﻿ 一些新的差族和几乎差族

# 一些新的差族和几乎差族Some New Difference Families and Almost Difference Families

Abstract: The characteristic sequences of a kind of difference families (DFs) or almost difference families (ADFs) form optical orthogonal codes which have many applications in a code division multiple access communication system. Moreover, either a DF or ADF corresponds to a kind of partially balanced incomplete block designs which arise in many combinatorial and statistical problems. In this paper, we obtain some new DFs and ADFs by cyclotomic classes of order 6.

1. 引言

2. 基础知识

2.1. 差族和几乎差族的定义

2.2. 分圆类的相关定义及性质

$q=ef+1$ 为一个奇素数， $\theta$$\text{GF}\left(q\right)$ 的一个本原元， $〈{\theta }^{e}〉=\left\{{\theta }^{ie}:0\le i\le f-1\right\}$ 是由 ${\theta }^{e}$ 生成的 ${\text{GF}}^{\text{*}}\left(q\right)$ 的乘法子群，它的陪集 ${C}_{j}^{\left(e,q\right)}={\theta }^{j}〈{\theta }^{e}〉\text{\hspace{0.17em}}\left(0\le j\le e-1\right)$ 称为在 $\text{GF}\left(q\right)$ 中的e阶分圆类。定义分圆数 ${\left(j,k\right)}_{e}$ 为方程 $x+1=y,x\in {C}_{j}^{\left(e,q\right)},y\in {C}_{k}^{\left(e,q\right)}$ 解的个数，即

${\left(j,k\right)}_{e}=|\left({C}_{j}^{\left(e,q\right)}+1\right)\cap {C}_{k}^{\left(e,q\right)}|.$

1) ${\left(i,j\right)}_{e}={\left({i}^{\prime },{j}^{\prime }\right)}_{e}$，其中 $i\equiv {i}^{\prime }\left(\mathrm{mod}e\right),j\equiv {j}^{\prime }\left(\mathrm{mod}e\right)$

2) ${\left(i,j\right)}_{e}={\left(e-i,j-i\right)}_{e}=\left\{\begin{array}{l}{\left(j,i\right)}_{e},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f为偶数,\\ {\left(j+\frac{e}{2},i+\frac{e}{2}\right)}_{e},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f为奇数.\end{array}$

3. 六阶分圆类构造差族或几乎差族

3.1. 六阶分圆数

Table 1. The relations of the cyclotomic numbers of order 6 when f is odd [9]

Table 2. The ten basic cyclotomic numbers of order 6 when f is odd [9]

3.2. 新的差族和几乎差族

①当f为奇数， $m\equiv 0\left(\mathrm{mod}3\right)$，且 $a-6b=-2$ 时，

$\left({C}_{0}\cup {C}_{1},{C}_{3}\cup {C}_{5}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p-1}{3}\right\},\frac{2p-8}{9}\right)$ -DF;

$\left({C}_{0}\cup {C}_{2},{C}_{0}\cup {C}_{1}\cup {C}_{5}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p-1}{2}\right\},\frac{13p-43}{36}\right)$ -DF.

②当f为奇数， $m\equiv 1\left(\mathrm{mod}3\right)$，且 $a+3b=7$ 时，

$\left({C}_{0}\cup \left\{0\right\},{C}_{1}\cup {C}_{2}\right)$ 是一个 $\left(p,\left\{\frac{p+5}{6},\frac{p-1}{3}\right\},\frac{5p-11}{36}\right)$ -DF.

③当f为奇数， $m\equiv 1\left(\mathrm{mod}3\right)$，且 $a+3b=1$ 时，

$\left({C}_{0}\cup \left\{0\right\},{C}_{1}\cup {C}_{2}\cup \left\{0\right\}\right)$ 是一个 $\left(p,\left\{\frac{p+5}{6},\frac{p+2}{3}\right\},\frac{5p+13}{36}\right)$ -DF.

$x\in {C}_{i}\left(0\le i\le 5\right)$，则

$\begin{array}{l}|\left(\left({C}_{0}\cup {C}_{1}\right)+x\right)\cap \left({C}_{0}\cup {C}_{1}\right)|\\ =|\left({C}_{0}+x\right)\cap {C}_{0}|+|\left({C}_{0}+x\right)\cap {C}_{1}|+|\left({C}_{1}+x\right)\cap {C}_{0}|+|\left({C}_{1}+x\right)\cap {C}_{1}|\\ =\left(-i,-i\right)+\left(-i,1-i\right)+\left(1-i,-i\right)+\left(1-i,1-i\right)\\ =\left(i,0\right)+\left(i,1\right)+\left(i-1,-1\right)+\left(i-1,0\right).\end{array}$

$\begin{array}{l}|\left(\left({C}_{3}\cup {C}_{5}\right)+x\right)\cap \left({C}_{3}\cup {C}_{5}\right)|\\ =|\left({C}_{3}+x\right)\cap {C}_{3}|+|\left({C}_{3}+x\right)\cap {C}_{5}|+|\left({C}_{5}+x\right)\cap {C}_{3}|+|\left({C}_{5}+x\right)\cap {C}_{5}|\\ =\left(3-i,3-i\right)+\left(3-i,5-i\right)+\left(5-i,3-i\right)+\left(5-i,5-i\right)\\ =\left(i-3,0\right)+\left(i-3,2\right)+\left(i-5,-2\right)+\left(i-5,0\right).\end{array}$

${\text{Δ}}_{i}=\left(i,0\right)+\left(i,1\right)+\left(i-1,-1\right)+\left(i-1,0\right)+\left(i-3,0\right)+\left(i-3,2\right)+\left(i-5,-2\right)+\left(i-5,0\right),$

$\text{Δ}\left({C}_{0}\cup {C}_{1},{C}_{3}\cup {C}_{5}\right)={\cup }_{i=0}^{5}{\text{Δ}}_{i}{C}_{i}.$

${\text{Δ}}_{0}={\text{Δ}}_{3}=-\frac{10}{9}-\frac{a}{9}+\frac{2b}{3}+\frac{2p}{9},$

${\text{Δ}}_{1}={\text{Δ}}_{2}={\text{Δ}}_{4}={\text{Δ}}_{5}=-\frac{7}{9}+\frac{a}{18}-\frac{b}{3}+\frac{2p}{9}.$

$\left({C}_{0}\cup {C}_{1},{C}_{3}\cup {C}_{5}\right)$ 为差族，当且仅当 ${\text{Δ}}_{0}={\text{Δ}}_{1}$，即 $-\frac{10}{9}-\frac{a}{9}+\frac{2b}{3}+\frac{2p}{9}=-\frac{7}{9}+\frac{a}{18}-\frac{b}{3}+\frac{2p}{9}$。化简，得 $a-6b=-2$，此时，参数 $\lambda =\frac{2p-8}{9}$，所以 $\left({C}_{0}\cup {C}_{1},{C}_{3}\cup {C}_{5}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p-1}{3}\right\},\frac{2p-8}{9}\right)$ -DF。

$\text{Δ}\left({C}_{0}\cup {C}_{2},{C}_{0}\cup {C}_{1}\cup {C}_{5}\right)={\cup }_{i=0}^{5}{\text{Δ}}_{i}{C}_{i},$

${\text{Δ}}_{0}={\text{Δ}}_{2}={\text{Δ}}_{3}={\text{Δ}}_{5}=-\frac{47}{36}-\frac{a}{18}+\frac{b}{3}+\frac{13p}{36},$

${\text{Δ}}_{1}={\text{Δ}}_{4}=-\frac{35}{36}+\frac{a}{9}-\frac{2b}{3}+\frac{13p}{36}.$

$-\frac{47}{36}-\frac{a}{18}+\frac{b}{3}+\frac{13p}{36}=-\frac{35}{36}+\frac{a}{9}-\frac{2b}{3}+\frac{13p}{36}.$

①当f为奇数， $m\equiv 0\left(\mathrm{mod}3\right)$ 时，

$a=4$，则 $\left({C}_{0}\cup \left\{0\right\},{C}_{0}\cup {C}_{1}\cup {C}_{4}\right)$ 是一个 $\left(p,\left\{\frac{p+5}{6},\frac{p-1}{2}\right\},\frac{5p-17}{18},\frac{2\left(p-1\right)}{3}\right)$ -ADF， $\left({C}_{0}\cup {C}_{3}\cup \left\{0\right\},{C}_{1}\cup {C}_{5}\right)$ 是一个 $\left(p,\left\{\frac{p+2}{3},\frac{p-1}{3}\right\},\frac{2p-5}{9},\frac{2\left(p-1\right)}{3}\right)$ -ADF；

$a=16$，则 $\left({C}_{0}\cup {C}_{3}\cup \left\{0\right\},{C}_{1}\cup {C}_{5}\right)$ 是一个 $\left(p,\left\{\frac{p+2}{3},\frac{p-1}{3}\right\},\frac{2p-8}{9},\frac{p-1}{3}\right)$ -ADF。

②当f为奇数， $m\equiv 1\left(\mathrm{mod}3\right)$ 时，若 $a=-2$，则 $\left({C}_{0}\cup {C}_{1}\cup \left\{0\right\},{C}_{1}\cup {C}_{5}\right)$$\left({C}_{0}\cup {C}_{2}\cup \left\{0\right\},{C}_{0}\cup {C}_{3}\right)$$\left(p,\left\{\frac{p+2}{3},\frac{p-1}{3}\right\},\frac{2p-5}{9},\frac{2\left(p-1\right)}{3}\right)$ -ADF；

$a=10$，则 $\left({C}_{0}\cup {C}_{2}\cup \left\{0\right\},{C}_{0}\cup {C}_{3}\right)$ 是一个 $\left(p,\left\{\frac{p+2}{3},\frac{p-1}{3}\right\},\frac{2p-8}{9},\frac{p-1}{3}\right)$ -ADF。

③当f为奇数， $m\equiv 2\left(\mathrm{mod}3\right)$ 时，

$a-3b=-5$，则 $\left({C}_{0}\cup {C}_{2},{C}_{0}\cup {C}_{1}\cup {C}_{3}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p-1}{2}\right\},\frac{13p-55}{36},\frac{2\left(p-1\right)}{3}\right)$ -ADF；

$a-3b=1$，则 $\left({C}_{0}\cup {C}_{2},{C}_{0}\cup {C}_{1}\cup {C}_{3}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p-1}{2}\right\},\frac{13p-67}{36},\frac{p-1}{3}\right)$ -ADF；

$a=-2$，则 $\left({C}_{0}\cup {C}_{3},{C}_{0}\cup {C}_{4}\cup \left\{0\right\}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p+2}{3}\right\},\frac{2p-5}{9},\frac{2\left(p-1\right)}{3}\right)$ -ADF；

$a=10$，则 $\left({C}_{0}\cup {C}_{3},{C}_{0}\cup {C}_{4}\cup \left\{0\right\}\right)$ 是一个 $\left(p,\left\{\frac{p-1}{3},\frac{p+2}{3}\right\},\frac{2p-8}{9},\frac{p-1}{3}\right)$ -ADF。

$\text{Δ}\left({C}_{0}\cup \left\{0\right\}\right)=\left(\left(0,0\right)+1\right){C}_{0}\cup \left(1,0\right){C}_{1}\cup \left(2,0\right){C}_{2}\cup \left(\left(3,0\right)+1\right){C}_{3}\cup \left(4,0\right){C}_{4}\cup \left(5,0\right){C}_{5}.$

$\begin{array}{l}|\left(\left({C}_{0}\cup {C}_{1}\cup {C}_{4}\right)+x\right)\cap \left({C}_{0}\cup {C}_{1}\cup {C}_{4}\right)|\\ =|\left({C}_{0}+x\right)\cap {C}_{0}|+|\left({C}_{0}+x\right)\cap {C}_{1}|+|\left({C}_{0}+x\right)\cap {C}_{4}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+|\left({C}_{1}+x\right)\cap {C}_{0}|+|\left({C}_{1}+x\right)\cap {C}_{1}|+|\left({C}_{1}+x\right)\cap {C}_{4}|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+|\left({C}_{4}+x\right)\cap {C}_{0}|+|\left({C}_{4}+x\right)\cap {C}_{1}|+|\left({C}_{4}+x\right)\cap {C}_{4}|\\ =\left(i,0\right)+\left(i,1\right)+\left(i,4\right)+\left(i-1,-1\right)+\left(i-1,0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\left(i-1,3\right)+\left(i-4,-4\right)+\left(i-4,-3\right)+\left(i-4,0\right).\end{array}$

$\begin{array}{c}\text{Δ}\left({C}_{0}\cup {C}_{1}\cup {C}_{4}\right)={\cup }_{i=0}^{5}\left(\left(i,0\right)+\left(i,1\right)+\left(i,4\right)+\left(i-1,-1\right)+\left(i-1,0\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(i-1,3\right)+\left(i-4,-4\right)+\left(i-4,-3\right)+\left(i-4,0\right)\right){C}_{i}.\end{array}$

$\text{Δ}\left({C}_{0}\cup \left\{0\right\},{C}_{0}\cup {C}_{1}\cup {C}_{4}\right)={\cup }_{i=0}^{5}{\text{Δ}}_{i}{C}_{i}$，则

$\begin{array}{c}{\text{Δ}}_{i}=2\left(i,0\right)+\left(i,1\right)+\left(i,4\right)+\left(i-1,-1\right)+\left(i-1,0\right)+\left(i-1,3\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(i-4,-4\right)+\left(i-4,-3\right)+\left(i-4,0\right)+{\delta }_{i},\end{array}$

${\text{Δ}}_{0}={\text{Δ}}_{3}=-\frac{1}{18}-\frac{2a}{9}+\frac{5p}{18},$

${\text{Δ}}_{1}={\text{Δ}}_{4}=-\frac{25}{18}+\frac{a}{9}+\frac{5p}{18},$

${\text{Δ}}_{2}={\text{Δ}}_{5}=-\frac{7}{18}+\frac{a}{9}+\frac{5p}{18}.$

$\left(p,\left\{\frac{p+5}{6},\frac{p-1}{2}\right\},\frac{5p-17}{18},\frac{2\left(p-1\right)}{3}\right)$ -ADF.

${\text{Δ}}_{0}={\text{Δ}}_{3}=\frac{8}{9}-\frac{a}{9}+\frac{2p}{9},$

${\text{Δ}}_{1}={\text{Δ}}_{2}={\text{Δ}}_{4}={\text{Δ}}_{5}=-\frac{7}{9}+\frac{a}{18}+\frac{2p}{9}.$

$\left({C}_{0}\cup {C}_{3}\cup \left\{0\right\},{C}_{1}\cup {C}_{5}\right)$ 为几乎差族，当且仅当 $|{\text{Δ}}_{0}-{\text{Δ}}_{1}|=1$，即

$|\frac{8}{9}-\frac{a}{9}+\frac{2p}{9}+\frac{7}{9}-\frac{a}{18}-\frac{2p}{9}|=1.$

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