Existence of Simple OA$_\lambda(3, 5, v)'$s

DOI：10.3770/j.issn:2095-2651.2015.03.004

 作者 单位 史册 上海立信会计学院数学与信息学院, 上海 201620 蒋领 苏州大学数学系, 江苏 苏州 215006 闻斌 常熟理工学院数学与统计学院, 江苏 常熟 215500

正交表是一个$\lambda v^t\times k$阵列, 每列上的元素取自一个$v$元符号集, 且满足每个$\lambda v^t\times t$子阵列包含每个$t$-元组恰好$\lambda$次, 记为OA$_\lambda(t,k,v)$, 其中$t$为强度, $k$为度, $v$为阶数, $\lambda$为指标. 一个正交表OA$_\lambda(t,k,v)$称为单纯的, 如果它不含有重复的行向量, 用SOA$_\lambda(t,k,v)$来表示. 本文证明了除去$v=6$和$\lambda \in \{3,7,11,13,15,17,19, 21,23,25,29,33\}$可能的例外, 单纯正交表SOA$_\lambda(3,5,v)$存在的必要条件也是充分的, 其中$\lambda \geq 2$.

An orthogonal array of strength $t$, degree $k$, order $v$ and index $\lambda$, denoted by OA$_\lambda(t,k,v)$, is a $\lambda v^t\times k$ array on a $v$ symbol set such that each $\lambda v^t\times t$ subarray contains each $t$-tuple exactly $\lambda$ times. An OA$_\lambda(t,k,v)$ is called simple and denoted by SOA$_\lambda(t,k,v)$ if it contains no repeated rows. In this paper, it is proved that the necessary conditions for the existence of an SOA$_\lambda(3,5,v)$ with $\lambda \geq 2$ are also sufficient with possible exceptions where $v=6$ and $\lambda \in \{3,7,11,13,15,17,19, 21,23,25,29,33\}$.