# Optimal Constructions of Low-Hit Zone Frequency-Hopping Sequence Set Based on _m_-Sequence
## 2 Preliminaries
### \(q\)-Ary \(m\)-Sequences and Its Decimated Sequence
It is easy to check that \(\mathbf{S_{a}}\) is an optimal \((511,160,16,49,32)\)-LHZ FHS set according to Peng-Fan bound (4)._  
_Compared with the construction of [PERSON] et al. (2015), different values are taken for \(L_{Z}\), and we can obtain more LHZ FHS sets with optimal MPHC by Example 1, which are listed in Table 2. Furthermore, under the same conditions as Example 1, the construction of [PERSON] et al. (2014) fails to yield the desired results._  
**Construction B: design based on any \(m\)-sequence**. First, let \(q\) be any prime power, and construct a \(q\)-ary \(m\)-sequence with degree \(n\), such as \(\mathbf{e}\). Second, select any integer \(L_{z}\) with \(2\leq L_{z}\leq\left\lfloor\frac{q^{n}-1}{2}\right\rfloor-1\), define a sequence set \(\mathbf{a}=(a_{0},a_{1},...,a_{\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor -1})\), \(a_{i}=(a_{i}(0),a_{i}(1),...,a_{i}(q^{n}-2))\), and \(a_{i}(t)\) is given as  
\[a_{i}(t)=e(t+(i+1)(L_{z}+1)). \tag{23}\] \begin{table}
\begin{tabular}{c c c} \hline
**Ref.** & **Parameters** & **Parameters** \\ \hline \multirow{8}{*}{this paper} & (511, 32, 16, 170–254, 32) & (511, 48, 16, 127–169, 32) \\ \cline{2-3}  & (511, 64, 16, 102–126, 32) & (511, 80, 16, 85–101, 32) \\ \cline{2-3}  & (511, 96, 16, 73–84, 32) & (511, 112, 16, 63–72, 32) \\ \cline{2-3}  & (511, 128, 16, 56–62, 32) & (511, 144, 16, 51–57, 32) \\ \cline{2-3}  & \(\ldots\) & \(\ldots\) \\ \cline{2-3}  & (511, 2032, 16, 3, 32) & (511, 2720, 16, 2, 32) \\ \hline
[15] & (511, 1168, 16, 6, 32) & (511, 112, 16, 72, 32) \\ \hline \end{tabular}
\end{table} 
Table 2: LHZ FHS sets with optimal MPHC constructed by Example 1 and existing ones. Figure 1: The MPHCC distribution and the MPHC distribution of \(\mathbf{S_{a}}\) for \(0\leq\tau\leq 510\). Last, define the desired FHS set \(\mathbf{S_{b}}=\{s_{b_{w}}s_{b_{1}},...,s_{b_{l}}|_{\frac{q^{n}-1}{L_{z}+1}}\}\) where \(s_{b_{w}}=(s_{b_{w}}(0),s_{b_{w}}(1),\)  
\[...,s_{b_{w}}(q^{n}-2)),0\leq w\leq\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor -1,\,s_{b_{w}}(t)\text{ is defined by}\]  
\[s_{b_{w}}(t)=a_{w}(t)+qa_{w}(t+1)+\cdots+q^{k-1}a_{w}(t+k-1), \tag{24}\]  
where \(0\leq t\leq q^{n}-2\) and \(1\leq k\leq n\). Theorem 3.2: _According to Peng-Fan bound (4), \(\mathbf{S_{b}}\) is a \((q^{n}-1,\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor,q^{k},L_{z},q^{n-k} -1)\text{-}LHZ\) FHS set with an optimal MPHC._  
Proof: Let \(\mathbf{s_{b_{w}}},\mathbf{s_{b_{l}}}\) be any two FHSs in \(\mathbf{S_{b}}\). According to the proof of Theorem 3.1, the periodic Hamming correlation \(H(\mathbf{s_{b_{w}}},\mathbf{s_{b_{l}}},\tau)\) of \(\mathbf{s_{b_{w}}}\) and \(\mathbf{s_{b_{l}}}\) at the time delay \(\tau\) can be calculated as follows. \[ \begin{split} H(\mathbf{s_{b_{w}}},\mathbf{s_{b_{l}}},\tau)=\sum \limits_{t=0}^{q^{n}-1}h[&(e(t+(w+1)(L_{z}+1)+\tau)-e(t+(l+1)(L_{ z}+1)),\\ & e(t+(w+1)(L_{z}+1)+\tau+1)-e(t+(l+1)(L_{z}+1)+1),\\ & e(t+(w+1)(L_{z}+1)+\tau+k-1)\\ &-e(t+(l+1)(L_{z}+1)+k-1)),(0,0,...,0)]\end{split}  \tag{25}\]  
where \(1\leq\tau<q^{n}-1\) if \(w=l\) and \(0\leq\tau<q^{n}-1\) if \(w\
eq l\). It is easy to check that \(H(\mathbf{S_{b}})=q^{n-k}-1\) for \(\tau\leq L_{z}\). Then, we input all the parameters of \(\mathbf{S_{b}}\) on the right side of Zhou-Peng bound (4), the value of which is as follows. \[ \begin{split}\lambda_{right}&=\left\lceil\frac{ \left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor(L_{z}+1)-q^{k}}{\left\lfloor \frac{q^{n}-1}{L_{z}+1}\right\rfloor(L_{z}+1)-1}\cdot\frac{q^{n}-1}{q^{k}} \right\rceil\\ &=\left\lceil\frac{q^{n}-1}{q^{k}}-\frac{q^{k}-1}{\left\lfloor \frac{q^{n}-1}{L_{z}+1}\right\rfloor(L_{z}+1)-1}\cdot\frac{q^{n}-1}{q^{k}} \right\rceil\\ &=\left\lceil q^{n-k}-1-\frac{(q^{k}-1)(q^{n}-\left\lfloor\frac{q^ {n}-1}{L_{z}+1}\right\rfloor(L_{z}+1))}{\left(\left\lfloor\frac{q^{n}-1}{L_{ z}+1}\right\rfloor(L_{z}+1)-1\right)q^{k}}\right\rceil.\end{split}