New sets of optimal low-hit-zone frequency-hopping sequences based on m-sequences

Abstract

In quasi-synchronous frequency-hopping multiple-access systems where relative delays are restricted within a certain correlation zone, low-hit-zone frequency-hopping sequences (LHZ-FHSs) are commonly employed to minimize multiple-access interferences. In this paper, we present two classes of optimal LHZ-FHS sets with respect to the Peng-Fan-Lee bound, which are obtained from an m-sequence and its decimated sequence, respectively. The parameters of these LHZ-FHS sets are new and flexible.

Appendices

Appendix A:

Proof Proof of Theorem 1

Clearly, $\mathcal {U}$ contains q ^k(q−1) sequences of length q ⁿ−1 over V _k . Let M _i ={m _i (l)} = b _i ⋅M, 0≤i<q−1, 0≤l<q ⁿ−1. For any two FHSs $U_{i_{1},j_{1}}, U_{i_{2},j_{2}}\in \mathcal {U}$, the periodic Hamming correlation $H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau )$ at time delay τ can be calculated as follows:

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau)=\sum\limits_{l=0}^{q^{n}-2}h[\phi(M_{i_{1}}(l,k)+\phi^{-1}(a_{j_{1}})),\phi(M_{i_{2}}(l+\tau,k)+\phi^{-1}(a_{j_{2}}))] \\ =|\{l|\phi(M_{i_{1}}(l,k)+\phi^{-1}(a_{j_{1}}))=\phi(M_{i_{2}}(l+\tau,k)+\phi^{-1}(a_{j_{2}})), 0\leq l< q^{n}-1\}|, \end{array} $$

where 0<τ<q ⁿ−1 when (i ₁,j ₁)=(i ₂,j ₂), and 0≤τ<q ⁿ−1 when (i ₁,j ₁)≠(i ₂,j ₂).

Since ϕ is one-to-one, then we have

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) =|\{l|M_{i_{1}}(l,k)-M_{i_{2}}(l+\tau,k)=\phi^{-1}(a_{j_{2}})-\phi^{-1}(a_{j_{1}}), 0\leq l< q^{n}-1\}|. \end{array} $$

Let $S_{i_{1},i_{2}} = \{s_{i_{1},i_{2}}(l)\} = \{m_{i_{1}}(l)\}-\{m_{i_{2}}(l+\tau )\}$ be a sequence of length q ⁿ−1 and $\mathbf {r}_{j_{1},j_{2}}=\phi ^{-1}(a_{j_{2}})-\phi ^{-1}(a_{j_{1}})$ a k-tuple. It then follows that

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) =|\{l|S_{i_{1},i_{2}}(l,k)=\mathbf{r}_{j_{1},j_{2}}, 0\leq l<q^{n}-1\}|. \end{array} $$

(7)

Equation (7) is equivalent to

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) = N_{S_{i_{1},i_{2}}}(\mathbf{r}_{j_{1},j_{2}}). \end{array} $$

(8)

Since α is a primitive element of $\mathbf {F}_{q^{n}}$, then α ^W is a primitive element of F _q . Let $b_{i_{1}}=\alpha ^{Wd_{1}}$, $b_{i_{2}}=\alpha ^{Wd_{2}}$, 0≤d ₁,d ₂<q−1, and τ = τ ₁ W + τ ₂, 0≤τ ₁<q−1, 0≤τ ₂<W. For all 0≤l<q ⁿ−1, we have

$$\begin{array}{@{}rcl@{}} s_{i_{1},i_{2}}(l) &=& m_{i_{1}}(l)-m_{i_{2}}(l+\tau) \\ &=&b_{i_{1}}tr_{q^{n}/q}(\eta \alpha^{l})-b_{i_{2}}tr_{q^{n}/q}(\eta \alpha^{l+\tau}) \\ &=&tr_{q^{n}/q}((b_{i_{1}}-b_{i_{2}}\alpha^{\tau})\eta \alpha^{l}) \\ &=&tr_{q^{n}/q}((1-\alpha^{(d_{2}-d_{1}+\tau_{1})W+\tau_{2}})\alpha^{Wd_{1}}\eta \alpha^{l}). \end{array} $$

It is easy to check that $S_{i_{1},i_{2}}$ is an m-sequence of length q ⁿ−1 when $1-\alpha ^{(d_{2}-d_{1}+\tau _{1})W+\tau _{2}}\neq 0$.

In order to compute $H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau )$, we divide the problem into four cases.

Case 1

i ₁ = i ₂, j ₁ = j ₂, 0<τ<q ⁿ−1. In this case, we have

$$\begin{array}{@{}rcl@{}} \mathbf{r}_{j_{1},j_{2}}=\textbf{0}_{k} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} s_{i_{1},i_{2}}(l) = tr_{q^{n}/q}((1-\alpha^{\tau})\alpha^{Wd_{1}}\eta \alpha^{l}). \end{array} $$

Since 0<τ<q ⁿ−1, it is obvious that 1−α ^τ≠0. From Lemma 3 and (8), we obtain

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) = q^{n-k}-1. \end{array} $$

Case 2

i ₁ = i ₂, j ₁≠j ₂, 0≤τ<q ⁿ−1. In this case, we have

$$\begin{array}{@{}rcl@{}} \mathbf{r}_{j_{1},j_{2}}\neq \textbf{0}_{k} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} s_{i_{1},i_{2}}(l) = tr_{q^{n}/q}((1-\alpha^{\tau})\alpha^{Wd_{1}}\eta \alpha^{l}). \end{array} $$

From Lemma 3 and (8), we get

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) = \left\{\!\!\!\begin{array}{lll} \ 0, \ \ \ \ \ \tau=0 \\ \ q^{n-k}, \ \ 0<\tau<q^{n}-1. \end{array}\right. \end{array} $$

Case 3

i ₁≠i ₂, j ₁ = j ₂, 0≤τ<q ⁿ−1. In this case, we have

$$\begin{array}{@{}rcl@{}} \mathbf{r}_{j_{1},j_{2}}= \textbf{0}_{k} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} s_{i_{1},i_{2}}(l) = tr_{q^{n}/q}((1-\alpha^{(d_{2}-d_{1}+\tau_{1})W+\tau_{2}})\alpha^{Wd_{1}}\eta \alpha^{l}). \end{array} $$

From Lemma 3 and (8), it follows that

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) = \left\{\!\!\!\begin{array}{lll} \ q^{n}-1, \ \ \ \ \ \text{if} \ d_{2}-d_{1}+\tau_{1} \equiv 0 \ \text{mod} \ (q-1) \ \text{and}\ \tau_{2}=0 \\ \ q^{n-k}-1, \ \ \text{otherwise}. \end{array}\right. \end{array} $$

Since 1≤|d ₂−d ₁|<q−1, there exist two FHSs $U_{i^{\prime }_{1},j^{\prime }_{1}}, U_{i^{\prime }_{2},j^{\prime }_{2}} \in \mathcal {U}$, whose periodic Hamming correlation $H_{U_{i^{\prime }_{1},j^{\prime }_{1}}, U_{i^{\prime }_{2},j^{\prime }_{2}}}(\tau )$ can be given by

$$\begin{array}{@{}rcl@{}} H_{U_{i^{\prime}_{1},j^{\prime}_{1}}, U_{i^{\prime}_{2},j^{\prime}_{2}}}(\tau) = \left\{\!\!\!\begin{array}{lll} \ q^{n}-1,\ \ \ \ \ \tau=W \\ \ q^{n-k}-1, \ \ \ \text{otherwise}. \end{array}\right. \end{array} $$

Case 4

i ₁≠i ₂, j ₁≠j ₂, 0≤τ<q ⁿ−1. In this case, we have

$$\begin{array}{@{}rcl@{}} \mathbf{r}_{j_{1},j_{2}}\neq \textbf{0}_{k} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} s_{i_{1},i_{2}}(l) = tr_{q^{n}/q}((1-\alpha^{(d_{2}-d_{1}+\tau_{1})W+\tau_{2}})\alpha^{Wd_{1}}\eta \alpha^{l}). \end{array} $$

From Lemma 3 and (8), it follows that

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) = \left\{\!\!\!\begin{array}{lll} \ 0, \ \ \ \ \ \ \text{if} \ d_{2}-d_{1}+\tau_{1} \equiv 0 \ \text{mod} \ (q-1) \ \text{and}\ \tau_{2}=0 \\ \ q^{n-k}, \ \ \text{otherwise}. \end{array}\right. \end{array} $$

By summarizing the results of the above four cases, we have

$$\begin{array}{@{}rcl@{}} H_{U_{i_{1},j_{1}}, U_{i_{2},j_{2}}}(\tau) \leq q^{n-k} \end{array} $$

for 0<τ<W when (i ₁,j ₁)=(i ₂,j ₂), and for 0≤τ<W when (i ₁,j ₁)≠(i ₂,j ₂). Thus, $\mathcal {U}$ is a (q ⁿ−1,q ^k(q−1),q ^k,W−1,q ^n−k) LHZ-FHS set.

We then check the optimality of $\mathcal {U}$ with respect to the Peng-Fan-Lee Bound,

$$\begin{array}{@{}rcl@{}} \varepsilon &\geq& \left\lceil\frac{(q^{k}(q-1)(W-1)+q^{k}(q-1)-q^{k})(q^{n}-1)}{(q^{k}(q-1)(W-1)+q^{k}(q-1)-1)q^{k}}\right\rceil \\ &=&\left\lceil\frac{(q^{n}-2)(q^{n}-1)}{q^{k}(q^{n}-1)-1}\right\rceil \\ &=&\left\lceil q^{n-k}-\frac{2q^{n}-q^{n-k}-2}{q^{n+k}-q^{k}-1}\right\rceil. \end{array} $$

Since

$$\begin{array}{@{}rcl@{}} 0< \frac{2q^{n}-q^{n-k}-2}{q^{n+k}-q^{k}-1} <1, \end{array} $$

it follows that

$$\begin{array}{@{}rcl@{}} \varepsilon \geq q^{n-k}. \end{array} $$

According to Definition 2, $\mathcal {U}$ is a (q ⁿ−1,q ^k(q−1),q ^k,W−1,q ^n−k) LHZ-FHS set with optimal maximum periodic Hamming correlation. □

Appendix B:

Proof Proof of Theorem 2

Clearly, $\mathcal {G}$ contains q−1 sequences of length $\frac {q^{n}-1}{t}$ over V _k . Similar to the proof of Theorem 1. Let H _i ={h _i (l)} = b _i ⋅H, 0≤i<q−1, $0\leq l< \frac {q^{n}-1}{t}$. For any two FHSs $G_{i_{1}}, G_{i_{2}}\in \mathcal {G}$, the periodic Hamming correlation $H_{G_{i_{1}}, G_{i_{2}}}(\tau )$ at time delay τ can be calculated as follows:

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau)&=&\sum\limits_{l=0}^{\frac{q^{n}-1}{t}-1}h[\phi(H_{i_{1}}(l,k)),\phi(H_{i_{2}}(l+\tau,k))] \\ &=&|\{l|\phi(H_{i_{1}}(l,k))=\phi(H_{i_{2}}(l+\tau,k)), 0\leq l< \frac{q^{n}-1}{t}\}|, \end{array} $$

where $0<\tau < \frac {q^{n}-1}{t}$ when i ₁ = i ₂, and $0\leq \tau < \frac {q^{n}-1}{t}$ when i ₁≠i ₂.

Since ϕ is one-to-one, then we have

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau) =|\{l|H_{i_{1}}(l,k)-H_{i_{2}}(l+\tau,k)=\textbf{0}_{k}, 0\leq l< \frac{q^{n}-1}{t}\}|. \end{array} $$

Let $P_{i_{1},i_{2}}= \{p_{i_{1},i_{2}}(l)\} = \{h_{i_{1}}(l)\}-\{h_{i_{2}}(l+\tau )\}$ be a sequence of length $\frac {q^{n}-1}{t}$. It then follows that

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau) =|\{l|P_{i_{1},i_{2}}(l,k)=\textbf{0}_{k}, 0\leq l<\frac{q^{n}-1}{t}\}|. \end{array} $$

(9)

Equation (9) is equivalent to

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau) = N_{P_{i_{1},i_{2}}}(\textbf{0}_{k}). \end{array} $$

(10)

Let $b_{i_{1}}=\alpha ^{Wd_{1}}$, $b_{i_{2}}=\alpha ^{Wd_{2}}$, 0≤d ₁,d ₂<q−1, and τ = τ ₁ W + τ ₂, $0\leq \tau _{1}<\frac {q-1}{t}$, 0≤τ ₂<W. For all $0\leq l<\frac {q^{n}-1}{t}$, we have

$$\begin{array}{@{}rcl@{}} p_{i_{1},i_{2}}(l) &=& h_{i_{1}}(l)-h_{i_{2}}(l+\tau) \\ &=&b_{i_{1}}tr_{q^{n}/q}(\gamma \delta^{l})-b_{i_{2}}tr_{q^{n}/q}(\gamma \delta^{l+\tau}) \\ &=&tr_{q^{n}/q}((b_{i_{1}}-b_{i_{2}}\delta^{\tau})\gamma \delta^{l}) \\ &=&tr_{q^{n}/q}((1-\alpha^{(d_{2}-d_{1}+t\tau_{1})W+t\tau_{2}})\alpha^{Wd_{1}}\gamma \delta^{l}). \end{array} $$

It can be seen that $P_{i_{1},i_{2}}$ is a t-decimated sequence of m-sequence when $1-\alpha ^{(d_{2}-d_{1}+t\tau _{1})W+t\tau _{2}}$ ≠0.

We distinguish the following two cases to calculate $H_{G_{i_{1}}, G_{i_{2}}}(\tau )$.

Case 1

i ₁ = i ₂, $0<\tau <\frac {q^{n}-1}{t}$. In this case, we have

$$\begin{array}{@{}rcl@{}} p_{i_{1},i_{2}}(l) = tr_{q^{n}/q}((1-\alpha^{t\tau})\alpha^{Wd_{1}}\gamma \delta^{l}). \end{array} $$

Since $0<\tau <\frac {q^{n}-1}{t}$, it is obvious that 1−α ^tτ≠0. From Lemma 4 and (10), we obtain

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau) = \frac{q^{n-k}-1}{t}. \end{array} $$

Case 2

i ₁≠i ₂, $0\leq \tau <\frac {q^{n}-1}{t}$. In this case, we have

$$\begin{array}{@{}rcl@{}} p_{i_{1},i_{2}}(l) = tr_{q^{n}/q}((1-\alpha^{(d_{2}-d_{1}+t\tau_{1})W+t\tau_{2}})\alpha^{Wd_{1}}\gamma \delta^{l}). \end{array} $$

From Lemma 4 and (10), it follows that

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau) = \left\{\!\!\!\begin{array}{lll} \ \frac{q^{n}-1}{t}, \ \ \ \text{if} \ d_{2}-d_{1}+t\tau_{1} \equiv 0 \ \text{mod} \ (q-1) \ \text{and}\ \tau_{2}=0 \\ \ \frac{q^{n-k}-1}{t}, \ \ \text{otherwise}. \end{array}\right. \end{array} $$

Since 1≤|d ₂−d ₁|<q−1 and t|(q−1), there exist two FHSs $G_{i^{\prime }_{1}}, G_{i^{\prime }_{2}} \in \mathcal {G}$ , whose periodic Hamming correlation $H_{G_{i^{\prime }_{1}}, G_{i^{\prime }_{2}}}(\tau )$ can be given by

$$\begin{array}{@{}rcl@{}} H_{G_{i^{\prime}_{1}}, G_{i^{\prime}_{2}}}(\tau) = \left\{\!\!\!\begin{array}{lll} \ \frac{q^{n}-1}{t},\ \ \ \ \ \tau=W \\ \ \frac{q^{n-k}-1}{t}, \ \ \ \text{otherwise}. \end{array}\right. \end{array} $$

By summarizing the results of the above two cases, we have

$$\begin{array}{@{}rcl@{}} H_{G_{i_{1}}, G_{i_{2}}}(\tau) = \frac{q^{n-k}-1}{t} \end{array} $$

for 0<τ<W when i ₁ = i ₂, and for 0≤τ<W when i ₁≠i ₂. Thus, $\mathcal {G}$ is a $(\frac {q^{n}-1}{t},q-1,q^{k},W-1,\frac {q^{n-k}-1}{t})$ LHZ-FHS set.

We then check the optimality of $\mathcal {G}$ with respect to the Peng-Fan-Lee Bound,

$$\begin{array}{@{}rcl@{}} \varepsilon &\geq& \left\lceil\frac{((q-1)(W-1)+(q-1)-q^{k})\frac{q^{n}-1}{t}}{((q-1)(W-1)+(q-1)-1)q^{k}}\right\rceil \\ &=&\left\lceil\frac{q^{n-k}-1}{t}-\frac{q^{k}-1}{(q^{n}-2)q^{k}t}\right\rceil. \end{array} $$

Since

$$\begin{array}{@{}rcl@{}} 0< \frac{q^{k}-1}{(q^{n}-2)q^{k}t} <1, \end{array} $$

it follows that

$$\begin{array}{@{}rcl@{}} \varepsilon \geq \frac{q^{n-k}-1}{t}. \end{array} $$

According to Definition 2, $\mathcal {G}$ is a $(\frac {q^{n}-1}{t},q-1,q^{k},W-1,\frac {q^{n-k}-1}{t})$ LHZ-FHS set with optimal maximum periodic Hamming correlation. □