# 8阶分圆类构造几乎差集偶Constructions of Almost Difference Set Pairs by Cyclotomic Classes of Order Eight

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Almost difference set pairs are widely used in many problems because they are closely related to almost optimal binary sequence pairs. Cyclotomic classes method is an important method to con-struct almost difference set pairs. In this paper, several new almost difference set pairs are con-structed by using cyclotomic classes of order eight.

1. 引言

2. 基础知识

${Z}_{q}^{*}={\cup }_{i=0}^{e-1}{C}_{i}$

3. 8阶分圆类构造几乎差集偶

1) 当f为偶数且2不是4次剩余，且 $b=-y,x-a=4$ 时，

1、 $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

Table 1. Relations of cyclotomic numbers of order 8 for even f

Table 2. The fifteen basic cyclotomic numbers of order 8 when f is even and 2 is not a quartic residue

2、 $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

3、 $\left(U,{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f,2f,6f\right)-\text{ADSP}$

4、 $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

2) 当f为偶数且2不是4次剩余，且 $b=4-y,x-a=4$ 时， $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

$\begin{array}{c}{\text{Δ}}_{i}=|\left({C}_{0}+{\theta }^{i}\right)\cap {C}_{0}|+|\left({C}_{0}+{\theta }^{i}\right)\cap {C}_{4}|+|\left({C}_{0}+{\theta }^{i}\right)\cap {C}_{5}|+|\left({C}_{0}+{\theta }^{i}\right)\cap {C}_{6}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\left({C}_{1}+{\theta }^{i}\right)\cap {C}_{0}|+|\left({C}_{1}+{\theta }^{i}\right)\cap {C}_{4}|+|\left({C}_{1}+{\theta }^{i}\right)\cap {C}_{5}|+|\left({C}_{1}+{\theta }^{i}\right)\cap {C}_{6}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\left({C}_{2}+{\theta }^{i}\right)\cap {C}_{0}|+|\left({C}_{2}+{\theta }^{i}\right)\cap {C}_{4}|+|\left({C}_{2}+{\theta }^{i}\right)\cap {C}_{5}|+|\left({C}_{2}+{\theta }^{i}\right)\cap {C}_{6}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\left({C}_{4}+{\theta }^{i}\right)\cap {C}_{0}|+|\left({C}_{4}+{\theta }^{i}\right)\cap {C}_{4}|+|\left({C}_{4}+{\theta }^{i}\right)\cap {C}_{5}|+|\left({C}_{4}+{\theta }^{i}\right)\cap {C}_{6}|\end{array}$

$\begin{array}{l}=\left(-i,-i\right)+\left(-i,4-i\right)+\left(-i,5-i\right)+\left(-i,6-i\right)+\left(1-i,-i\right)+\left(1-i,4-i\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left(1-i,6-i\right)+\left(1-i,5-i\right)+\left(2-i,-i\right)+\left(2-i,4-i\right)+\left(2-i,5-i\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left(2-i,6-i\right)+\left(4-i,-i\right)+\left(4-i,4-i\right)+\left(4-i,5-i\right)+\left(4-i,6-i\right)\end{array}$

${\Delta }_{0}={\Delta }_{4}=\frac{16q-4x+16y+4a+16b-64}{64}$

${\Delta }_{1}={\Delta }_{5}={\Delta }_{2}={\Delta }_{6}=\frac{16q+4x-4a-32}{64}$

${\Delta }_{3}={\Delta }_{7}=\frac{16q-4x-16y+4a-16b}{64}$

${\Delta }_{0}={\Delta }_{1},|{\Delta }_{1}-{\Delta }_{3}|=1$，即要满足

$\frac{16q-4x+16y+4a+16b-64}{64}=\frac{16q+4x-4a-32}{64}$

$|\frac{16q+4x-4a-32}{64}-\frac{16q-4x-16y+4a-16b}{64}|=1$

${\Delta }_{0}={\Delta }_{3},|{\Delta }_{1}-{\Delta }_{3}|=1$，即

$\frac{16q-4x+16y+4a+16b-64}{64}=\frac{16q-4x-16y+4a-16b}{64}$

$|\frac{16q+4x-4a-32}{64}-\frac{16q-4x-16y+4a-16b}{64}|=1$

${\Delta }_{1}={\Delta }_{3},|{\Delta }_{0}-{\Delta }_{1}|=1$，即

$\frac{16q+4x-4a-32}{64}=\frac{16q-4x-16y+4a-16b}{64}$

$|\frac{16q-4x+16y+4a+16b-64}{64}-\frac{16q+4x-4a-32}{64}|=1$

1) 当f为偶数且2不是4次剩余，且 $b=-y,x-a=4$ 时，

1、 $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

2、 $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

3、 $\left(U,{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f,2f,6f\right)-\text{ADSP}$

4、 $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

2) 当f为偶数且2不是4次剩余，且 $b=4-y,x-a=4$ 时， $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

1) 当f为偶数且2不是4次剩余，且 $b=-y,x-a=4$ 时，

1、 $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

2、 $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

3、 $\left(U,{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f,2f,6f\right)-\text{ADSP}$

4、 $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

2) 当f为偶数且2不是4次剩余，且 $b=4-y,x-a=4$ 时， $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

1) 当f为偶数且2不是4次剩余，且 $b=y,x-a=4$ 时，

1、 $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

2、 $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

3、 $\left(U,{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f,2f,6f\right)-\text{ADSP}$

4、 $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

2) 当f为偶数且2不是4次剩余类，且 $b=y-4,x-a=4$ 时， $\left({U}^{\prime },W\right)$ 构成

1) 当f为偶数且2不是4次剩余， $b=y,x-a=4$ 时，

1、 $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

2、 $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

3、 $\left(U,{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f,2f,6f\right)-\text{ADSP}$

4、 $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

2) 当f为偶数且2不是4次剩余，且 $b=y-4,x-a=4$ 时， $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

1) 当f为偶数且2不是4次剩余，且 $b=y,x-a=4$ 时，

1、 $\left(U,W\right)$ 构成 $\left(8f+1,4f,4f,2f,2f-1,2f\right)-\text{ADSP}$

2、 $\left({U}^{\prime },W\right)$ 构成 $\left(8f+1,4f+1,4f,2f,2f,6f\right)-\text{ADSP}$

3、 $\left(U,{W}^{\prime }\right)$ 构成 $\left(8f+1,4f,4f+1,2f,2f,6f\right)-\text{ADSP}$

4、 $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

2) 当f为偶数且2不是4次剩余类，且 $b=y-4,x-a=4$ 时， $\left({U}^{\prime },{W}^{\prime }\right)$ 构成 $\left(8f+1,4f+1,4f+1,2f+1,2f,2f\right)-\text{ADSP}$

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