# Optimal Constructions of Low-Hit Zone Frequency-Hopping Sequence Set Based on _m_-Sequence
## 2 Preliminaries
### \(q\)-Ary \(m\)-Sequences and Its Decimated Sequence
 \tag{26}\]  
It can be easily verified that \(0<\frac{(q^{k}-1)(q^{n}-\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor(L_{ z}+1))}{\left(\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor(L_{z}+1)-1 \right)q^{k}}<1\) which leads to  
\[\lambda_{right}=q^{n-k}-1. \tag{27}\]  
The results show that the theorem holds. **Construction C: design based on \(d\)-decimated sequence of \(q\)-ary \(m\)-sequence, \(q\
eq 2\). **  
Step 1: Let \(q\) be any prime power with \(q\
eq 2\), and construct a \(d\)-decimated sequence set \(\mathbf{a}=\{a_{0},a_{1},...,a_{d-1}\}\) of any \(q\)-ary \(m\)-sequence with degree \(n\), such as  
\[a_{i}=\left(T_{q}^{q^{n}}\left(\beta\alpha^{i}\right),T_{q}^{q^{n}}\left(\beta \alpha^{i+d}\right),...,T_{q}^{q^{n}}\left(\beta\alpha^{i+(\frac{q^{n}-1}{d}-1) d}\right)\right) \tag{28}\]  
where \(0\leq i\leq d-1\), \(d\mid(q-1)\) and \(\gcd(d,n)=1\). Step 2: Select any integer \(L_{z}\) with \(2\leq L_{z}\leq\left\lfloor\frac{q^{n}-1}{2d}\right\rfloor-1\) and define a sequence set \(\textbf{b}=\{b_{0},b_{1},...,b_{\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)}\right\rfloor d -1}\},b_{j}=(b_{j}(0),b_{j}(1),...,b_{j}(\frac{q^{n}-1}{d}-1))\), where \(b_{j}(t)\) is defined by  
\[b_{j}(t)=a_{\left\|\frac{q^{n}-1}{d(L_{z}+1)}\right\rceil}\left(t+(\langle j \rangle_{\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)}\right\rfloor}+1)(L_{z}+1)\right) \tag{29}\]  
where \(0\leq j\leq\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)}\right\rfloor d-1\). Step 3: Design the FHS set \(\textbf{S}_{\textbf{c}}=\{s_{c_{0}},s_{c_{1}},...,s_{c_{\left\lfloor\frac{q^{ n}-1}{d(L_{z}+1)}\right\rfloor d-1}\}\), \(s_{c_{k}}=(s_{c_{k}}(0),s_{c_{k}}(1),\)\(...,s_{c_{k}}(\frac{q^{n}-1}{d}-1))\), where \(s_{c_{k}}(t)\) is defined by  
\[s_{c_{k}}(t)=b_{h}(t)+qb_{h}(t+1)+...+q^{k-1}b_{h}(t+k-1) \tag{30}\]  
where \(0\leq h\leq\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)}\right\rfloor d-1,1\leq k\leq n\). **Theorem 3**. : _According to Peng-Fan bound (4), \(\textbf{S}_{\textbf{c}}\) is an optimal LHZ FHS set with parameters \((\frac{q^{n}-1}{d},\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)}\right\rfloor d,q^{ k},L_{z},\frac{q^{n}-k-1}{d})\)-LHZ FHS set under the MPHC._  
Proof. : Let \(\alpha^{\prime}=\left[\alpha\right]_{\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)} \right\rfloor},\)\(\alpha_{0}=\left\langle\alpha\right\rangle_{\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)} \right\rfloor},\)\(\beta^{\prime}=\left[\beta\right]_{\left\lfloor\frac{q^{n}-1}{d(L_{z}+1)} \right\rfloor},\)\(\beta_  
where \(1\leq\tau\leq\frac{q^{n}-1}{d}-1\) if \(\alpha=\beta\) and \(0\leq\tau\leq\frac{q^{n}-1}{d}-1\) if \(\alpha\
eq\beta\). We then analyze \(H(\mathbf{s}_{\mathbf{c}_{\alpha}},\mathbf{s}_{\mathbf{c}_{\beta}},\tau)\) from the perspective of the shift-and-add property of \(m\)-sequences. According to the definition of **a** (28), it is easy to verify that \(L^{(\tau+(\alpha_{0}+1)(L_{z}+1))}(a_{\alpha^{\prime}})\) is a \(d\)-decimated sequence of \(L^{(q^{\prime}+(\tau+(\alpha_{0}+1)(L_{z}+1))d)}(e)\) and \(L^{((\beta_{0}+1)(L_{z}+1))}(a_{\beta^{\prime}})\) is a \(d\)-decimated sequence of \(L^{(\beta^{\prime}+(\beta_{0}+1)(L_{z}+1)d)}(e)\). Thus,  
\[\mathbf{S}_{\boldsymbol{\pi}}=L^{(\tau+(\alpha_{0}+1)(L_{z}+1))}(a_{\alpha^{ \prime}})-L^{((\beta_{0}+1)(L_{z}+1))}(a_{\beta^{\prime}}) \tag{35}\]  
is a \(d\)-decimated sequence of  
\[L^{(\alpha^{\prime}+(\tau+(\alpha_{0}+1)(L_{z}+1))d)}(e)-L^{(\beta^{\prime}+( \beta_{0}+1)(L_{z}+1)d)}(e)=L^{(\alpha^{\prime}-\beta^{\prime}+(\tau+(\alpha_{ 0}-\beta_{0})(L_{z}+1))d)}(e)-e. \tag{36}\]  
Therefore, \((a_{\alpha^{\prime}}(t+(\alpha_{0}+1)(L_{z}+1)+\tau)-a_{\beta^{\prime}}(t+( \beta_{0}+1)(L_{z}+1)),a_{\alpha^{\prime}}(t+(\alpha_{0}+1)(L_{z}+1)+\tau+1)-a_ {\beta^{\prime}}(t+(\beta_{0}+1)(L_{z}+1)+1),...,a_{\alpha^{\prime}}(t+( \alpha_{0}+1)(L_{z}+1)+\tau+k-1)-a_{\beta^{\prime}}(t+(\beta_{0}+1)(L_{z}+1)+k- 1))\) is the consecutive \(k\)-tuples in \(\mathbf{S}_{\boldsymbol{\pi}}\). Hence, (34) can be rewritten as (37). \[