Two new classes of binary sequence pairs with three-level cross-correlation

A pair of binary sequences is generalized from the concept of a 
two-level autocorrelation function of single binary sequence. In 
this paper, we describe two classes of binary sequence pairs of 
period $N=2q$, where $q=4f+1$ is an odd prime and $f$ is an even integer. 
Those classes of binary sequence pairs are based on cyclic almost difference set pairs. They 
have optimal three-level cross-correlation, and either balanced or 
almost balanced.

When sequences s and t are identical, their correlation is called an auto-correlation and denoted by R s (τ ). Generally, a periodic binary sequence is said to have twolevel auto-correlation if all its out-of-phase correlation coefficients are some fixed constant which is different from the in-phase correlation value, that is (1.2) and when E = 0 we say that s is a perfect binary sequence. Sequences with good auto-correlation properties have wide applications. When the absolute value of E is as small as possible, it is quite useful for many applications in measurement, digital communication, and radar. An important problem in sequence design is to find sequences with optimal auto-correlation. However, it is believed that no binary perfect sequences of length other than 4 exist [9]. As a further possible remedy, Zhao in [18] introduced a new class of discrete signal sequence pair (s, t). The desired information is extracted from the received signal using the periodic cross-correlation of the transmitter signal s and the receiver signal t. In such way, many more useful signals can be used in many areas of engineering and sciences.
Let (s, t) be a sequence pairs and H be a subset of Z * N = Z N \ {0}. Let the periodic cross-correlation function of the sequence pair (s, t) where the set Z * − H is formed by the elements that are present in Z * , but not in H. When E 1 = E 2 = 0, the sequence pair (s, t) is called a perfect binary sequence pair, otherwise we call it the sequence pair with two-level or three-level correlation [11,12]. The concept of perfect binary sequence pair was first introduced in [18]. Jin and Song in [8] constructed a class of perfect binary sequence pair whose in-phase correlation is 4 and period is every multiple of 4. When E 1 = E 2 = −1, the binary sequence pair is called the ideal two-level correlation binary sequence pair which was first introduced in [12] and constructed in [7] based on cyclotomic classes of order 2, 4, and 6. From Lemma 2.2 in Section 2, we can prove that the even period binary sequence pair has even out-of-phase correlation values and the odd period binary sequence pair has odd out-of-phase correlation values. When N is even, it is concluded that the smallest difference number of the two out-of-phase correlations |E 1 − E 2 | is 4 and the smallest out-of-phase correlation values are {0, −4}, {0, 4} or {2, −2}. In this case, we say the sequence pair with optimal correlation value. Peng et. al in [13] gave two new constructions of binary sequence pairs with outof-phase correlations {0, −4} and period N ≡ 0 (mod 4) based on cyclotomy and interleaving technique.
In this paper, we will present two new classes of binary sequence pairs with outof-phase correlations {2, −2} and period N = 2q, where q ≡ 1 (mod 16) is prime and has a quadratic partition of form q = 1 + 4y 2 . The sequence will be either balanced or almost balanced, that is, two more zeros than ones in each period. See Table I for all the lengths N ≤ 15000 our construction applies for. This paper is organized as follows. Section 2 introduces the notation and the related results required for our constructions, including the important notion of difference set pair, almost difference set pair, and cyclotomy. In Section 3, we present the idea of our constructions for the binary sequence pairs with three-level correlation. In Sections 4 and 5, we give two new constructions of binary sequence pairs with out-of-phase correlations {2, −2} based on cyclotomy.

Preliminaries
It is well known that a binary sequence with two-level auto-correlation is equivalent to a difference set [1], and a binary sequence with three-level auto-correlation is equivalent to an almost difference set [4,5]. As an analogy, Xu in [17] proposed the concept of difference set pair (DSP) and established the relationship between binary sequence pair with two-level cross-correlation and difference set pair. Li and Ke in [10] introduced the concept of almost difference set pair (ADSP) and the relationship between binary sequence pair with three-level cross-correlation and almost difference set pair was also built. Many more examples can be seen in [7,11,16].
be two subsets of Z N with k and k elements, respectively. Let e = |U ∩ V |, where |A| denotes the number of elements in the set A. Then (U, V ) is called an (N, k, k , e, λ)-difference set pair (DSP) if every nonzero element g ∈ Z * N can be expressed in exactly λ ways in the form u i − v j ≡ g (mod N ), where u i ∈ U and v i ∈ V . Furthermore, let H be the nonzero subset of Z N , then (U, V ) is called an (N, k, k , e, λ 1 , λ 2 )-almost difference set pair (ADSP) if the list of differences u i − v j : u i ∈ U , and v j ∈ V contains each nonzero element of H exactly λ 1 times and each element of Z * N − H exactly λ 2 times. If λ 1 = λ 2 , (U, V ) is called a difference set pair. And if U = V , an almost difference set pair is called an almost difference set. We define the difference function Let s = (s(0), s(1), · · · , s(N − 1)) be an N -period binary sequence, and U be a subset of Z N . If U = {j : s(j) = 1, 0 ≤ j < N }, the set U is called the characteristic set of s, and s is called the characteristic sequence of U .
The following lemma establishes the connection between the correlation of binary sequence pair with three-level cross-correlation and almost difference set pair.

Lemma 2.2 ([10]
). Let U and V be two subsets of Z N , and H be the nonzero subset of Z N , s = (s(0), s(1), · · · , s(N − 1)) and t = (t(0), t(1), · · · , t(N − 1)) be two characteristic sequences of U and V respectively, then the relationship between the parameters (N, k, k , e, λ 1 , λ 2 ) of ADSP (U, V ) and periodic cross-correlation of binary sequence pair (s, t) is Cyclotomy is a powerful method for constructing almost difference set pairs. We introduce a number of results related to cyclotomy, which will be needed in the sequel.
Let q be a power of an odd prime, and let α be a generator of GF (q) * . Assume that q − 1 = ef , where e > 1 and f > 1 are integers. Define D (e) 0 to be the subgroup of GF (q) * generated by α e , and let D are called cyclotomic classes of order e with respect to GF (q). The cyclotomic numbers of order e, denoted (i, j) e , is the number of solutions of the equation The following lemma concludes several well-known properties of cyclotomic numbers. 3,14,15]). Let symbols and notations be the same as before. Then . Let symbols and notations be the same as before and g ∈ D (e) k . Then the number of solutions (x, y) of the equation  Table 2 together with the relations where q = x 2 + 4y 2 , x ≡ 1 (mod 4) is the proper representation of q = p m if p ≡ 1 (mod 4); the sign of y is ambiguously determined. Table 2. Relations of cyclotomic numbers of order 4 (f even) 3. The idea of our construction In this paper, we will give several new families of binary sequence pairs of period N = 2q with optimal correlation {2, −2}, where q is an odd prime. Finding sequence pairs with three-level correlation will be equivalent to constructing almost difference set pairs, as made clear before.
By the Chinese Remainder Theorem, , ω (mod q)) (see, [6]). Therefore, construction of almost difference set pairs over Z N is equivalent to that of almost difference set pairs over Then we may evaluate the difference function as follows: , and s, t be two characteristic sequences of U and V , respectively. If (U, V ) is an almost difference set pair and (s, t) is a binary sequence pair with optimal correlation {2, −2} of length N = 2q, then by Lemma 2.2, This gives us some hint about how we should choose our C i . In the following two sections, we shall use cyclotomic classes to form our C i and then look for conditions to ensure that our (U, V ) is an almost difference set pair. Such an almost difference set pair will give us a binary sequence pair with optimal correlation {2, −2}.

Construction of almost balance sequence pairs with optimal three-level cross-correlation
In the remainder of this paper, we consider cyclotomic classes D (4) i with respect to GF (q) and cyclotomic numbers of order 4. For simplicity, let D i denote D (4) i . And all symbols and notations are the same as in Scetions 2 and 3. Let |U | = k, |V |= k , s, t be two characteristic sequences of U and V . If k = k = N/2, we say that s, t are balanced. If k = k = N/2 − 1, we say that s, t are almost balanced. For almost balance sequence pairs, by (3.9) we have or N −6 4 , since N = 2q, which is equivalent to q−1 2 or q−3 2 . Now we shall use cyclotomic classes to form our C i and then look for conditions to ensure that our (U, V ) is an almost difference set pair.
Let q = 4f + 1 be a prime and f be even, and let and i, j, l, k are four pairwise distinct integers between 0 and 3. It is clear that We now consider the number of times each element appears in differences of two elements of U and V . Recall Eq 3.7, we have: For ω 1 = 0, ω 2 = 0, it is clear that For ω 1 = 1, ω 2 = 0, it is clear that For ω 1 = 0, ω 2 = 0, let ω 2 ∈ D h , then we have (4.14) Note that Eq 4.13 and Eq 4.14 are from Lemmas 2.4 and 2.3, respectively.
For ω 1 = 1, ω 2 = 0, let ω 2 ∈ D h , then we have We have defined that U and V are the characteristic sets for s and t. The quaternion (i, j, l, k) will be called the defining set for the sequences s and t. Recall Lemma 2.2 that the cross-correlation values are dependent on the difference function. The next result gives us the evaluation of this function for a certain defining set (i, j, l, k).
This completes the proof.

Construction of balance sequence pairs with optimal three-level cross-correlation
The sequence pairs with optimal three-level cross-correlation constructed in Section 4 are almost balanced. In this section, we modify the construction and give a class of balanced binary sequence pairs with optimal three-level cross-correlation.
This completes the proof.
For (U q , V q ), the conclusion is similar as Theorem 5.1.

Conclusion
In this paper, we presented several classes of almost difference set pairs based on cyclotomic class of order 4. And we got several classes of binary sequence pairs with three-level cross-correlation. As mentioned earlier, finding binary sequence pairs with three-level cross-correlation values is equivalent to finding almost difference set pairs with corresponding parameters. We also constructed almost difference set pairs by using cyclotomic classes of order 2. It is necessary to point out that the existed almost difference set pair would correspond to a binary sequence pair with bad cross-correlation. For example, let N = 2q, where q is an odd prime, we can construct almost difference set pair (U, V ) = ({0} × D ). The corresponding binary sequence pair has in-phase cross-correlation value 2, and out-phase cross-correlation value {2, −2}. It is clear that this kind of almost difference set pairs are unsuitable to get binary sequence pairs.