Autocorrelation Values of Generalized Cyclotomic Sequences with Period pⁿ⁺¹

Abstract

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period $p^{n+1}$, where $p=dR+1$ is an odd prime, $d,R$ are two non-negative integers, and $n>0$ is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period $p^{n+1}$ $(n\ge 0)$ with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences.

1. Introduction

Pseudorandom sequences with good randomness properties are widely applied in simulation, radar systems, spread-spectrum communication systems, ranging systems, software testing, global positioning systems, channel coding, code-division multiple-access (CDMA) systems, and stream ciphers [1,2,3,4,5]. If a sequence $s^{\infty }=(s_0,s_1,\cdots )$ over a field $\mathbb{F}$ satisfies $s_n=s_{n+T}$, $n\ge 0$, then $s^{\infty }$ is said to be T-periodic. For a binary sequence $s^{\infty }$ over the binary field ${\mathbb{F}}_2$, if $s_n=1$ if and only if $n\in D$, then set D is called the characteristic set or support set of $s^{\infty }$. Two important tools for measuring the randomness properties of pseudorandom sequences are autocorrelation values and linear complexity. The periodic autocorrelation value $C_s\left(\tau \right)$ of a T-periodic binary sequence $s^{\infty }$ is defined by:

$$C_s\left(\tau \right)=\sum _{n=0}^{T-1}{(-1)}^{s_{n+\tau }+s_n},$$

where $0\le \tau \le T-1$. The periodic autocorrelation value reflects global randomness. The linear complexity $L\left(s^{\infty }\right)$ of the sequence $s^{\infty }$ is the length of the shortest linear feedback shift register which can generate $s^{\infty }$. It is defined to be the least positive integer L satisfying:

$$s_t=c_1{s}_{t-1}+c_2{s}_{t-2}+\cdots +c_L{s}_{t-L}\mathrm{for}\mathrm{all}t\ge L,$$

where $c_1,\cdots ,c_L\in \mathbb{F}$.

The construction and randomness properties analysis of pseudorandom sequences are the core problem for pseudorandom sequences theory. Ding and Helleseth [6] introduced a generalized cyclotomy with respect to $p_1^{e_1}{p}_2^{e_2}\cdots p_t^{e_t}$, and introduced a class of new binary sequences whose characteristic set is selected as $\left\{0\right\}\cup [{\cup }_{d\mid n,d>1}\frac{n}{d}{D}_1^{\left(d\right)}]$. Several generalized cyclotomic sequences were constructed based on this generalized cyclotomy. Ding [7] obtained lower bounds on the linear complexity of generalized cyclotomic sequences with period $p^2$. He also determined the exact values of autocorrelation of these sequences by using certain formulas for generalized cyclotomic numbers. Later Ding [8] calculated the linear complexity of these sequences. In contrast to [7], this time he obtained the exact linear complexity, and the results did not require any special requirement for the prime. Park, Hong, and Eun [9] found technical errors in [8], so they corrected the errors and re-established the main results on the linear complexity. Yan, Sun, and Xiao [10] studied new generalized cyclotomic binary sequences with respect to $p^2$, which are a special case of those whose characteristic set are $\left\{0\right\}\cup [{\cup }_{d\mid n,d>1}\frac{n}{d}{D}_1^{\left(d\right)}]$ in [6]. Results indicate that these sequences possess high linear complexity. The exact values of autocorrelation are five-valued for $p\equiv 1(mod4)$, and three-valued for $p\equiv 3(mod4)$. The exact same results on the linear complexity and the exact values of autocorrelation of these new binary sequences were presented in [11,12], respectively. Kim, Jin, and Song [13] calculated the exact values of autocorrelation and linear complexity of prime cube sequences with period $p^3$. Results show that the autocorrelation values of these prime cube sequences are seven-valued for $p\equiv 1(mod4)$, and four-valued for $p\equiv 3(mod4)$. The linear complexity of generalized cyclotomic sequences of period $p^n$ for any $n>0$ is calculated in [14,15,16,17]. The autocorrelation values of those sequences of order 2 and period $p^n$ are calculated in [18]. In this paper the described results on autocorrelation values are a generalization of the known ones from [10,11,12,13,19]. In contrast to [11,12,13], we present a simpler proof by using certain identities involving character sums. The proof of the results on autocorrelation values in [11,12,13] are all based on generalized cyclotomic numbers. The method for computing the autocorrelation values of the binary sequences in [10] is based on their characteristic polynomials. The autocorrelation values of our theorem are entirely consistent with those in [18], but the described results in this paper do not require the restriction $d=2$. In addition, the parameters of our sequences are more complicated than those in [18], but the proof of our results is shorter and simpler.

In 2011 Edemskiy [20] proposed a method for computing the linear complexity of $p^{n+1}$-periodic generalized cyclotomic binary sequences. For details, suppose g is a primitive root of $p^{n+1}$, where $p=dR+1$ is an odd prime, $d,R$ are two non-negative integers, and $n>0$ is a positive integer. Let $D_0^{\left(p^{n+1}\right)}=\left(g^d\right)$ be a cyclic subgroup of the multiplicative group ${\mathbb{Z}}_{p^{n+1}}^{*}$. Define $D_i^{\left(p^{n+1}\right)}=g^i{D}_0^{\left(p^{n+1}\right)}$, $i=0,1,\cdots ,d-1$, $p^m{D}_i^{\left(p^{n+1}\right)}=\left\{p^{m}a:a\in D_i^{\left(p^{n+1}\right)}\right\}$, and $m=0,1,\cdots ,n$. Then:

$${\mathbb{Z}}_{p^{n+1}}^{*}=\bigcup _{i=0}^{d-1}{D}_i^{\left(p^{n+1}\right)}\mathrm{and}{\mathbb{Z}}_{p^{n+1}}=\bigcup _{m=0}^n\bigcup _{i=0}^{d-1}{p}^m{D}_i^{\left(p^{n+1}\right)}\cup \left\{0\right\}.$$

Edemskiy defined the binary sequence $s^{\infty }$ with period $p^{n+1}$ as follows:

$$\begin{array}{c}\hfill s_i=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}(imod{p}^{n+1})\in C,\hfill \\ 0,\hfill & \mathrm{if}(imod{p}^{n+1})\notin C,\hfill \end{array}\right.\end{array}$$

(1)

where the characteristic set of $s^{\infty }$ is selected as:

$$C=\bigcup _{m=0}^n\bigcup _{i\in I_m}{p}^m{D}_i^{\left(p^{n+1}\right)}\cup \left\{0\right\},I_m\subset \left\{0,1,\cdots ,d-1\right\}.$$

He proposed a method for computing the linear complexity of $s^{\infty }$ and considered some special given $I_m$. In this paper for even d we shall choose special subsets:

$$\begin{array}{c}\hfill I_0=I_1=\cdots =I_n=\left\{1,3,5,\cdots ,d-1\right\},\end{array}$$

(2)

and compute the exact values of autocorrelation of this special generalized cyclotomic sequence with period $p^{n+1}$ $(n\ge 0)$.

2. Sums Involving Legendre Symbol

We need the following lemma from [21].

Lemma 1.

Let $p>2$ be a prime, $l\in \mathbb{Z}$, and ${\displaystyle \left(\frac{·}{p}\right)}$ denote the Legendre symbol modulo p. Then we have:

$$\sum _{t=0}^{p-1}\left(\frac{t(t+l)}{p}\right)=\left\{\begin{array}{cc}p-1,\hfill & p\mid l,\hfill \\ -1,\hfill & p\nmid l,\hfill \end{array}\right.$$

where p∣l indicates that p is a divisor of l.

Lemma 2.

Let ${\displaystyle \left(\frac{·}{p}\right)}$ denote the Legendre symbol modulo p. For $1\le \tau \le p^{n+1}-1$ with ${\displaystyle gcd(\tau ,p^{n+1})=p^{m_0}}$, we have:

$$\begin{array}{c}\sum _{m_1=0}^n\sum _{m_2=0}^n\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)=p^{n+1}-p^{n-m_0}(p+1).\hfill \end{array}$$

Proof. 

It is convenient to divide the relations between $m_1$ and $m_2$ into three cases according as $m_1>m_2$ or $m_1<m_2$ or $m_1=m_2$. From properties of the Legendre symbols modulo p and complete residue systems, we deduce:

$$\begin{array}{c}\underset{m_1>m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)=\underset{m_1>m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(p^{m_1}t+\tau ,p^{n+1})=p^{m_2}}{\sum _{t=0}^{p^{n+1-m_1}-1}}\left(\frac{t}{p}\right)\left(\frac{\frac{p^{m_1}t+\tau }{p^{m_2}}}{p}\right)\hfill \\ =\underset{m_1>m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(p^{m_1}t+\tau ,p^{n+1})=p^{m_2}}{\sum _{t=0}^{p^{n+1-m_1}-1}}\left(\frac{t}{p}\right)\left(\frac{\frac{\tau }{p^{m_2}}}{p}\right)=\underset{p^{m_2}\Vert \tau }{\underset{m_1>m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}}\sum _{t=0}^{p^{n+1-m_1}-1}\left(\frac{t}{p}\right)\left(\frac{\frac{\tau }{p^{m_2}}}{p}\right)=0,\hfill \end{array}$$

and,

$$\begin{array}{c}\underset{m_1<m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)=\underset{m_1<m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t,p^{n+1})=p^{m_2}}{\underset{gcd(t-\tau ,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t-\tau }{p^{m_1}}}{p}\right)\left(\frac{\frac{t}{p^{m_2}}}{p}\right)\hfill \\ =\underset{m_1<m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(p^{m_2}t+\tau ,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1-m_2}-1}}\left(\frac{t}{p}\right)\left(\frac{\frac{p^{m_2}t-\tau }{p^{m_1}}}{p}\right)=\underset{m_1<m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(p^{m_2}t+\tau ,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1-m_2}-1}}\left(\frac{t}{p}\right)\left(\frac{\frac{-\tau }{p^{m_1}}}{p}\right)\hfill \\ =\underset{p^{m_1}\Vert \tau }{\underset{m_1<m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}}\sum _{t=0}^{p^{n+1-m_2}-1}\left(\frac{t}{p}\right)\left(\frac{\frac{-\tau }{p^{m_1}}}{p}\right)=0,\hfill \end{array}$$

and,

$$\begin{array}{c}\underset{m_1=m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)=\sum _{m=0}^n\underset{gcd(p^{m}t+\tau ,p^{n+1})=p^m}{\sum _{t=0}^{p^{n+1-m}-1}}\left(\frac{t}{p}\right)\left(\frac{\frac{p^{m}t+\tau }{p^m}}{p}\right)\hfill \\ =\underset{p^m\Vert \tau }{\sum _{m=0}^n}\underset{gcd(t+\frac{\tau }{p^m},p)=1}{\sum _{t=0}^{p^{n+1-m}-1}}\left(\frac{t}{p}\right)\left(\frac{t+\frac{\tau }{p^m}}{p}\right)+\underset{p^{m+1}\mid \tau }{\sum _{m=0}^n}\underset{gcd(t,p)=1}{\sum _{t=0}^{p^{n+1-m}-1}}\left(\frac{t}{p}\right)\left(\frac{t+\frac{\tau }{p^m}}{p}\right)\hfill \\ =\underset{p^m\Vert \tau }{\sum _{m=0}^n}\sum _{t=0}^{p^{n+1-m}-1}\left(\frac{t}{p}\right)\left(\frac{t+\frac{\tau }{p^m}}{p}\right)+\underset{p^{m+1}\mid \tau }{\sum _{m=0}^n}{p}^{n-m}(p-1)\hfill \\ =\underset{p^m\Vert \tau }{\sum _{m=0}^n}(-p^{n-m})+\underset{p^{m+1}\mid \tau }{\sum _{m=0}^n}{p}^{n-m}(p-1)\hfill \\ =\left\{\begin{array}{cc}-p^n,\hfill & \mathrm{if}{m}_0=0,\hfill \\ -p^{n-m_0}+p^{n-m_0+1}(p^{m_0}-1),\hfill & \mathrm{if}1\le m_0\le n,\hfill \end{array}\right.\hfill \\ =p^{n+1}-p^{n-m_0}(p+1),\hfill \end{array}$$

by Lemma 1. The combined results in these three cases yield:

$$\begin{array}{c}\sum _{m_1=0}^n\sum _{m_2=0}^n\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)=\underset{m_1>m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)\hfill \\ +\underset{m_1<m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)+\underset{m_1=m_2}{\sum _{m_1=0}^n\sum _{m_2=0}^n}\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)\hfill \\ =p^{n+1}-p^{n-m_0}(p+1).\hfill \end{array}$$

□

3. Autocorrelation Values

The object of this section is to compute the autocorrelation values of $s^{\infty }$ and the main results are stated as follows.

Theorem 1.

Let $s^{\infty }$ be defined as in Equation (1) with $I_m=\left\{1,3,5,\cdots ,d-1\right\}$ for $m=0,1,\cdots ,n$. For $0\le \tau \le p^{n+1}-1$, the autocorrelation values of $s^{\infty }$ are:

$$\begin{array}{c}\hfill C_s\left(\tau \right)=\left\{\begin{array}{cc}{p}^{n+1},\hfill & \mathrm{if}\tau =0,\hfill \\ p^{n+1}-p^{n-m_0}(p+1)-\left(\frac{\frac{-\tau }{p^{m_0}}}{p}\right)-\left(\frac{\frac{\tau }{p^{m_0}}}{p}\right),\hfill & \mathrm{if}gcd(\tau ,p^{n+1})=p^{m_0},\hfill \end{array}\right.\end{array}$$

where $\left(\frac{·}{p}\right)$ denotes the Legendre symbol modulo p.

Proof. 

For $t\in \mathbb{Z}$, let $gcd(t,p^{n+1})=p^m$, $0\le m\le n$ and write $t=p^m{t}^{\prime }$, $gcd(t^{\prime },p)=1$. Observe that by ${\displaystyle C=\bigcup _{m=0}^n\bigcup _{i\in I_m}{p}^m{D}_i^{\left(p^{n+1}\right)}\bigcup \left\{0\right\}}$ and make use of the orthogonality relation for characters modulo $p^{n+1}$, we have:

$$\begin{array}{cc}\hfill t\in C& ⟺\mathrm{there}\mathrm{exists}i\in I_m\mathrm{satisfying}t\in p^m{D}_i^{\left(p^{n+1}\right)}\hfill \\ & ⟺\mathrm{there}\mathrm{exists}i\in I_m\mathrm{satisfying}{t}^{\prime }\in D_i^{\left(p^{n+1}\right)}\hfill \\ & ⟺\mathrm{there}\mathrm{exist}i\in I_m,0\le s\le p^{n}R-1\mathrm{satisfying}{t}^{\prime }\equiv g^{ds+i}(mod{p}^{n+1})\hfill \\ & {\displaystyle ⟺\frac{1}{\varphi \left(p^{n+1}\right)}\sum _{i\in I_m}\sum _{s=0}^{p^{n}R-1}\sum _{\chi mod{p}^{n+1}}\chi \left(t^{\prime }\right)\overline{\chi }\left(g^{ds+i}\right)=1}\hfill \\ & ⟺\frac{1}{d}\underset{\chi \left(g^d\right)=1}{\sum _{\chi mod{p}^{n+1}}}\chi \left(t^{\prime }\right)\sum _{i\in I_m}\overline{\chi }\left(g^i\right)=1,\hfill \end{array}$$

(3)

where the sum ${\displaystyle \sum _{\chi mod{p}^{n+1}}}$ is over all multiplicative characters modulo $p^{n+1}$. With the aid of Equation (3) and the definition of $I_m$ we now get:

$$\begin{array}{cc}\hfill {(-1)}^{s_t}& =-\frac{2}{d}\underset{\chi \ne {\chi }_0}{\underset{\chi \left(g^d\right)=1}{\sum _{\chi mod{p}^{n+1}}}}\left(\sum _{i\in I_m}\overline{\chi }\left(g^i\right)\right)\chi \left(t^{\prime }\right)=-\frac{2}{d}\underset{\chi \ne {\chi }_0}{\underset{\chi \left(g^d\right)=1}{\sum _{\chi mod{p}^{n+1}}}}\left(\sum _{i=0}^{\frac{d}{2}-1}\overline{\chi }\left(g^{2i+1}\right)\right)\chi \left(t^{\prime }\right)\hfill \\ & =-\underset{\chi \ne {\chi }_0}{\underset{\chi \left(g^2\right)=1}{\underset{\chi \left(g^d\right)=1}{\sum _{\chi mod{p}^{n+1}}}}}\overline{\chi }\left(g\right)\chi \left(t^{\prime }\right)=\left(\frac{t^{\prime }}{p}\right),\hfill \end{array}$$

where ${\displaystyle \left(\frac{·}{p}\right)}$ is the Legendre symbol modulo p. Then for $0\le t\le p^{n+1}-1$, we have:

$$\begin{array}{c}\hfill {(-1)}^{s_t}=\left\{\begin{array}{cc}{\displaystyle \left(\frac{t^{\prime }}{p}\right),}\hfill & t=p^m{t}^{\prime },0\le m\le n,gcd(t^{\prime },p)=1,\hfill \\ -1,\hfill & p^{n+1}\mid t.\hfill \end{array}\right.\end{array}$$

(4)

For $1\le \tau \le p^{n+1}-1$ with ${\displaystyle gcd(\tau ,p^{n+1})=p^{m_0}}$, by means of Equation (4) we have:

$$\begin{array}{c}{C}_s\left(\tau \right)=\sum _{t=0}^{p^{n+1}-1}{(-1)}^{s_{t+\tau }+s_t}\hfill \\ =\sum _{m_1=0}^n\sum _{m_2=0}^n\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)-\sum _{m=0}^n\underset{p^{n+1}\mid t+\tau }{\underset{gcd(t,p^{n+1})=p^m}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^m}}{p}\right)\hfill \\ -\sum _{m=0}^n\underset{gcd(t+\tau ,p^{n+1})=p^m}{\underset{p^{n+1}\mid t}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t+\tau }{p^m}}{p}\right)+\underset{p^{n+1}\mid t+\tau }{\underset{p^{n+1}\mid t}{\sum _{t=0}^{p^{n+1}-1}}}1\hfill \\ =\sum _{m_1=0}^n\sum _{m_2=0}^n\underset{gcd(t+\tau ,p^{n+1})=p^{m_2}}{\underset{gcd(t,p^{n+1})=p^{m_1}}{\sum _{t=0}^{p^{n+1}-1}}}\left(\frac{\frac{t}{p^{m_1}}}{p}\right)\left(\frac{\frac{t+\tau }{p^{m_2}}}{p}\right)-\left(\frac{\frac{-\tau }{p^{m_0}}}{p}\right)-\left(\frac{\frac{\tau }{p^{m_0}}}{p}\right).\hfill \end{array}$$

It follows from Lemma 2 that:

$$\begin{array}{c}\hfill C_s\left(\tau \right)=\left\{\begin{array}{cc}{p}^{n+1},\hfill & \mathrm{if}\tau =0,\hfill \\ p^{n+1}-p^{n-m_0}(p+1)-\left(\frac{\frac{-\tau }{p^{m_0}}}{p}\right)-\left(\frac{\frac{\tau }{p^{m_0}}}{p}\right),\hfill & \mathrm{if}gcd(\tau ,p^{n+1})=p^{m_0},\hfill \end{array}\right.\end{array}$$

which proves Theorem 1. □

In the case of $d=2$, that is, if $s^{\infty }$ is defined by the quadratic residue classes, then the autocorrelation values are entirely consistent with those when $n=0$ in [19], $n=1$ in [10,11,12], and $n=2$ in [13]. As a consequence, we get the following two corollaries for two special cases.

Corollary 1.

If $d=2$, $n=1$, $I_m=\left\{1\right\}$, $m=0,1$, that is:

$$\begin{array}{c}\hfill s_i=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}(imod{p}^2)\in D_1^{\left(p^2\right)}\cup p{D}_1^{\left(p^2\right)}\cup \left\{0\right\},\hfill \\ 0,\hfill & \mathrm{if}(imod{p}^2)\in D_0^{\left(p^2\right)}\cup p{D}_0^{\left(p^2\right)}.\hfill \end{array}\right.\end{array}$$

Then for $p\equiv 1(mod4)$ we have:

$$\begin{array}{c}\hfill C_s\left(\tau \right)=\left\{\begin{array}{cc}{p}^2,\hfill & \mathrm{if}\tau =0,\hfill \\ -p-2,\hfill & \mathrm{if}\tau \in D_0^{\left(p^2\right)},\hfill \\ -p+2,\hfill & \mathrm{if}\tau \in D_1^{\left(p^2\right)},\hfill \\ p^2-p-3,\hfill & \mathrm{if}\tau \in p{D}_0^{\left(p^2\right)},\hfill \\ p^2-p+1,\hfill & \mathrm{if}\tau \in p{D}_1^{\left(p^2\right)},\hfill \end{array}\right.\end{array}$$

and for $p\equiv 3(mod4)$ we have:

$$\begin{array}{c}\hfill C_s\left(\tau \right)=\left\{\begin{array}{cc}{p}^2,\hfill & \mathrm{if}\tau =0,\hfill \\ -p,\hfill & \mathrm{if}\tau \in {\mathbb{Z}}_{p^2}^{*},\hfill \\ p^2-p-1,\hfill & \mathrm{if}\tau \in p{\mathbb{Z}}_p^{*}.\hfill \end{array}\right.\end{array}$$

Corollary 2.

If $d=2$, $n=2$, $I_m=\left\{1\right\}$, $m=0,1,2$, that is:

$$\begin{array}{c}\hfill s_i=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}(imod{p}^3)\in D_1^{\left(p^3\right)}\cup p{D}_1^{\left(p^3\right)}\cup p^2{D}_1^{\left(p^3\right)}\cup \left\{0\right\},\hfill \\ 0,\hfill & \mathrm{if}(imod{p}^2)\in D_0^{\left(p^3\right)}\cup p{D}_0^{\left(p^3\right)}\cup p^2{D}_0^{\left(p^3\right)}.\hfill \end{array}\right.\end{array}$$

Then for $p\equiv 1(mod4)$ we have:

$$\begin{array}{c}\hfill C_s\left(\tau \right)=\left\{\begin{array}{cc}{p}^3,\hfill & \mathrm{if}\tau =0,\hfill \\ -p^2-2,\hfill & \mathrm{if}\tau \in D_0^{\left(p^3\right)},\hfill \\ -p^2+2,\hfill & \mathrm{if}\tau \in D_1^{\left(p^3\right)},\hfill \\ p^3-p^2-p-2,\hfill & \mathrm{if}\tau \in p{D}_0^{\left(p^3\right)},\hfill \\ p^3-p^2-p+2,\hfill & \mathrm{if}\tau \in p{D}_1^{\left(p^3\right)},\hfill \\ p^3-p-3,\hfill & \mathrm{if}\tau \in p^2{D}_0^{\left(p^3\right)},\hfill \\ p^3-p+1,\hfill & \mathrm{if}\tau \in p^2{D}_1^{\left(p^3\right)},\hfill \end{array}\right.\end{array}$$

and for $p\equiv 3(mod4)$ we have:

$$\begin{array}{c}\hfill C_s\left(\tau \right)=\left\{\begin{array}{cc}{p}^3,\hfill & \mathrm{if}\tau =0,\hfill \\ -p^2,\hfill & \mathrm{if}\tau \in D_0^{\left(p^3\right)}\cup D_1^{\left(p^3\right)},\hfill \\ p^3-p^2-p,\hfill & \mathrm{if}\tau \in p{D}_0^{\left(p^3\right)}\cup p{D}_1^{\left(p^3\right)},\hfill \\ p^3-p-1,\hfill & \mathrm{if}\tau \in p^2{D}_0^{\left(p^3\right)}\cup p^2{D}_1^{\left(p^3\right)}.\hfill \end{array}\right.\end{array}$$

4. Conclusions

In this paper we computed the exact values of autocorrelation of generalized cyclotomic binary sequences of any order d and period $p^{n+1}$ $(n\ge 0)$. Theorem 1 included the results of the autocorrelation values of Legendre sequences, prime-square sequences, and prime cube sequences from [10,11,12,13,19]. The autocorrelation values of our theorem were entirely consistent with those in [18]. In contrast to [18], our main results did not need the restriction $d=2$, and the proof of our theorem was based on certain identities involving character sums while the proof in [18] used the generalized cyclotomic numbers.