) 4 ( )( ) 3 4 ( )( ± ) 5 Correlation Distribution of Families V (0) and V (1) In this section, we use Corollary 8 to establish the correlation distribution of Families V (0) and V (1). The correlation distribution of Family V (0) has been derived in [THJ08]. Based on Corollary 8, we give a short alternative proof for this result. Theorem 10. The distribution of the correlation values is given in Table. C (a,γ),(b,δ) (u) : a, b Γ(0), γ, δ K, 0 u < ( 1)} (17) Proof. Let ξ(a, c, γ) be as defined in Corollary 8. The Z 4 -valued uadratic form Q a (x) = Tr(ax) is nonalternating and has rank m for each a L \ K. (Alternatively, the last fact can be deduced from Theorem 4 with t = 0.) Application of Theorem 3 then gives the number of occurrences of the correlation values ± and ± ±, identified in Corollary 8 (a). The number of occurrences of the correlation values ( 1), 0, and ±, identified in Corollary 8 (b), are easily established using the fact ( 1) tr(cx) 0 for c L \ 0} = for c = 0 (see Lemma 1) and that 0 H. x L In what follows, we establish the correlation distribution of Family V (1). We treat the respective exponential sums arising in Corollary 8 (a) and in Corollary 8 (b) in Lemmas 11 and 1 below. Lemma 11. The distribution of the correlation values is given in Table 3. C (a,γ),(b,δ) (u) : a, b Γ(1), γ, δ K, 0 u < ( 1), u 0, 1}} (18) 11

12 Table 3: Correlation Distribution of Family V (t) at Shifts not in 0, 1}. value 0 ± ± ± ± ± ± 3 6 freuency ( ) ( ) ( ) ( )( + 4) (each) 1 ( )( + 4)( ) 4 ( )( + 4)( ± ) 3 1 ( ) (each) 6 ( ) ( 8 8 ) ( ) ( 8 ± 8 ) Proof. We adopt the notation of Corollary 8. The Z 4 -valued uadratic form Q a is nonalternating for each a (L \ K) L. Therefore, in view of Theorem 3, in order to establish the distribution of the values in (18), it is sufficient to determine the numbers C j := Q a : a (L \ K) L, rank(q a ) = j or j 1} for j = 0, 1,..., m+1. Let A j be the number of elements in Q a : a L L} having rank j, and let B j be the number of elements in Q a : a L L} that are alternating and have rank j. Since Q a is alternating if and only if a 0 = 0 and for a 0 0 we have rank(q (a0,a 1 )) = rank(q (1,a1 /a 3 0 )), we readily verify that C j = 1 (A j + A j 1 B j ) for j = 0, 1,..., m+1. From Theorem 4, A 0 = B 0 = 1, and for j > 0, we have A j = B j = 0 except for Therefore, C j is eual to zero, except for A m = 1 3 ( 1)( + 4), A m 1 = ( 1), A m = 1 3 ( 1)( 1), B m 1 = 1. C m+1 = 1 3 ( )( + 4), C m 1 = 3 ( ). Now the lemma follows from Theorem 3 and Corollary 8 (a). 1

13 Table 4: Correlation Distribution of Family V (t) at Shifts in 0, 1}. value freuency ( 1) 0 4 ± ± 4 8 ( ) 3 4 ( 4 8 ) ( 3 4 1)( 4 ± 8 ) Lemma 1. The distribution of the correlation values is given in Table 4. C (a,γ),(b,δ) (u) : a, b Γ(1), γ, δ K, u 0, 1}} Proof. We use the notation of Corollary 8. Note that for a 0 = 0 we have Q a (x) = Tr(a 1 x 3 ) = tr(a 1 x 3 ). For b, c L define τ(b, c) := x L( 1) tr(bx3 )+tr(cx), so that ξ((0, b), c, γ) = Re(i γ [ 1+τ(b, c)]) for all b, c L and all γ Z 4. In view of Corollary 8 (b), to prove the lemma, it is sufficient to determine the distribution of and the distribution of τ(b, c) : b L, c H} (19) τ(b, c) : b L, c L \ H}. (0) For a 0 = 0 the Z 4 -valued uadratic form Q a (x) = tr(a 1 x 3 ) is alternating, and Theorem 4 can be used to verify that its rank euals m 1 for a 1 0 and euals zero for a 1 = 0. Therefore, by Theorem, the distribution of τ(b, c) : b, c L} (1) is given by: occurs 1 time 0 occurs ( 1)( + 1) times ± occurs ( 1)( 4 ± 8 ) times. 13

14 For c 0 we have τ(b, c) = τ(b/c 3, 1) for each b L. Therefore, for fixed c L, the distribution of τ(b, c) : b L} depends only on whether c = 0 or c 0. Since we know the distribution of (1), in order to obtain the distributions of (19) and (0), it is sufficient to know τ(b, 0) for all b L. Clearly, τ(0, 0) =. When m is odd, gcd(3, 1) = 1, hence for b 0, the mapping x bx 3 is a permutation on L. By Lemma 1, applied with E = L, we then have for b = 0 τ(b, 0) = 0 for b 0. Since 0 L is contained in H, the distribution of (19) is given by: and the distribution of (0) is given by: occurs 1 time 0 occurs ( 1) + 1 times ± occurs ( 1)( 4 ± 8 ) times, 0 occurs ± occurs 4 times ( 4 ± 8 ) times. Now the lemma is a straightforward conseuence of Corollary 8 (b). Combination of Lemmas 11 and 1 gives the correlation distribution of Family V (1). Theorem 13. The distribution of the correlation values is given in Table 5. C (a,γ),(b,δ) (u) : a, b Γ(1), γ, δ K, 0 u < ( 1)} 6 Remarks on the Case When t > 1 This paper grew out in an attempt to establish the correlation distribution of Family V (t) for all t satisfying 0 t m 1. However, it turned out to be difficult to handle the cases when t > 1. In this section, we comment on this issue. It is not hard to establish, for general t, the distribution of the correlation values between seuences in Family V (t) for shifts u not in 0, 1}. This can be done by adapting the proof of Lemma 11. The difficulty arises in the analysis of the distribution of the remaining correlation values C (a,γ),(b,δ) (u) : a, b Γ(t), γ, δ K, u 0, 1}}. () Given b L t, write b = (b 1, b,..., b t ). For b L t and c L define f b (x) := t tr(b j x j +1 ) j=1 14

15 Table 5: Correlation Distribution of Family V (1). value freuency ( 1) 0 ± ± ± ± ± ± 3 ( ) ( ) ( ) + ( ) 3 6 ( )( + 4) (each) 1 ( )( + 4)( ) 4 ( )( + 4)( ± ) 3 1 ( ) (each) 6 ( ) ( 8 8 ) ( 4 8 ) ( ) ( 8 ± 8 ) + ( 3 4 1)( 4 ± 8 ) and τ(b, c) := x L( 1) f b(x)+tr(cx). By Corollary 8 (b), in order to compute the distribution of the values in (), we need to determine the distribution of τ(b, c) when (b, c) ranges over L t H and when (b, c) ranges over L t L \ H. Using Theorems 4 and, we can establish the distribution of τ(b, c) when (b, c) ranges over L t L. For c 0 the variable substitution x x c gives τ(b, c) = τ(b, 1), where b = (b 1, b,..., b t) and b j = b j c j +1 for j = 1,,..., t. It is therefore sufficient to determine the distribution of τ(b, 0) when b ranges over L t. For t = we have f b (x) = tr(b 1 x 3 ) + tr(b x 5 ). In this case, one can deduce the distribution of τ(b, 0) : b L t } from a recent result by Johansen and Helleseth [JH09] on the crosscorrelation between the two m-seuences tr(β 3k )} and tr(β 5k )}, where β is a primitive element in L. In this way, the correlation distribution of Family V () can be established. The result however is uite complex and is omitted. In the general case, the computation of the distribution of the values in the multiset τ(b, 0) : b L t } is left as a challenging open problem. 15

16 References [HK98] T. Helleseth and P. V. Kumar. Seuences with low correlation. In V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory. Amsterdam, The Netherlands: Elsevier, [JH09] [MS77] A. Johansen and T. Helleseth. A family of m-seuences with five-valued cross correlation. IEEE Trans. Inf. Theory, 55(): , Feb F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North Holland, [Nec91] A. A. Nechaev. Kerdock code in a cyclic form. Discrete Math. Appl., 1(4): , [Sch08] [Sch09] K.-U. Schmidt. Symmetric bilinear forms over finite fields of even characteristic. Submitted for publication, 008. K.-U. Schmidt. Z 4 -valued uadratic forms and uaternary seuence families. IEEE Trans. Inf. Theory, 55(1): , Dec [THJ08] X. Tang, T. Helleseth, and A. Johansen. On the correlation distribution of Kerdock seuences. In Proc. of Seuences and Their Applications (SETA), volume 503 of Lecture Notes in Computer Science, pages New York: Springer Verlag,