Binary Sequences of Even Period with 5-Level Autocorrelation and Their Variations for Optimum Odd Autocorrelation

Ⅰ. Introduction

Binary sequences with good autocorrelation properties are advantageous for synchronization in various communication systems^[1-4]. There have been a lot of results on the constructions of sequences(binary, almost binary, ternary, non-binary, polyphase, almost polyphase, etc) for the last half century or more for improved performance of various communications systems. Most of the sequences in this paper are over the binary alphabet F₂ = {0, 1} but the correlation is computed over with the correspondence

When we are given a binary sequence s = s(i) ∈ F₂ | i = {0, 1, ..., L − 1} of length L, we may consider its (usual) periodic expansion for computing its periodic autocorrelation. In that sense, we will use the term ‘length’ and ‘period’ of a binary sequence interchangeably. Then, the periodic autocorrelation of s at shift τ , denoted by C_s(τ), is given by

where i + τ is computed mod L. There is an alternative way of expanding the sequence s of length L periodically. Let s′ be a complement of s defined by

[TeX:] $$\begin{equation}s^{\prime}(i)=s(i)+1, \quad i=0,1, \ldots, L-1\end{equation}$$

Then, the alternative periodic expansion, calledodd-periodic expansion, is to repeat s in concatenationwith s′ of total length 2L. The autocorrelation of s with this type of expansion is called the odd autocorrelation of s. The odd autocorrelation at shift τ with 0 ≤ τ < L, denoted by [TeX:] $$\begin{equation}C_{\mathrm{s}}^{\text {odd }}(\tau)\end{equation}$$, is given by

where i + τ is computed mod L. In fact, C_s(τ) can be said to be an even autocorrelation.

The binary sequence s of even period L is said to have optimal autocorrelation^[3,5] if

[TeX:] $$\begin{equation}C_{\mathbf{s}}(\tau)= \begin{cases}0 \text { or }-4 & \text { if } L \equiv 0(\bmod 4) \\ 2 \text { or }-2 & \text { if } L \equiv 2(\bmod 4) .\end{cases}\end{equation}$$

for any τ ≠ 0. A lot of binary sequences of even period L above with (non-perfect) optimal autocorrelation have been constructed^[6-11], which would be best possible in terms of its periodic autocorrelation, since the perfect binary sequence is known only for L = 4^[3].

Instead of suppressing all the out-of-phase autocorrelation magnitudes, one started to consider having all-zero out-of-phase autocorrelations except for one non-zero value at some τ ≠ 0. It is called almost perfect sequences12 and investigated immediately by many others^[13-16] and further generalized into some non-binary zero-correlation zone sequences^[17-22]. We would like to mention that Pott and Bradley^[13,14] established some fundamental relation between cyclic relative difference sets and almost perfect binary sequences, which is very much similar to the relation between cyclic difference sets and binary sequences with two-level autocorrelation. For example, binary NTU sequences15 are closely related with binary sequences from a cyclic relative difference set^[16,23-25].

In search of sequences with better autocorrelation property, on the other hand, almost binary sequences or ternary sequences have been studied a lot^[13,26,27]. Here, an almost binary sequence is a ternary sequence over {0, +1, −1} but the term 0 occurs only once or a few times. Such sequences with ‘perfect’ autocorrelation have been found, for example, in [27].

Some reviews on the binary and almost binary sequences with good odd autocorrelation follows now. In [28], especially in Section Ⅳ. A. 4 there, a binary sequence of even period is said to have an odd optimal autocorrelation if the magnitude of out of phase odd autocorrelation is no larger than 2. The binary sequences with low or optimal odd periodic autocorrelation have also been proposed a lot^[15,26-29].

In this paper, we propose (Theorem 1) a balanced binary sequence of length 2u for some integer u with 5-level autocorrelation from a cyclic relative difference set. The out-of-phase autocorrelation magnitudes are all zero except for three indices at which the value is either 2u(τ = u, once) or 4 (at someτ ≠ 0, u twice). We observe the half period of this sequence and found that it has optimal odd autocorrelation (Corollary 1). We also observe that changing one specific bit of this balanced binary sequence results in an almost perfect sequence(Corollary 2). Furthermore, we explain some of previously known constructions for sequences with good (even or odd) autocorrelation are closely related with the two variations of the main result in this paper.

This paper is organized as follows. Section 2 introduces some preliminaries. Section 3 discusses main results of this paper. We propose a construction for a binary 5-level autocorrelation sequence and its two variations. Section 4 explains as concluding remarks the relation between our constructions andother known constructions, especially in [15],[26].

Ⅱ. Preliminaries

2.1 Notation

We will fix the following notation throughout thepaper.

· [TeX:] $$\begin{equation}\mathbb{Z}\end{equation}$$ is the set of integers and [TeX:] $$\begin{equation}\mathbb{Z}_L\end{equation}$$ is the integersmod L.

· [TeX:] $$\begin{equation}\mathbb{C}\end{equation}$$ is the set of complex numbers and [TeX:] $$\begin{equation}\mathbb{F}_q\end{equation}$$ is thefinite field of size q.

· Given a binary sequence [TeX:] $$\begin{equation}\mathbf{s}=\{s(i) \in \mathbb{F}_2 \mid i=0,1, …, L-1, \ldots, L-1\}\end{equation}$$ of length L, the periodic autocorrelation of s at shift τ, denoted by C_s(τ), is givenby(2) and the odd autocorrelation [TeX:] $$\begin{equation}C_{\mathrm{s}}^{\text {odd }}(\tau)\end{equation}$$ is givenby (3), both in the beginning of Introduction.

· For a subset X of [TeX:] $$\begin{equation}\mathbb{Z}_L\end{equation}$$ and an element [TeX:] $$\begin{equation}\tau \in \mathbb{Z}_L\end{equation}$$, we define

[TeX:] $$\begin{equation}\Delta_X(\tau) \triangleq|(\tau+X) \cap X|\end{equation}$$

where [TeX:] $$\begin{equation}\tau+X=\{\tau+x \mid x \in X\}\end{equation}$$. Note that

[TeX:] $$\begin{equation}\Delta_X(\tau)=\Delta_X(-\tau)\end{equation}$$

for any subset X and any τ.

2.2 Relative Difference Sets

Definition 1 (Relative Difference Sets^[13,30,31]). Let u, v, k, λ be positive integers. A (u, v, k, λ) relative difference set (RDS) D in the (additive) cyclic group Zuv relative to its subgroup [TeX:] $$\begin{equation}(u)=u\mathbb{Z}_{u v}\end{equation}$$ is a k-subset [TeX:] $$\begin{equation}\left\{d_1, \quad d_2, \quad \ldots, \quad d_k\right\} \subset \mathbb{Z}_{u v}\end{equation}$$, satisfying the following condition:

for any [TeX:] $$\begin{equation}d \in \mathbb{Z}_{u v}\end{equation}$$.

It is well-known that a (u, k, λ)-cyclic difference set (CDS) in [TeX:] $$\begin{equation}\mathbb{Z}_u\end{equation}$$ is a (u, v = 1, k, λ)-RDS in [TeX:] $$\begin{equation}\mathbb{Z}_u\end{equation}$$(relative to its trivial subgroup {0}). We are mostly interested in the case where v = 2^[31] and k = u − 1 so that the parameters become (u, v = 2, k = [TeX:] $$\begin{equation}\frac{u}{2}\end{equation}$$ − 1, λ = − 1), since the existence of a cyclic (u, v, k, λ )-RDS implies the relation

[TeX:] $$\begin{equation}k(k-1)=\lambda v(u-1)\end{equation}$$.

This set of parameters further implies that u itself must be even. The following provides an equal-size partition of [TeX:] $$\begin{equation}\mathbb{Z}_2u\end{equation}$$ so that a binary sequence can be constructed from such RDS D

Proposition 1. Let D be a (u, 2, u − 1, [TeX:] $$\begin{equation}\frac{u}{2}\end{equation}$$ − 1)-RDS in [TeX:] $$\begin{equation}\mathbb{Z}_2u\end{equation}$$ relative to its subgroup [TeX:] $$\begin{equation}\mathbb{uZ}_2u\end{equation}$$. Then, [TeX:] $$\begin{equation}\mathbb{Z}_2u\end{equation}$$ can be decomposed into the following disjoint union:

[TeX:] $$\begin{equation}\mathbb{Z}_{2 u}=D \cup(u+D) \cup\{z\} \cup\{u+z\},\end{equation}$$

for some z.

Proof. By (4), δ_D(u) = 0. Therefore,

[TeX:] $$\begin{equation}D \cap(u+D)=\emptyset\end{equation}$$

Therefore,

[TeX:] $$\begin{equation}\left|\mathbb{Z}_{2 u} \backslash(D \cup(u+D))\right|=2 u-2 k=2 .\end{equation}$$

Let [TeX:] $$\begin{equation}z \in \mathbb{Z}_{2 u} \backslash(D \cup(u+D))\end{equation}$$. Claim that u + z canbein neither D nor u + D. If [TeX:] $$\begin{equation}u+z \in D\end{equation}$$, then [TeX:] $$\begin{equation}z=u+u+z\in u+D\end{equation}$$. If [TeX:] $$\begin{equation}u+z \in u+D\end{equation}$$, then [TeX:] $$\begin{equation}z \in D\end{equation}$$. Therefore, two elements in [TeX:] $$\begin{equation}\mathbb{Z}_{2 u}(D \cup(u+D))\end{equation}$$ are z and u +z for some z.

Ⅲ. Binary Sequences with Favourable Autocorrelation from RDS

In this section, we propose a balanced binary sequence [TeX:] $$\begin{equation}\mathbf{s}=\left\{s(i) \mid i \in \mathbb{Z}_{2 u}\right\}\end{equation}$$ of length 2u with 5-level autocorrelation from a (u, 2, u − 1, [TeX:] $$\begin{equation}\frac{u}{2}\end{equation}$$ − 1)-RDS and discuss its two variations with (somewhat) better correlation property: the first is its half period portion of length u which has 4-level optimal oddautocorrelation; the second is its one-bit-changed version so that the result is almost balanced but with3-level autocorrelation so that it is almost perfect.

Theorem 1 (Main Construction). Let D be a (u, v= 2, k = u − 1, λ = [TeX:] $$\begin{equation}\frac{u}{2}\end{equation}$$ − 1)-RDS. Let [TeX:] $$\begin{equation}z \in \mathbb{Z}_{2 u}\end{equation}$$ so that [TeX:] $$\begin{equation}\mathbb{Z}_{2 u}\end{equation}$$ is partitioned as in Proposition 1

Define a binary sequence [TeX:] $$\begin{equation}\mathbf{s}=\{s(i) \mid i=0,1, \ldots, 2 u-1\}\end{equation}$$ as follows:

Then, the periodic (even) autocorrelation of s becomes:

Proof. The binary sequence [TeX:] $$\begin{equation}\mathbf{s}=\{s(i) \mid i=0,1, \ldots, 2 u-1\}\end{equation}$$ is balanced since

[TeX:] $$\begin{equation}|D \cup\{z\}|=|(u+D) \cup\{u+z\}| .\end{equation}$$

Note also that

This explains its special periodic property. When it is (cyclically) shifted by half the period, then the result is a complement of the original sequence. Therefore, its half period portion of length u is expanded odd-periodically, the result is the same as the (even) periodic expansion of the original sequence s of length 2u. This gives the value

[TeX:] $$\begin{equation}C_{\mathbf{s}}(\tau=u)=-2 u\end{equation}$$

For the other cases, we calcualte the autocorrelation as follows.

The first sum in (9) can be split into the following three cases: (a) [TeX:] $$\begin{equation}i \in D\end{equation}$$ and [TeX:] $$\begin{equation}i+\tau \in D\end{equation}$$ so that s(i) + s(i + τ) = 0, (b) [TeX:] $$\begin{equation}i \in D\end{equation}$$ and [TeX:] $$\begin{equation}i+\tau \in u+D\end{equation}$$ so that s(i) + s(i + τ) = 1 and (c) [TeX:] $$\begin{equation}i \in D\end{equation}$$ and [TeX:] $$\begin{equation}i+\tau \in\{z, u+z\}\end{equation}$$ so that s(i) + s(i + τ) = s(i + τ) which is 1 if i + τ = z and 0 if i + τ = u + z.

Then the case (a) becomes

[TeX:] $$\begin{equation}\sum_{\substack{i \in D \\ i+\tau \in D}}(+1)=|D \cap(-\tau+D)|=|\tau+D \cap D|=\Delta_D(\tau) .\end{equation}$$

Similarly, the case (b) becomes

[TeX:] $$\begin{equation}\sum_{\substack{i \in D \\ i+\tau \in u+D}}(-1)=-|D \cap(-\tau+u+D)|=-\Delta_D(u-\tau)\end{equation}$$

Similarly, the second sum can be split into the following three cases: (a) [TeX:] $$\begin{equation}i \in u+D\end{equation}$$ and [TeX:] $$\begin{equation}i+\tau \in D\end{equation}$$ so that s(i) + s(i + τ) = 1, (b) [TeX:] $$\begin{equation}i \in u+D\end{equation}$$ and [TeX:] $$\begin{equation}i+\tau \in u+D s(i)+s(i+\tau)=0\end{equation}$$, and (c) [TeX:] $$\begin{equation}i \in u+D\end{equation}$$ and [TeX:] $$\begin{equation}i+\tau \in\{z, u+z\}\end{equation}$$. Then, similar to the first two cases of the first sum, the cases (a) and (b) become:

[TeX:] $$\begin{equation}\sum_{\substack{i \in u+D \\ i+\tau \in D}}(-1)=-|(u+D) \cap(-\tau+D)|=-\Delta_D(u+\tau)\end{equation}$$

and

[TeX:] $$\begin{equation}\sum_{\substack{i \in u+D \\ i+\tau \in u+D}}(+1)=|(u+D) \cap(u-\tau+D)|=\Delta_{u+D}(\tau)=\Delta_D(\tau) .\end{equation}$$

Therefore, the autocorrelation of s at shift τ becomes:

Assume that τ ≠ 0 and τ ≠ u. Then, the first lineof C_s>(u) in (10) becomes

[TeX:] $$\begin{equation}2 \Delta_D(\tau)-\Delta_D(u+\tau)-\Delta_D(u-\tau)=0\end{equation}$$

since [TeX:] $$\begin{equation}\Delta_D(\tau)=\Delta_D(u \pm \tau)=\lambda=\frac{u}{2}-1\end{equation}$$. Now, the middlesum in (10) becomes the sum of only two termscorresponding to i = z − τ and i = u + z − τ :

[TeX:] $$\begin{equation} \begin{aligned} & \sum_{\substack{i \in D \cup(u+D) \\ i+\tau=z, u+z}}(-1)^{s(i)+s(i+\tau)} \\ = & (-1)^{s(z-\tau)+s(z)}+(-1)^{s(u+z-\tau)+s(u+z)} \\ = & (-1)^{s(z-\tau)}+(-1)^{s(u+z-\tau)+1} . \end{aligned} \end{equation}$$

Therefore, (10) finally becomes

[TeX:] $$\begin{equation} \begin{aligned} C_{\mathbf{s}}(\tau)= & (-1)^{s(z-\tau)}+(-1)^{s(u+z-\tau)+1} \\ & +(-1)^{s(z+\tau)}+(-1)^{s(u+z+\tau)+1} . \end{aligned} \end{equation}$$

Therefore,

[TeX:] $$\begin{equation}C_{\mathbf{s}}(\tau)= \begin{cases}-4, & z-\tau, z+\tau \in D \\ 4, & z-\tau, z+\tau \in u+D \\ 0, & \text { otherwise. }\end{cases}\end{equation}$$

This proves the theorem.

From the half-odd-periodic property of s> in the proof of Theorem 1, we may identify its half period as a binary sequence having an optimal odd autocorrelation:

Corollary 1. Let s be the binary sequence of period 2u constructed from Theorem 1 with some (u, 2, u − 1, u − 1)-RDS and an integer z satisfying the relation (5). Let the binary sequence t> of length u be any half period of s>. Then,

[TeX:] $$\begin{equation}C_{\mathbf{t}}^{o d d}(\tau)= \begin{cases}u, & \tau=0 \\ 2, & \tau,-\tau \in-z+u+D \\ -2, & \tau,-\tau \in-z+D \\ 0, & \text { otherwise. }\end{cases}\end{equation}$$

Remark 1. The binary sequence t constructed from Corollary 1 is optimal in the sense of minimizing the maximum magnitude of out of phase odd autocorrelation described in Section IV. A. 4 of [28], as mentioned in Introduction.

Remark 2. In Corollary 1, since s is balanced and u is even, a suitably chosen half period always allows t to be balanced.

Remark 3. All the known parameters of a (u, v = 2, k = u − 1, τ )-RDS are (u = q + 1, v = 2, k = q, [TeX:] $$\begin{equation}\lambda=\frac{q-1}{2}\end{equation}$$) for some odd prime power q^[13,32-34]. Indeed, our construction in Corollary 1 gives some binary sequences of period q + 1 with optimal odd autocorrelation. We may conjecture that it is the only way of getting a binary sequence of period q + 1 with optimal odd autocorrelation for some odd prime power q.

Now, we construct a sequence by changing the only one term of s> from the construction in Theorem 1 at index z. The resulting sequence is no longer balanced (we may call this almost balanced) but withbetter autocorrelation property which is only 3-level as in the following.

Corollary 2. Let s be the binary sequence of period 2u constructed from Theorem 1 with some (u, 2, u− 1, u − 1)-RDS and an integer z satisfying the relation (5). Let the binary sequence r be exactly the same as s except that it is complemented only at the index z. That is, r(z) = s(z) + 1. Then,

[TeX:] $$\begin{equation}C_{\mathbf{r}}(\tau)= \begin{cases}2 u, & \tau=0 \\ -2 u+4, & \tau=u \\ 0, & \text { otherwise }\end{cases}\end{equation}$$

Proof. We assume τ ≠ 0. Then, since r(i) and s(i) differ only at i = z, we have

When τ = u, then

[TeX:] $$\begin{equation}(-1)^{s(z+\tau)}+(-1)^{s(z-\tau)}=-2 .\end{equation}$$

When τ ≠ u, both z + τ and z − τ are in [TeX:] $$\begin{equation}D \cup(u+D)\end{equation}$$. If both z + τ and z − τ are in D, then

[TeX:] $$\begin{equation}(-1)^{s(z+\tau)}+(-1)^{s(z-\tau)}=+2\end{equation}$$

If both z + τ and z − τ are in u + D, then

[TeX:] $$\begin{equation}(-1)^{s(z+\tau)}+(-1)^{s(z-\tau)}=-2 .\end{equation}$$

If z + τ and z − τ are such that one is in D and the other is in u + D, then

[TeX:] $$\begin{equation}(-1)^{s(z+\tau)}+(-1)^{s(z-\tau)}=0 .\end{equation}$$

Therefore, we complete the proof from (7) and (11).

Remark 4. The binary sequence r = {r(i) | i = 0, 1, ..., 2(q + 1) − 1} in Cor. 2 is given in fact as

[TeX:] $$\begin{equation}r(i)= \begin{cases}0, & i \in D \\ 1, & i \notin D\end{cases}\end{equation}$$

Compare this with the definition of s in (6).

Remark 5. The value z (for one-bit change) in Cor. 2 can be in the range 0 ≤ z < u (first half) or in the range u ≤ z < 2u (second half). When we choose it to be the second half, we may identify the first half of r (which is exactly the same as the first half of s) to be the binary sequence t in Cor. 1

Remark 6. The binary sequence r in Cor. 2 has the property that its autocorrelation value is zero except for the only two time shifts. This type of binary sequences has been defined to be the almost perfect sequence^[12,14]. Pott and Bradley proved^[14] that they are equivalent to some (u, 2, u − 1, λ)-RDS in [TeX:] $$\begin{equation}\mathbb{Z}_{2 u}\end{equation}$$ relative to its subgroup [TeX:] $$\begin{equation}u \mathbb{Z}_{2 u}\end{equation}$$

Ⅳ. Some Interesting Relations and Concluding Remarks

We now conclude this paper by discussing the relation between our construction in Corollaries 1 and 2 from an RDS of parameters (u, 2, u − 1, [TeX:] $$\begin{equation}\frac{u}{2}\end{equation}$$ − 1) and other previous constructions, for example, those in [15], [26], [28] which were given without mentioning any RDS structure.

Nogami, Tada and Uehara^[15] in 2014 proposed some binary sequences as in the following. Let q be an odd prime power and [TeX:] $$\begin{equation}a \in \mathbb{F}_{q^2}\end{equation}$$ be a primitive element. Let [TeX:] $$\begin{equation}\operatorname{Tr}(a)=a+a^q\end{equation}$$ be the trace of [TeX:] $$\begin{equation}a \in \mathbb{F}_{q^2}\end{equation}$$ to [TeX:] $$\begin{equation}\mathbb{F}_q\end{equation}$$, and [TeX:] $$\begin{equation}\beta \triangleq \alpha^{q+1}\end{equation}$$ be primitive in [TeX:] $$\begin{equation}\mathbb{F}_q\end{equation}$$ The binary NTU sequence [TeX:] $$\begin{equation}\mathbf{x}=\{x(i) \mid i=0,1, \ldots, 2(q+1)-1\}\end{equation}$$ of length2(q + 1) is defined as [15]

We just note the values of i above so that x(i) =1. It is not difficult to observe that the set of thesevalues of i forms an RDS of parameters (q +1, 2, q, [TeX:] $$\begin{equation}\frac{q-1}{2}\end{equation}$$)^[30]. That is, claim that

is a (q + 1, 2, q, [TeX:] $$\begin{equation}\frac{q-1}{2}\end{equation}$$)-RDS in [TeX:] $$\begin{equation}\mathbb{Z}_{2(q+1)}\end{equation}$$ relative toitssub-group [TeX:] $$\begin{equation}(q+1) \mathbb{Z}_{2(q+1)}\end{equation}$$. For the proof, see Cor. 5.1.1in [30] or Sec. 2.2 in [13].

From Remark 4 and (13), we have the following conclusion:

Remark 7. The binary NTU sequence x of length2(q+ 1) in (12) is equivalent to the binary sequenceconstructed from Cor. 2 with the RDS Din (13).

A construction of some binary sequences with optimal odd autocorrelation is given in 2003 by Luke et.al.[28], where the binary odd optimal sequence is obtained from a ternary odd perfect sequence. To explain this technique, we must be careful of representing the sequence whether it is over the complex or its phases. A binary sequence uses the complex values +1 or −1 and its phase notation would be 0 or 1, corresponding to the relation in (1). Similar situation happens for ternary sequences. However, there are many different ways to consider the ternary alphabet. One choice is the complex values {0, +1, −1} and its phase notation would be {∗, 0, 1}. Using this phase notation for tenary sequences, Krengel defined in 2004 the odd perfect ternary sequence y = {y(i) | i = 0, 1, …, q} of length q + 1 as follows^[26]:

When we apply the technique by Luke et.al. to the ternary Krengel sequence above, we found that we obtain a binary sequence with odd optimal autocorrelation property. The resulting binary Krengel sequence y′ = {y′(i) | i = 0, 1, ..., q} of length q + 1 is given as follows:

It is interesting that the only difference between the binary Krengel sequence in (15) and the binary NTU sequence is the range of i at which the sequence is defined. The range of i for the binary Krengel sequence is half of those for the binary NTU sequence. From Remark 5, we see that the half period of the binary NTU sequence (which is called the modified Krengel sequence) is equivalent to the binary sequence in Cor. 1:

Remark 8. The modified Krengel sequence y′ of length q + 1 in (15) is equivalent to those constructed from Cor. 1 with the RDS D in (13).

Example 1. Let q = 3², α be a primitive element of [TeX:] $$\begin{equation}\mathbb{F}_3{ }^4\end{equation}$$ and [TeX:] $$\begin{equation}\beta \triangleq \alpha^{10}\end{equation}$$ be primitive in [TeX:] $$\begin{equation}\mathbb{F}_{32}\end{equation}$$. Then, the (10, 2, 9, 4)-RDS D in (13) becomes:

D = {4, 8, 10, 11, 12, 13, 16, 17, 19}.

Then, [TeX:] $$\begin{equation}\mathbb{Z}_{20}\end{equation}$$ is partitioned as follows:

[TeX:] $$\begin{equation}\mathbb{Z}_{20}=\{5\} \cup\{15\} \cup D \cup(10+D) .\end{equation}$$

Using the above RDS D and an integer [TeX:] $$\begin{equation}z \triangleq 5\end{equation}$$, the sequence s of period 20 constructed from Theorem 1 becomes:

s = (1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0).

The above sequence s is actually the same as the binary NTU sequence x of period 20 in (12). The first half period of s is the same as the binary Krengel sequence y′ of period 10 in (15). In this case, the binary Krengel sequence y′ is not balanced. By choosing the half period of s as specified in the underlined part, we obtain the following balanced binary sequence t of period 10 constructed from Corollary 1, as Remark 2 states that a half period of s can always be chosen for t to be balanced.

t = (1, 1, 0, 0, 1, 1, 0, 1, 0, 0).

Table 1 summarizes well-known constructions and the proposed construction of even-length balanced or almost balanced binary sequences. Here, an almost balanced binary sequence is defined as a binary sequence in which the number of 0s and 1s differs by 2. The proposed sequence has a period that conventional balanced or almost balanced binary sequences with ideal autocorrelation in [6], [9], [35], [36] do not typically possess. Instead, it exhibits alternative correlation properties such as 5-level autocorrelation, almost perfect autocorrelation, or ideal odd autocorrelation.

- q is an odd prime power - p₁ and p₂ are odd primes with p₁ ≡5 (mod 8) and p₂ ≡3 (mod 4) - p_t is an odd prime such that (p_t, p_t + 2) is a twin prime pair