New constructions of optimal frequency hopping sequences

Abstract

Frequency hopping sequences (FHSs) are employed to mitigate the interferences caused by the hits of frequencies in frequency hopping spread spectrum systems. In this paper, four new constructions for FHSs are presented, including a direct construction of FHSs by using cyclotomic cosets and three extension constructions of FHSs by using units of rings and multiplicative groups of finite fields. By these constructions, a number of series of new FHSs are then produced. These FHSs are optimal with respect to the Lempel-Greenberger bound.

Appendices

Appendix A. Proof of Lemma 16

Proof

Since $a\not \equiv b \pmod p$, we have $Nj+a\not \equiv 0\pmod p$ or $Nj+b\not \equiv 0\pmod p$. By Lemma 14, $Y(Nj+a)=Y(Nj+b)$ only if $Nj+a\not \equiv 0\pmod p$ and $Nj+b\not \equiv 0\pmod p$. When $Y(Nj+a)=Y(Nj+b)$ for some j, $0\le j<pv$, we have

$$\begin{aligned} Y(Nj+a)=Y(Nj+b)=\psi (c_j, d_j)\ \mathrm{for\ some}\ c_j\in {\mathbb {Z}}_N, d_j\in {\mathbb {Z}}_v. \end{aligned}$$

Firstly, we show that there exist at most $\frac{p-1}{N}$ solutions $(j_a, j_b)$ with $j_a, j_b\in \{0,1,\ldots ,p-2\}$ and $j_a\ne j_b$ that satisfy the following equations:

$$\begin{aligned} \begin{aligned} \langle Nj+a \rangle _p=\alpha ^{j_a}, \ \langle Nj+b \rangle _p=\alpha ^{j_b}, \ \langle Nj+a \rangle _N=\langle j_a+c_j\rangle _N \ \mathrm{and} \ \langle Nj+b \rangle _N=\langle j_b+c_j\rangle _N. \end{aligned} \end{aligned}$$

(4)

In view of Eq. (4), we have $\langle {j_a}-{j_b} \rangle _N=\langle a-b \rangle _N$. Then there exists some integer s, $0\le s<\frac{p-1}{N}$, such that $\langle {j_a}-{j_b} \rangle _{p-1}=\langle a-b+Ns \rangle _{p-1}$ since $N|p-1$. By Eq. (4), we have

$$\begin{aligned} \begin{aligned} \langle a-b \rangle _p=\alpha ^{j_b}\big ( \alpha ^{j_a-j_b}-1\big )=\alpha ^{j_b}\big ( \alpha ^{a-b+Ns}-1\big ). \end{aligned} \end{aligned}$$

(5)

For each integer s with $\langle a-b+Ns\rangle _{p-1}\ne 0$, there exists unique element $j_b\in {\mathbb {Z}}_{p-1}$ that satisfies Eq. (5). Thus, there are at most $\frac{p-1}{N}$ pairs of elements $(s, j_b)$ satisfying Eq. (5). Since $\langle {j_a}-{j_b} \rangle _{p-1}=\langle a-b+Ns \rangle _{p-1}$, there are at most $\frac{p-1}{N}$ pairs of elements $(j_a, j_b)$ satisfying the following equations:

$$\begin{aligned} \langle Nj+a \rangle _p=\alpha ^{j_a}, \ \langle Nj+b \rangle _p=\alpha ^{j_b} \ \mathrm{and} \ \langle a-b \rangle _N=\langle j_a-j_b\rangle _N. \end{aligned}$$

Hence, there exist at most $\frac{p-1}{N}$ solutions $(j_a, j_b)$ with $j_a, j_b\in \{0,1,\ldots ,p-2\}$ and $j_a\ne j_b$ that satisfy Eq. (4).

Secondly, we show that there exists a unique element $j\in {\mathbb {Z}}_v$ that satisfies the following Equations

$$\begin{aligned} \begin{aligned} \langle Nj+a \rangle _v=d\theta _{j_a} \ \mathrm{and}\ \langle Nj+b \rangle _v=d\theta _{j_b} \end{aligned} \end{aligned}$$

(6)

for each pair of elements $(j_a, j_b)$ with $j_a\ne j_b.$

Since $j_a\ne j_b$, then it holds that

$$\begin{aligned} \theta _{j_a}-\theta _{j_b}\in U({\mathbb {Z}}_v). \end{aligned}$$

Thus, we can get the unique elements $j\in {\mathbb {Z}}_v$ and $d\in {\mathbb {Z}}_v$ that satisfies Eq. (6), where

$$\begin{aligned} d=\langle a-b \rangle _v\cdot \big (\theta _{j_a}-\theta _{j_b}\big )^{-1}\ \mathrm{and}\ \langle j \rangle _v= \big (d\theta _{j_a}-\langle a\rangle _v \big )\cdot N^{-1}. \end{aligned}$$

Finally, we show that

$$\begin{aligned} \sum \limits _{i=0}^{pv-1}h[Y(Ni+a), Y(Ni+b)]\le \frac{p-1}{N}. \end{aligned}$$

For each pair of elements $(j_a, j_b)$ with $j_a\ne j_b$, there exists at most a $j\in {\mathbb {Z}}_p$ satisfying Equation (4) and a $j'\in {\mathbb {Z}}_{v}$ satisfying Equations (6). Then there exists at most a $i\in {\mathbb {Z}}_{pv}$ satisfying $Y(Ni+a)=Y(Ni+b)$ since ${\mathbb {Z}}_{pv}\cong {\mathbb {Z}}_{p}\times {\mathbb {Z}}_{v}$. Since there exist at most $\frac{p-1}{N}$ pairs of elements $(j_a, j_b)$ that satisfy Equation (4), then there exist at most $\frac{p-1}{N}$ elements $i\in {\mathbb {Z}}_{pv}$ such that $Y(Ni+a)= Y(Ni+b)$. Hence, we have

$$\begin{aligned} \sum \limits _{i=0}^{pv-1}h[Y(Ni+a), Y(Ni+b)]\le \frac{p-1}{N}. \end{aligned}$$

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Appendix B. Proof of Lemma 19

Proof

Since e=lcm$(2e_1,2e_2)$ and $e_1, e_2$ are two odds, we have $(2e_1)|e$ and $(2e_2)|e$. By the definition of $B_{(y,i)}^1$, $B_{(y,j)}^2$ and Eq. (3), it holds that

$$\begin{aligned} \begin{array}{l} \bigcup \limits _{y\in S} \bigcup \limits _{0\le i<\frac{e}{2e_1}}B_{(y,i)}^1 \cup \bigcup \limits _{y\in S} \bigcup \limits _{0\le j<\frac{e}{2e_2}}B_{(y,j)}^2\\ \quad =\bigcup \limits _{y\in S} \bigcup \limits _{0\le i<\frac{e}{2e_1}}\{yg^{2i+\frac{e}{e_1}k}:~0\le k<e_1\} \cup \bigcup \limits _{y\in S} \bigcup \limits _{0\le j<\frac{e}{2e_2}}\{yg^{1+2j+\frac{e}{e_2}k}:~0\le k<e_2\}\\ \quad =\bigcup \limits _{y\in S} \{yg^{2i}:~0\le i<\frac{e}{2}\} \cup \bigcup \limits _{y\in S} \{yg^{1+2j}:~0\le j<\frac{e}{2}\}\\ \quad =\bigcup \limits _{y\in S} \{yg^{i}:~0\le i <e\} \\ \quad ={\mathbb {Z}}_v{\setminus } \{0\}. \end{array} \end{aligned}$$

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Appendix C. Proof of Lemma 21

Proof

In the following, we only prove the first case. Another case can be handled similarly. By the definition of g and $(2e_1)|e$, we have

$$\begin{aligned} \begin{array}{l} \sum \limits _{y\in S} \sum \limits _{0\le i<\frac{e}{2e_1}}|B_{(y,i)}^1\cap \big (a+ B_{(y,i)}^1\big )|\\ \quad =\sum \limits _{y\in S} \sum \limits _{0\le i<\frac{e}{2e_1}}|\{yg^{2i+\frac{e}{e_1}k}:~0\le k<e_1\}\cap \{a+yg^{2i+\frac{e}{e_1}k}:~0\le k<e_1\}|\\ \quad =\sum \limits _{y\in S} \sum \limits _{0\le i<\frac{e}{2e_1}}|\{yg^{2i+\frac{e}{e_1}k}(g^{\frac{e}{e_1}k'}-1):~0\le k<e_1,\ 1\le k'<e_1\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1):~0\le i<\frac{e}{2},\ 1\le k'<e_1\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1),\ yg^{2i}(g^{e-\frac{e}{e_1}k'}-1):~0\le i<\frac{e}{2},\ 1\le k' \le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1),\ yg^{2i+e-\frac{e}{e_1}k'}(1-g^{\frac{e}{e_1}k'}):~0\le i<\frac{e}{2},\ 1\le k' \le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1),\ yg^{2i}(1-g^{\frac{e}{e_1}k'}):~0\le i<\frac{e}{2},\ 1\le k' \le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1),\ -yg^{2i}(g^{\frac{e}{e_1}k'}-1):~0\le i <\frac{e}{2},\ 1\le k'\le \frac{e_1-1}{2}\}\cap \{a\}|. \end{array} \end{aligned}$$

Since $-1=g^{\frac{e}{2}}$, $e\equiv 2\pmod 4$ and Equation (3) , we have

$$\begin{aligned} \begin{array}{l} \sum \limits _{y\in S} \sum \limits _{0\le i<\frac{e}{2e_1}}|B_{(y,i)}^1\cap \big (a+ B_{(y,i)}^1\big )|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1),\ -yg^{2i}(g^{\frac{e}{e_1}k'}-1):~0\le i<\frac{e}{2},\ 1\le k'\le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{2i}(g^{\frac{e}{e_1}k'}-1),\ yg^{\frac{e}{2}+2i}(g^{\frac{e}{e_1}k'}-1):~0\le i<\frac{e}{2},\ 1\le k'\le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =\sum \limits _{y\in S}|\{yg^{i}(g^{\frac{e}{e_1}k'}-1):~0\le i <e,\ 1\le k'\le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =|\{x(g^{\frac{e}{e_1}k'}-1):~x\in {\mathbb {Z}}_v{\setminus } \{0\},\ 1\le k'\le \frac{e_1-1}{2}\}\cap \{a\}|\\ \quad =\frac{e_1-1}{2}. \end{array} \end{aligned}$$

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