# Optimal Constructions of Low-Hit Zone Frequency-Hopping Sequence Set Based on _m_-Sequence
## 2 Preliminaries
### \(q\)-Ary \(m\)-Sequences and Its Decimated Sequence  
Using the trace function \(T_{q}^{q^{n}}(x)\) from \(\mathbb{F}_{q^{n}}\) to \(\mathbb{F}_{q}\), any \(q\)-ary \(m\)-sequence of degree \(n\) over \(\mathbb{F}_{q}\), such as \(\mathbf{e}\), can be presented as  
\[\mathbf{e}=(T_{q}^{q^{n}}(\beta),T_{q}^{q^{n}}(\beta\alpha),...,T_{q}^{q^{n}}( \beta\alpha^{q^{n}-2})) \tag{5}\]  
where \(\beta\in\mathbb{F}_{q^{n}}/\{0\}\), and \(\alpha\) is a primitive element of \(\mathbb{F}_{q^{n}}\). Any \(q\)-ary \(m\)-sequences have many good properties, and scholars such as [PERSON] et al. [29]. have conducted in-depth research on them. Among these properties, the ideal \(k\)-tuple distribution property and shift-and-add property will be utilized in the designs of LHZ FHS sets in this paper. **Lemma 2** (Ideal \(k\)-tuple distribution property). : _For any \(q\)-ary \(m\)-sequence of degree \(n\) such as \(\mathbf{e}\), for any \(k\) with \(1\leq k\leq n\), the zero \(k\)-tuple \((0,0,...,0)\) occurs in \(\mathbf{e}\) exactly \(q^{n-k}-1\) times, and each nonzero \(k\)-tuple \((\lambda_{0},\lambda_{1},...,\lambda_{k-1})\in\mathbb{F}_{q}^{k}\) occurs exactly \(q^{n-k}\) times._  
**Lemma 3** (Shift-and-add property). : _Let \(\mathbf{e}\) be any \(q\)-ary \(m\)-sequence of degree \(n\). Then, for each pair  
**Theorem 1**. : _According to the Peng-Fan bound (4), \(\mathbf{S_{a}}\) is a \((q^{n}-1,q^{k}\left\lfloor\frac{q^{n}-1}{\sum_{i}+1}\right\rfloor,q^{k},L_{z},q^ {n-k})\)-LHZ FHS set with optimal MPHC._  
Proof. : Let \(\mathbf{s}_{a_{p}},\mathbf{s}_{a_{\gamma}}\) be any two FHSs in \(\mathbf{S_{a}}\). \(v^{\prime}=\left[\rho\right]_{q^{k}},l^{\prime}=\left[\gamma\right]_{q^{k}  
We then analyze the MPHAC, MPHCC and MPHC of \(\mathbf{S}\), and the following cases exist here. Case 1: \(\rho=\gamma\). Obviously, \((l_{0}-v_{0},l_{1}-v_{1},...,l_{k-1}-v_{k-1})=(0,0,...,0)\). In this case, according to Lemma 3 and (13), \(\mathbf{S}_{\theta}=L^{(\tau)}(\mathbf{e})-\mathbf{e}\) is still a \(q\)-ary \(m\)-sequence of degree \(n\) for any \(\tau\). Consequently, according to Lemma 2, \(H(\mathbf{s}_{a_{\rho}},\mathbf{s}_{a_{\gamma}},\tau)=q^{n-k}-1\) for any \(\rho=\gamma\). The MPHAC can be obtained as follows. \[H_{a}(\mathbf{S_{a}})=q^{n-k}-1. \tag{15}\]  
Case 2: \(\rho\
eq\gamma\). Case 2. 1: \(v^{\prime}=l^{\prime}\). \((l_{0}-v_{0},l_{1}-v_{1},...,l_{k-1}-v_{k-1})\
eq(0,0,...,0)\) because of \(\rho\
eq\gamma\). Based on Lemma 3, \(\mathbf{S}_{\theta}=L^{(\tau)}(\mathbf{e})-\mathbf{e}\) is still a \(q\)-ary \(m\)-sequence of degree \(n\) for any \(\tau\). Thus, it follows that \(H(\mathbf{s}_{a_{\rho}},\mathbf{s}_{a_{\gamma}},\tau)=q^{n-k}\) from Lemma 2. Case 2. 2: \(v^{\prime}\
eq l^{\prime}\). \(\mathbf{S}_{\theta}\) changes to a sequence whose all items are zero when \(\tau=(l^{\prime}-v^{\prime})(L_{z}+1)\). Consequently, from (14), it can be deduced that  
\[H(\mathbf{s}_{a_{\rho}},\mathbf{s}_{a_{\gamma}},\tau)=\left\{ \begin{array}{ ll}q^{n}-1&\quad\text{if}(l_{0}-v_{0},...,l_{k-1}-v_{k-1})=(0,...,0),\\ 0&\quad\text{otherwise}.\end{array} \right. \tag{16}\]  
\(\mathbf{S}_{\theta}\) is still a \(q\)-ary \(m\)-sequence of degree \(n\) for any \(\tau\) when \(\tau\
eq(l^{\prime}-v^{\prime})(L_{z}+1)\) from Lemma 3. It then follows, based on (14), that  
\[H(\mathbf{s}_{a_{\rho}},\mathbf{s}_{a_{\gamma}},\tau)=\left\{ \begin{array}{ ll}q^{n-k}-1&\quad\text{if}(l_{0}-v_{0},...,l_{k-1}-v_{k-1})=(0,...,0),\\ q^{n-k}&\quad\text{otherwise}.\end{array} \right. \tag{17}\]  
Based on the above discussions, for \(\tau\leq L_{z}\), the MPHCC of \(\mathbf{S}\) is  
\[H_{c}(\mathbf{S_{a}})=q^{n-k}. \tag{18}\]  
Therefore, \(\mathbf{S_{a}}\) is an LHZ FHS set, the MPHC of which is equal to  
\[H(\mathbf{S_{a}})=q^{n-k}. \tag{19}\]  
for \(\tau\leq L_{z}\).