Unit time-phase signal sets with the explicit maximum cross ambiguity amplitudes

Abstract

Unit time-phase signal sets have many important applications in radar or sonar systems. Upper bounds or lower bounds on the maximum cross ambiguity amplitudes of $(n, M)$ unit time-phase signal sets with $M \ge 2$ have been presented in the literature. In this paper, we use Gauss sums to determine the explicit maximum cross ambiguity amplitudes of some infinite series of unit time-phase signal sets which were constructed by Ding et al. (Cryptogr Commun 5:209–227, 2013).

1 Introduction

Let $\mathbb {H}_n=\mathbb {C}(\mathbb {Z}_n)$ be a set of all complex-valued functions on $\mathbb {Z}_n=\{0, 1, \ldots , n-1\}$ for an integer $n>1$, which is a Hilbert space with the Hermitian product given by

$$\begin{aligned} \langle \phi , \varphi \rangle =\sum _{t\in \mathbb {Z}_n}\phi (t)\overline{\varphi (t)},\;\text{ for }\; \phi , \varphi \in \mathbb {H}_n. \end{aligned}$$

Digital signals are complex-valued functions on $\mathbb {Z}_n$. The function $\phi \in \mathbb {H}_n$ can be viewed as a sequence via the following mapping

$$\begin{aligned} \phi \mapsto (\phi (0), \phi (1), \ldots , \phi (n-1)). \end{aligned}$$

A subset $S \subset \mathbb {H}_n$ is called a signal set, and a unit signal set if the norm $\parallel \phi \parallel \triangleq \sqrt{\langle \phi , \phi \rangle }=1$ for every signal $\phi \in S$. In this paper, we only consider unit signal sets because every signal can be normalized into a unit signal.

Signal sets with certain properties are required in some communication systems (see [1–8, 11, 12, 15]). During the transmission process, a signal $\phi$ might be distorted in various ways. Two basic types of distortion are the time shift

$$\begin{aligned} \phi (t)\mapsto \mathbf{L}_\tau \phi (t)=\phi (t+\tau ) \end{aligned}$$

and the phase shift

$$\begin{aligned} \phi (t)\mapsto \mathbf{M}_\omega \phi (t)=e^{{\frac{2\pi \sqrt{-1}}{n}}\omega t}\phi (t), \end{aligned}$$

where $\tau , \omega \in \mathbb {Z}_n$. To measure the capability of anti-distortion of a signal set $S$ with respect to the time and phase shift, Gurevich, Hadani and Sochen [7] defined the maximum cross ambiguity amplitude $\lambda$ of an $(n, M)$ signal set by

$$\begin{aligned} \lambda = \text{ max }\{|\langle \phi , \mathbf{M}_\omega \mathbf{L}_\tau \varphi \rangle |: \text{ either } \phi \ne \varphi \text{ or } (\tau ,\omega )\ne (0,0)\}. \end{aligned}$$

Then we call $S$ an $(n, M,\lambda )$ time-phase signal set when both time and phase shifts are considered, where $M$ denotes the total number of signals in $S$. If only time shift is considered (i.e., $\omega =0$), we call $S$ an $(n, M)$ time signal set or codebook (see [1, 8, 15]).

In [7], the authors used the group representation theory to design signal sets which were given by an algorithm. Unfortunately, the explicit form of these signal sets was unknown. Based on their results, Wang and Gong [12] gave an elegant expression for these time-phase signal sets by both multiplicative characters and additive characters of finite fields and presented upper bounds on the maximum cross ambiguity amplitudes. Schmidt [11] presented more constructions of such signal sets and easily obtained the upper bounds by using Weil bound. Moreover, the upper bounds on the maximum cross ambiguity amplitudes of some families of signal sets designed by multiplicative characters or additive characters were given in [13]. Ding et al. [2] proved that the famous Welch’s bound and the Levenstein’s bound on $\lambda$ are not good for time-phase signal sets. Moreover, they presented some better bounds from two one-way bridges between time-phase signal sets and time signal sets.

Lemma 1

[2] For any $(n, M, \lambda )$ unit time-phase signal set $S$ with $\lambda <1$ and $M>1$, we have the improved Levenstein’s bound:

$$\begin{aligned} \lambda \ge \sqrt{\frac{2nM-n-1}{(n+1)(nM-1)}}. \end{aligned}$$

Let $\mathbb {H}_{(n,q)}$ be the set of all complex-valued functions $f$ on $\mathbb {Z}_n$ such that $\sqrt{n} f(i)$ is a $q$th root of unity for all $i \in \mathbb {Z}_n$. In [2], we have some better linear programming bounds on $M$ and $\lambda$ from Levenstein’s results as follows.

Lemma 2

[2] Let $S\subset \mathbb {H}_{(n,q)}$ be any $(n, M,\lambda )$ unit time-phase signal set, where $q=2$. Then

$$\begin{aligned} nM \le \left\{ \begin{array}{ll} {\frac{1-\lambda ^2}{1-n\lambda ^2}}, &{} \text{ if }\; 0\le \lambda ^2 \le \frac{n-2}{n^2}, \\ {\frac{n^2(1-\lambda ^2)}{3n-2-n^2\lambda ^2}}, &{} \text{ if }\; \frac{n-2}{n^2} \le \lambda ^2 \le \frac{3n-8}{n^2}, \\ {\frac{n(1-\lambda ^2)[(n-2)(n^2-3n+8)-(n^2-n+2)n^2\lambda ^2]}{6n(n-2)-4(3n-4)n^2\lambda ^2+2n^4\lambda ^4}}, &{} \text{ if }\; \frac{3n-8}{n^2} \le \lambda ^2 \le \frac{3n-10+\sqrt{6n^2-42n+76}}{n^2}, \\ \frac{n^2(1-\lambda ^2)}{6} \frac{3n^3-23n^2+90n-136-(n^2-3n+8)n^2\lambda ^2}{15n^2-50n+24-10(n-2)n^2\lambda ^2+n^4\lambda ^4}, &{} \text{ if }\; \frac{3n-10+\sqrt{6n^2-42n+76}}{n^2} \le \lambda ^2 \\ &{}\quad \le \frac{5(n-4)+\sqrt{10n^2-90n+216}}{n^2} \end{array}\right. \end{aligned}$$

Lemma 3

[2] Let $S \subset \mathbb {H}_{(n,q)}$ be any $(n, M,\lambda )$ unit time-phase signal set, where $q \ge 3$. Then

$$\begin{aligned} nM\le \left\{ \begin{array}{ll} {\frac{1-\lambda ^2}{1-n\lambda ^2}}, &{} \text{ if }\; 0\le \lambda ^2 \le \frac{n-1}{n^2}, \\ {\frac{n^2(1-\lambda ^2)}{2n-1-n^2\lambda ^2}}, &{} \text{ if }\; \frac{n-1}{n^2}\le \lambda ^2 \le \frac{2n^2-5n+4}{n^2(n-1)},\\ {\frac{n(1-\lambda ^2)[(n^2-n+1)n^2\lambda ^2-n^3+3n^2-5n+4]}{n[4(n-1)n^2\lambda ^2-n^4\lambda ^4-2n^2+3n]}}, &{} \text{ if }\; \frac{2n^2-5n+4}{n^2(n-1)} \le \lambda ^2 \le \frac{2n-2+\sqrt{2n^2-5n+4}}{n^2}. \end{array}\right. \end{aligned}$$

When $q \ge 3$ and $\frac{n-1}{n^2} \le \lambda ^2 \le \frac{2n^2-5n+4}{n^2(n-1)}$, an infinite series of optimal $(n, 1)$ unit time-phase signal sets were firstly constructed by Ding et al. [2], who also presented $(n, M>1)$ unit time-phase signal sets and obtained the upper bounds on the maximum cross ambiguity amplitudes. Based on the construction in [2], we study some infinite series of $(n, M>1)$ unit time-phase signal sets for all cases in Lemmas 2 and 3 with the exception of the second case in Lemma 3 and use the Stickelberger’s Theorem and index 2 Gauss sums to give the explicit maximum cross ambiguity amplitudes.

The paper is organized as follows. In Sect. 2, we introduce some basic concepts and results about Gauss sums. In Sect. 3, we use Gauss sums to determine the explicit maximum cross ambiguity amplitudes of some infinite series of $(n, M>1)$ unit time-phase signal sets.

For convenience, we introduce the following notations:

${\mathbb Z}_N, {\mathbb Z}_N^*$ :

the ring of integers modulo $N$, the multiplicative group of ${\mathbb Z}_N$,

$\langle p \rangle$ :

the cyclic subgroup of ${\mathbb Z}_N^*$ generated by $p$,

$\varPhi (N)$ :

the number of integers $k$ with $1\le k\le N$ such that $\gcd (k, N)=1$,

$\text{ ord }_N(p)$ :

the order of $p$ modulo $N$,

${\mathbb F}_q$ :

the finite field of order $q$,

$\text{ Tr }$ :

the absolute trace from ${\mathbb F}_q$ to ${\mathbb F}_p$,

$\psi$ :

the additive character of ${\mathbb F}_q$,

$\chi$ :

the multiplicative character of ${\mathbb F}_q$,

$o(\chi )$ :

the order of the multiplicative character $\chi$,

$G(\chi )$ :

the Gauss sum over ${\mathbb F}_q$,

$(\frac{p_1}{p_2})$ :

the Legendre symbol.

2 Gauss sums

Let ${\mathbb F}_q$ be a finite field with $q=p^f$ elements, $p$ a prime, and $f$ a positive integer. Define an additive character of ${\mathbb F}_q$ as follows:

$$\begin{aligned} \psi _b:{\mathbb F}_q \rightarrow {\mathbb C}^*, \quad \psi _b(x)=\zeta _p^{\text{ Tr }(bx)}, \;\text{ for }\; b\in {\mathbb F}_q, \end{aligned}$$

(1)

where $\zeta _p=e^{{\frac{2\pi \sqrt{-1}}{p}}}$ is a $p$th primitive root of unity and $\text{ Tr }$ denotes the absolute trace from ${\mathbb F}_q$ to ${\mathbb F}_p$. For $b=1, \psi _1$ is called the canonical additive character of $\mathbb {F}_q$. Let $\chi : {\mathbb F}_q ^*\rightarrow {\mathbb C}^*$ be a multiplicative character of ${\mathbb F}_q ^*$. We have the Gauss sum:

$$\begin{aligned} G(\psi _b, \chi ) = \sum _{x \in {\mathbb F}_q ^*} \psi _b(x)\chi (x). \end{aligned}$$

Now we recall some properties of Gauss sums.

Lemma 4

[10] Let $\psi$ and $\chi$ be an additive character and a multiplicative character of $\mathbb {F}_q$, respectively. Then

$$\begin{aligned} G(\psi , \chi )= \left\{ \begin{array}{ll} q-1, &{} \text{ if }\; \psi =1 \;\text{ and }\; \chi =1, \\ -1, &{} \text{ if }\; \psi \ne 1 \;\text{ and }\;\chi =1, \\ 0, &{} \text{ if }\; \psi =1 \; \text{ and }\; \chi \ne 1. \end{array}\right. \end{aligned}$$

If $\psi =\psi _b \ne 1$ (i.e., $b\ne 0$) and $\chi \ne 1$, then

$$\begin{aligned} |G(\psi , \chi )|=\sqrt{q}, \end{aligned}$$

and

$$\begin{aligned} G(\psi _b,\chi )=\overline{\chi }(b)G( \chi ), \end{aligned}$$

where

$$\begin{aligned} G(\chi )=G(\psi _1, \chi )=\sum _{x \in \mathbb {F}_q^*}\psi _1(x)\chi (x) = \sum _{x \in \mathbb {F}_q^*}\chi (x)\zeta _p^{\text{ Tr }(x)}. \end{aligned}$$

While it is easy to know the absolute value of a nontrivial Gauss sum $G(\psi , \chi )$ is equal to $\sqrt{q}$, the explicit determination of Gauss sum is a difficult problem. However, the Gauss sums can be explicitly evaluated in a few cases. For future use, we state the Stickelberger’s Theorem.

Lemma 5

(Stickelberger’s Theorem [10]) Let $q=p^{2l}$ with $p$ a prime and $l$ a positive integer, let $\chi$ be a nontrivial multiplicative character of $\mathbb {F}_q$ of order $m$ dividing $p^l+1$, and let $\psi _1$ be the canonical additive character of $\mathbb {F}_q$. Then

$$\begin{aligned} G(\chi )=G(\psi _1, \chi )= \left\{ \begin{array}{ll} p^l, &{} \text{ if }\; m \text{ odd } \text{ or } \frac{p^l+1}{m} \text{ even, } \\ -p^l, &{} \text{ if }\; m \text{ even } \text{ and } \frac{p^l+1}{m} \text{ odd. } \end{array}\right. \end{aligned}$$

Below we introduce a result on one case of the index 2 Gauss sums which involves class numbers of number fields. The definition of the class number can be found in any algebraic number theory and we refer the readers to [6] and [9].

Lemma 6

[14] Let $N=p_1^{r_1}p_2^{r_2}$, where $p_1, p_2$ are distinct odd primes with $p_1 \equiv 3 \pmod 4$. Assume that $p$ is a prime such that $-1 \not \in \langle p \rangle \le {\mathbb Z}_N^*$ and $f=\text{ ord }_N(p)=\frac{\varPhi (N)}{2}$, where $f$ is the smallest positive integer such that $p^f \equiv 1 \pmod N$. Let $q=p^f$ and $\chi$ a multiplicative character of order $N$ over ${\mathbb F}_q$. Suppose that $\text{ ord }_{p_1^{r_1}}(p)={\frac{\varPhi (p_1^{r_1})}{2}},\text{ ord }_{p_2^{r_2}}(p)=\varPhi (p_2^{r_2})$. For $0 \le t_1 < r_1,0\le t_2 < r_2$, we have

$$\begin{aligned}&G(\chi ^{p_1^{t_1}p_2^{t_2}})=\left\{ \begin{array}{ll} p^{\frac{f}{2}}, &{} \text{ if } (\frac{p_2}{p_1})=1, \\ p^{{\frac{f}{2}}-hp_1^{t_1}p_2^{t_2}} (\frac{b+c \sqrt{-p_1}}{2})^{2p_1^{t_1}p_2^{t_2}}, &{} \text{ if } (\frac{p_2}{p_1})=-1; \end{array}\right. \\&G(\chi ^{p_1^{r_1}p_2^{t_2}})=p^{\frac{f}{2}};\\&G(\chi ^{p_1^{t_1}p_2^{r_2}})=-p^{{\frac{1}{2}} (f-hp_1^{t_1}\varPhi (p_2^{r_2}))} \left( \frac{b+c \sqrt{-p_1}}{2}\right) ^{p_1^{t_1}\varPhi (p_2^{r_2})}, \end{aligned}$$

where $h$ is the class number of ${\mathbb Q}(\sqrt{-p_1})$ and $b,c$ are determined by

$$\begin{aligned} \left\{ \begin{array}{ll} (1)\ 4p^h=b^2+p_1c^2, \\ (2)\ b \equiv 2p^{\frac{p_1-1+2h}{4}}\pmod {p_1}. \end{array}\right. \end{aligned}$$

3 $(n, M>1)$ unit time-phase signal sets

In this section, we use Gauss sums to determine the explicit maximum cross ambiguity amplitudes of some infinite series of unit time-phase signal sets which were constructed by Ding et al. [2]. Now we introduce the results on their constructions of $(n, 1)$ and $(n, M>1)$ unit time-phase signal sets.

The construction of case $(n, 1)$ can be described as follows [2]. Let $q=p^f, p$ a prime, $n=q-1, \text{ Tr } :\mathbb {F}_q \rightarrow \mathbb {F}_p$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. Let

$$\begin{aligned} \phi =\frac{1}{\sqrt{n}}(\phi (0), \phi (1), \ldots , \phi (n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi (i)=\zeta _p^{\text{ Tr }(\gamma ^i)}, \quad \ 0 \le i \le n-1. \end{aligned}$$

Then $S=\{\phi \}$ is an $(n, 1, \frac{\sqrt{n+1}}{n})$ unit time-phase set. Moreover, $S$ is optimal if $p\ge 3$. This is the first time that an infinite family of optimal time-phase signal sets was constructed.

The construction of the case $(n, M>1)$ can be described in the following lemma.

Lemma 7

[2] Let $q=p^f, p$ a prime, ${q-1}=en \ (e \ge 2), \text{ Tr }: \mathbb {F}_q\rightarrow \mathbb {F}_p$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0\le i \le e-1$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=\zeta _p^{\text{ Tr }(\gamma ^{i+et})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _i:0 \le i \le e-1\}$ is a $(\frac{q-1}{e}, e, \lambda )$ unit time-phase signal set with $\lambda \le \frac{\sqrt{en+1}}{n}$.

Proof

For completeness, we give a proof here. For $0 \le i, j \le e-1, 0\le \omega , \tau \le n-1, (i-j, \omega , \tau ) \ne (0, 0, 0)$, we have

$$\begin{aligned} \langle \phi _i, \mathbf{M}_\omega \mathbf{L}_\tau (\phi _j) \rangle&= \frac{1}{n} \sum _{t=0}^{n-1}\zeta _p^{\text{ Tr }(\gamma ^{i+et})} \overline{\zeta }_p^{\text{ Tr }(\gamma ^{j+e(t+\tau )})}\overline{\zeta }_n^{\omega t} \\&= \frac{1}{n} \sum _{t=0}^{n-1}\zeta _p^{\text{ Tr }(\gamma ^{et}(\gamma ^i- \gamma ^{j+e\tau }))}\overline{\zeta }_n^{\omega t} \\&= \frac{1}{n} \sum _{t=0}^{n-1}\zeta _p^{\text{ Tr }(\beta \gamma ^{et})}\chi ^\omega (\gamma ^{et}), \end{aligned}$$

where $\beta =\gamma ^i-\gamma ^{j+e\tau }$ and $\chi$ is the multiplicative character of $\mathbb {F}_q^*$ defined by $\chi (\gamma )=\overline{\zeta }_{q-1}$. Note that for $0 \le r \le q-2$, we have

$$\begin{aligned} \sum _{s=0}^{e-1}\chi ^{ns}(\gamma ^r)=\sum _{s=0}^{e-1}\overline{\zeta }_e^{rs}= \left\{ \begin{array}{ll} e, &{} \text{ if } e \mid r\\ 0, &{} \text{ otherwise } . \end{array}\right. \end{aligned}$$

Thus

$$\begin{aligned} \frac{1}{n} \sum _{t=0}^{n-1}\zeta _p^{\text{ Tr }(\beta \gamma ^{et})}\chi ^\omega (\gamma ^{et})&= \frac{1}{en} \sum _{x \in \mathbb {F}_q^*}\zeta _p^{\text{ Tr }(\beta x)}\chi ^\omega (x) \sum _{s=0}^{e-1}\chi ^{ns}(x) \\&= \frac{1}{en} \sum _{s=0}^{e-1}\sum _{x \in \mathbb {F}_q^*}\chi ^{ns+\omega }(x)\zeta _p^{\text{ Tr }(\beta x)}. \end{aligned}$$

If $(j-i, \tau )=(0, 0)$, we have $1 \le \omega \le n-1, \beta =\gamma ^i-\gamma ^{j+\tau e}=0$, and $\chi ^{ns+\omega }\ne 1$ for all $s, 0 \le s \le e-1$. Then

$$\begin{aligned} \langle \phi _i, \mathbf{M}_\omega \mathbf{L}_\tau (\phi _j) \rangle =\frac{1}{en} \sum _{s=0}^{e-1}\sum _{x \in \mathbb {F}_q^*}\chi ^{ns+\omega }(x) =\frac{1}{en} \sum _{s=0}^{e-1}0=0. \end{aligned}$$

If $(j-i, \tau )\ne (0, 0)$, i.e., $\beta \ne 0$, then

$$\begin{aligned} \langle \phi _i, \mathbf{M}_\omega \mathbf{L}_\tau (\phi _j) \rangle&= \frac{1}{en} \sum _{s=0}^{e-1}\overline{\chi }^{ns+\omega }(\beta )G(\chi ^{ns+\omega }) \\&\le \frac{1}{en} \sum _{s=0}^{e-1}|G(\chi ^{ns+\omega })| \le \frac{e\sqrt{q}}{en}=\frac{\sqrt{en+1}}{n}. \end{aligned}$$

Therefore the upper bound on $\lambda$ follows. $\square$

It is very difficult to compute the explicit value $\lambda$ for the time-phase signal set $S$ in Lemma 7, and thus unable to know if it is optimal. In the following, we determine the explicit values of $\lambda$ in some special cases. For $p \ge 3$, the case that $\lambda ^2 \in [\frac{n-1}{n^2}, \frac{2n^2-5n+4}{n^2(n-1)}]$ had been studied by Ding et al. [2]. In the following, we shall consider all cases stated in Lemmas 2 and 3 with the exception of the above case.

(1) When $p\ge 3$.

For the case $\lambda ^2 \in [0, \frac{n-1}{n^2}]$, we have

$$\begin{aligned} \lambda ^2 \ge \frac{nM-1}{n^2M-1} \end{aligned}$$

from the first bound of Lemma 3. Thus it is easily verified that $n=M=1$, it is trivial. Now we study the $(n, 2)$ and $(n, 3)$ unit time-phase signal sets for the case $\lambda ^2 \in [\frac{2n^2-5n+4}{n^2(n-1)}, \frac{2n-2+\sqrt{2n^2-5n+4}}{n^2}]$.

Theorem 1

Let $q=p^{2l}, p \equiv 3 \pmod 4$ a prime, $l$ odd, $n=\frac{q-1}{2}, \text{ Tr } : \mathbb {F}_q\rightarrow \mathbb {F}_p$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $i=0, 1$, let

$$\begin{aligned} \phi _i = \frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t) = \zeta _p^{\text{ Tr }(\gamma ^{i+2t})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _0, \phi _1\}$ is a $(\frac{q-1}{2}, 2, \frac{\sqrt{2n+1}}{n})$ unit time-phase signal set, which falls into the third case of Lemma 3.

Proof

From the proof of Lemma 7, we have

$$\begin{aligned} \langle \phi _i, \mathbf{M}_\omega \mathbf{L}_\tau (\phi _j) \rangle&= \frac{1}{2n}(\overline{\chi }^\omega (\beta )G(\chi ^\omega )+ \overline{\chi }^{n+\omega }(\beta )G(\chi ^{n+\omega })) , \end{aligned}$$

for $i, j \in \{0, 1\}, 0\le \omega , \tau \le n-1, (i-j, \omega , \tau )\ne (0, 0, 0)$, where $\beta =\gamma ^i-\gamma ^{j+2\tau }$.

For arbitrary $i\in \{0,1\},j\in \{0,1\}$ and $\tau \in \{0,\ldots , n-1\}$, we can always take some $(i_0, j_0, \tau _0)$ such that $\beta =\gamma ^{i_0}-\gamma ^{j_0+2\tau _0}=1$, thus $\overline{\chi }^{\omega }(\beta )=\overline{\chi }^{n+\omega }(\beta )=1$. Take $\omega =\frac{q-1}{4}$, then the orders of both $\chi ^\omega$ and $\chi ^{n+\omega }$ are equal to $4$. Since $p \equiv 3 \pmod 4$ and $l$ is odd, $4 \mid (p^l+1)$. By Stickelberger’s Theorem, we have

$$\begin{aligned} G(\chi ^{\frac{q-1}{4}})=G(\chi ^{n+\frac{q-1}{4}}) \end{aligned}$$

and

$$\begin{aligned} \lambda \ge |\langle \phi _{i_0}, \mathbf{M}_\omega \mathbf{L}_{\tau _0}(\phi _{j_0}) \rangle | =\frac{2\sqrt{q}}{2n}=\frac{\sqrt{2n+1}}{n}. \end{aligned}$$

By Lemma 7, we have $\lambda =\frac{\sqrt{2n+1}}{n}$ which falls into the third case of Lemma 3. Hence $S=\{\phi _0, \phi _1\}$ is a $(\frac{q-1}{2}, 2, \frac{\sqrt{2n+1}}{n})$ unit time-phase signal set. $\square$

Example 1

Let $q=3^2, n=4$, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $i=0, 1$, let

$$\begin{aligned} \phi _i=\frac{1}{2}(\phi _i(0), \phi _i(1), \phi (2), \phi _i(3)) \in \mathbb {C}^4, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=\zeta _3^{\text{ Tr }(\gamma ^{i+2t})}, \quad 0 \le t \le 3. \end{aligned}$$

Then $S=\{\phi _0, \phi _1\}$ is an $(4, 2, \frac{3}{4})$ unit time-phase signal set, which falls into the third case of Lemma 3.

Theorem 2

Let $q=p^{2l}, p \equiv 2 \pmod 3$ an odd prime, $l$ odd, $n=\frac{q-1}{3}, \text{ Tr } : \mathbb {F}_q\rightarrow \mathbb {F}_p$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 2$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=\zeta _p^{\text{ Tr }(\gamma ^{i+3t})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _0, \phi _1, \phi _2 \}$ is a $(\frac{q-1}{3}, 3, \frac{\sqrt{3n+1}}{n})$ unit time-phase signal set, which falls into the third case of Lemma  3.

Proof

From the proof of Lemma 7, we have

$$\begin{aligned} \langle \phi _i, \mathbf{M}_\omega \mathbf{L}_\tau (\phi _j) \rangle&= \frac{1}{3n}(\overline{\chi }^\omega (\beta )G(\chi ^\omega )+ \overline{\chi }^{n+\omega }(\beta )G(\chi ^{n+\omega }) +\overline{\chi }^{2n+\omega }(\beta )G(\chi ^{2n+\omega })) , \end{aligned}$$

for $0 \le i, j \le 2, 0\le \omega , \tau \le n-1, (i-j, \omega , \tau ) \ne (0, 0, 0)$, where $\beta =\gamma ^i-\gamma ^{j+3\tau }$.

For arbitrary $i, j\in \{0,1,2\}$ and $\tau \in \{0,\ldots , n-1\}$, we can always take some $(i_0, j_0, \tau _0)$ such that $\beta =\gamma ^{i_0}-\gamma ^{j_0+3\tau _0}=1$, thus $\overline{\chi }^\omega (\beta )=\overline{\chi }^{n+\omega }(\beta )=\overline{\chi }^{2n+\omega }(\beta )=1$. Take $\omega =\frac{q-1}{6}$, then there are three orders:

$$\begin{aligned} o(\chi ^\omega )=6, o(\chi ^{2n+\omega })=6 \quad \text{ and } \quad o(\chi ^{n+\omega })=2. \end{aligned}$$

Since $p \equiv 2 \pmod 3$ is an odd prime and $l$ is odd, $6 \mid (p^l+1)$ and $2\mid (p^l+1)$. By Stickelberger’s Theorem, we have

$$\begin{aligned} G(\chi ^{\frac{q-1}{6}})=G(\chi ^{n+\frac{q-1}{6}}) =G(\chi ^{2n+\frac{q-1}{6}}) \end{aligned}$$

and

$$\begin{aligned} \lambda \ge |\langle \phi _{i_0}, \mathbf{M}_\omega \mathbf{L}_{\tau _0}(\phi _{j_0}) \rangle | =\frac{3\sqrt{q}}{3n}=\frac{\sqrt{3n+1}}{n}. \end{aligned}$$

By Lemma 7, we know that $\lambda =\frac{\sqrt{3n+1}}{n}$ which falls into the third case of Lemma 3. Hence $S=\{\phi _0, \phi _1, \phi _2\}$ is a $(\frac{q-1}{3}, 3, \frac{\sqrt{3n+1}}{n})$ unit time-phase signal set. $\square$

Example 2

Let $q=5^2, n=8$, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 2$, let

$$\begin{aligned} \phi _i=\frac{1}{2\sqrt{2}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(7)) \in \mathbb {C}^8, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=\zeta _5^{\text{ Tr }(\gamma ^{i+3t})}, \quad 0 \le t \le 7. \end{aligned}$$

Then $S=\{\phi _0, \phi _1, \phi _2\}$ is an $(8, 3, \frac{5}{8})$ unit time-phase signal set, which falls into the third case of Lemma 3.

(2) When $p=2$.

For the case $\lambda ^2 \in [0, \frac{n-2}{n^2}]$, we have

$$\begin{aligned} \lambda ^2 \ge \frac{nM-1}{n^2M-1} \end{aligned}$$

from the fist bound of Lemma 2. Thus it is easily verified that $n=M=1$, it is trivial. The $(n, 1, \frac{\sqrt{n+1}}{n})$ unit time-phase signal set constructed in [2] falls into the second case of Lemma 2 when $p=2$. In the following, we study the other cases in Lemma 2 using Stickelberger’s Theorem and index 2 Gauss sums.

For the third case of Lemma 2, we use Stickelberger’s Theorem to give the explicit maximum cross ambiguity amplitude.

Theorem 3

Let $q=2^{2l}, l \equiv 5 \pmod {10}, n=\frac{q-1}{3}, \text{ Tr }: \mathbb {F}_q\rightarrow \mathbb {F}_2$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 2$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=(-1)^{\text{ Tr }(\gamma ^{i+3t})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _0, \phi _1, \phi _2 \}$ is a $(\frac{q-1}{3}, 3, \frac{\sqrt{3n+1}}{n})$ unit time-phase signal set, which falls into the third case of Lemma 2.

Proof

Since $l\equiv 5 \pmod {10}, l=5l_1$ and $l_1$ is odd. Then $q-1=2^{2l}-1=(2^{10}-1)d_1$ and $2^l+1=(2^5+1)d_2$, where $d_1, d_2$ are integers, so $33 \mid (q-1)$ and $33 \mid (2^l+1)$. Take $\omega =\frac{q-1}{33}$, then there are three orders:

$$\begin{aligned} o(\chi ^w)=o(\chi ^{2n+w})=33 \quad \text{ and } \quad o(\chi ^{n+w})=11. \end{aligned}$$

By Lemma 7 and Stickelberger’s Theorem, we have $\lambda =\frac{\sqrt{3n+1}}{n}$ which falls into the third case of Lemma 2. Then we finish the proof. $\square$

Example 3

Let $q=2^{10}, n=\frac{q-1}{3}=341$, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 2$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{341}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(340)) \in \mathbb {C}^{341}, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=(-1)^{\text{ Tr }(\gamma ^{i+3t})}, \quad 0 \le t \le 340. \end{aligned}$$

Then $S=\{\phi _0, \phi _1, \phi _2\}$ is a $(341, 3, \frac{32}{341})$ unit time-phase signal set, which falls into the third case of Lemma 2.

Theorem 4

Let $q=2^{2l}, l \equiv 6 \pmod {12}, n=\frac{q-1}{5}, \text{ Tr } : \mathbb {F}_q\rightarrow \mathbb {F}_2$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 4$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=(-1)^{\text{ Tr }(\gamma ^{i+5t})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _i: 0\le i \le 4 \}$ is a $(\frac{q-1}{5}, 5, \frac{\sqrt{5n+1}}{n})$ unit time-phase signal set, which falls into the third case of Lemma 2.

Proof

Since $l\equiv 6\pmod {12}, l=6l_1$ and $l_1$ is odd. Then $q-1=2^{2l}-1=(2^{12}-1)d_1$ and $2^l+1=(2^6+1)d_2$, where $d_1,d_2$ are integers, so $65|(q-1)$ and $65|2^l+1$. Take $\omega =\frac{q-1}{65}$, then there are five orders:

$$\begin{aligned} o(\chi ^{sn+w})=65(s=0,1, 2, 4) \quad \text{ and } \quad o(\chi ^{3n+w})=13. \end{aligned}$$

By Lemma 7 and Stickelberger’s Theorem, we have $\lambda = \frac{\sqrt{5n+1}}{n}$ which falls into the third case of Lemma 2. Then we finish the proof. $\square$

Example 4

Let $q=2^{12},n=\frac{q-1}{5}=819$, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 4$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{819}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(818)) \in \mathbb {C}^{819}, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=(-1)^{\text{ Tr }(\gamma ^{i+5t})}, \quad 0 \le t \le 818. \end{aligned}$$

Then $S=\{\phi _i: 0 \le i\le 4\}$ is an $(819, 5, \frac{64}{819})$ unit time-phase signal set, which falls into the third case of Lemma 2.

In the following, we use the index 2 Gauss sums to study the fourth case of Lemma 2.

Theorem 5

Let $q=2^f, 30 \mid f,n=\frac{q-1}{7}, \text{ Tr } : \mathbb {F}_q\rightarrow \mathbb {F}_2$ the trace mapping, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 6$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=(-1)^{\text{ Tr }(\gamma ^{i+7t})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _i: 0\le i \le 6 \}$ is a $(\frac{q-1}{7}, 7, \frac{\sqrt{7n+1}}{n})$ unit time-phase signal set, which falls into the fourth case of Lemma 2.

Proof

Since $30 \mid f$, we have $f=30f_1$ and $f_1$ is an integer. Then $q-1=2^f-1=(2^{30}-1)d$, where $d$ is an integer, so $77 \mid (2^{30}-1)$ and $77 \mid (q-1)$.

From the proof of Lemma 7, we have

$$\begin{aligned} \langle \phi _i, \mathbf{M}_\omega \mathbf{L}_\tau (\phi _j) \rangle = \frac{1}{7n} \sum _{s=0}^6\overline{\chi }^{ns+\omega }(\beta )G(\chi ^{ns+\omega }) \end{aligned}$$

for $0 \le i, j \le 6, 0\le \omega , \tau \le n-1, (i-j, \omega , \tau ) \ne (0, 0, 0)$, where $\beta =\gamma ^i-\gamma ^{j+7\tau }$.

For arbitrary $i, j,\tau$, we can always take some $(i_0, j_0, \tau _0)$ such that $\beta =\gamma ^{i_0}-\gamma ^{j_0+7\tau _0}=1$, thus $\overline{\chi }^{ns+\omega }(\beta )=1$ for $0\le s\le 6$. Take $\omega =\frac{q-1}{77}$, then

$$\begin{aligned} o(\chi ^{ns+\omega })=77 (s=0, 1, 2, 3, 4, 6) \quad \text{ and } \quad o(\chi ^{5n+\omega })=11. \end{aligned}$$

Note that $\text{ ord }_7(2)=\frac{\phi (7)}{2}, \text{ ord }_{11}(2)=\phi (11), \text{ ord }_{77}(2)=\frac{\phi (77)}{2}$, and the Legendre symbol $(\frac{11}{7})=1$. By Lemma 6 and Davenport–Hasse Lifting Theorem (see [10]), we have

$$\begin{aligned} G(\chi ^{ns+\frac{q-1}{6}})=(-1)^{f_1-1}\sqrt{q}, s=0, 1, \ldots , 6. \end{aligned}$$

Therefore

$$\begin{aligned} \lambda \ge |\langle \phi _{i_0}, \mathbf{M}_\omega \mathbf{L}_{\tau _0}(\phi _{j_0}) \rangle | =\frac{7\sqrt{q}}{7n}=\frac{\sqrt{7n+1}}{n}. \end{aligned}$$

By Lemma 7, we have $\lambda =\frac{\sqrt{7n+1}}{n}$ which falls into the fourth case of Lemma 2. Hence $S=\{\phi _i: 0\le i \le 6\}$ is a $(\frac{q-1}{7}, 7, \frac{\sqrt{7n+1}}{n})$ unit time-phase signal set. $\square$

Example 5

Let $q=2^{30}, n=\frac{2^{30}-1}{7}$, and $\gamma$ a primitive element of $\mathbb {F}_q$. For $0 \le i \le 6$, let

$$\begin{aligned} \phi _i=\frac{1}{\sqrt{n}}(\phi _i(0), \phi _i(1), \ldots , \phi _i(n-1)) \in \mathbb {C}^n, \end{aligned}$$

where

$$\begin{aligned} \phi _i(t)=(-1)^{\text{ Tr }(\gamma ^{i+7t})}, \quad 0 \le t \le n-1. \end{aligned}$$

Then $S=\{\phi _i: 0\le i \le 6\}$ is an $(n, 7, \frac{2^{15}}{n})$ unit time-phase signal set, which falls into the fourth case of Lemma 2.