# Optimal Constructions of Low-Hit Zone Frequency-Hopping Sequence Set Based on _m_-Sequence
## 2 Preliminaries
### \(q\)-Ary \(m\)-Sequences and Its Decimated Sequence
Let \(\lambda_{right}\) denote the right side of Peng-Fan bound (4), input all the parameters of \(\mathbf{S_{a}}\), and \(\lambda_{right}\) can be given by  
\[\lambda_{right} =\left[\frac{q^{k}\Big{\lfloor}\frac{q^{n}-1}{L_{z}+1}\Big{\rfloor} (L_{z}+1)-q^{k}}{q^{k}\Big{\lfloor}\frac{q^{n}-1}{L_{z}+1}\Big{\rfloor}(L_{z}+ 1)-1}\cdot\frac{q^{n}-1}{q^{k}}\right] \tag{20}\] \[=\left[q^{n-k}-\frac{(q^{k}\Big{\lfloor}\frac{q^{n}-1}{L_{z}+1} \Big{\rfloor}(L_{z}+1)-1)+(q^{k}-1)(q^{n}-1)}{(q^{k}\Big{\lfloor}\frac{q^{n}- 1}{L_{z}+1}\Big{\rfloor}(L_{z}+1)-1)q^{k}}\right].\]It is obvious that \(\frac{(q^{k}\left\lfloor\frac{q^{n}-1}{k+1}\right\rfloor(L_{z}+1)-1)+(q^{k}-1)(q ^{n}-1)}{(q^{k}\left\lfloor\frac{q^{n}-1}{k+1}\right\rfloor(L_{z}+1)-1)q^{k}}>0\). Furthermore, let \(q^{n}-1=g(L_{z}+1)+R\), \(0\leq R\leq L_{z}\leq\left\lfloor\frac{q^{n}-1}{2}\right\rfloor-1\leq\frac{q^{ n}-1}{2}-1\), and we have  
\[ \begin{split}(q^{k}\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right \rfloor(L_{z}+1)-1)q^{k}-(q^{k}\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor (L_{z}+1)-1)-(q^{k}-1)(q^{n}-1)\\ =(q^{k}-1)(q^{k}\left\lfloor\frac{q^{n}-1}{L_{z}+1}\right\rfloor (L_{z}+1)-q^{n})\\ =(q^{k}-1)(q^{k}(q^{n}-1-R)-q^{n})\\ >(q^{k}-1)(q^{k}\frac{q^{n}+1}{2}-q^{n})\\ >0.\end{split}  \tag{21}\]  
Thus, \(0<\frac{(q^{k}\left\lfloor\frac{q^{n}-1}{k+1}\right\rfloor(L_{z}+1)-1)+(q^{k }-1)(q^{n}-1)}{(q^{k}\left\lfloor\frac{q^{n}-1}{k+1}\right\rfloor(L_{z}+1)-1) q^{k}}<1\). One can have  
\[\lambda_{right}=q^{n-k}. \tag{22}\]  
The equal sign in Peng-Fan bound (4) is true. Thus, \(\mathbf{S_{a}}\) is an optimal LHZ FHS set. The proof is then completed. **Remark 1**. : _When \(L_{z}=\frac{q^{n}-1}{q-1}\cdot 1\), \(\mathbf{S_{a}}\) is a \((q^{n}-1,q^{k}(q-1),q^{k},\frac{q^{n}-1}{q-1}-1,q^{n-k})\)-LHZ FHS set which is exactly the sequence set constructed by [PERSON] et al. (2014). And when \(T(L_{z}+1)=q^{n}-1\), \(\mathbf{S_{a}}\) is a \((q^{n}-1,q^{k}T,q^{k},L_{z},q^{n-k})\)-LHZ FHS set which is exactly the sequence set constructed by [PERSON] et al. (2015). However, their constructions have minor issues. For instance, we do not get the desired results in the construction of [PERSON] et al. when \(q=2\) and in the design of [PERSON] et al. when \(q^{n}-1\) is Mersenne prime. It is obvious that \(\mathbf{S_{a}}\) in this paper can be designed by any \(m\)-sequence including a binary \(m\)-sequence under the condition of \((L_{z}+1)\
mid(q^{n}-1)\). That is, the designs of [PERSON] et al. and [PERSON] et al. are all the special cases of construction \(A\)._  
We then illustrate the construction \(A\) by the following example, which is designed by a binary \(m\)-sequence. **Example 1**. : _Let \(q=2\), \(n=9\). A binary \(m\)-sequence \(\mathbf{e}\) with degree 9, which is generated by \(f(x)=x^{7}+x^{6}+x^{5}+x^{4}+1\), is listed below._  
\(\mathbf{e}=(1,1,1,1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,0,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0,0,0,...)._  
_Let \(L_{z}=49\). After performing a shift operation on \(\mathbf{e}\), we obtain \(\mathbf{C}=\{c_{0},c_{1},...,c_{9}\}\) as follows._  
\(\mathbf{C}=\{c_{0}=(0,0,1,1,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,1,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,1,1,0\(s_{a_{159}}\) = (3, 7, 15, 14, 12, 8, 1, 2, 5, 11, 6, 13, 11, 7, 14, 12, 9, 2, 5, 10, 4, 9, 2, 4, 8, 0, 0, 1, 2, 4, 9, 3, 6, 12, 9, 3, 7, 14, 13, 10, 4, 8, 1, 3, 7, 15, 15, 14, 13, 11, 7, 15, 14, 12, 8,...)._  
_Computer experiments show that the MPHC of \(\mathbf{S_{a}}\) is equal to 31 for \(\mathbf{1\leq\tau\leq 510}\), and the MPHCC distribution is identical to the MPHC distribution, as illustrated in Figure 1. Thus, the MPHC of \(\mathbf{S_{a}}\) equals 32 for \(\mathbf{\tau\leq 49}\).