Constructions of irredundant orthogonal arrays

Abstract

<p style='text-indent:20px;'>An <inline-formula><tex-math id="M1">\begin{document}$ N \times k $\end{document}</tex-math></inline-formula> array <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math></inline-formula> with entries from <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>-set <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> is said to be an <i>orthogonal array</i> with <inline-formula><tex-math id="M5">\begin{document}$ v $\end{document}</tex-math></inline-formula> levels, strength <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula> and index <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, denoted by OA<inline-formula><tex-math id="M8">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if every <inline-formula><tex-math id="M9">\begin{document}$ N\times t $\end{document}</tex-math></inline-formula> sub-array of <inline-formula><tex-math id="M10">\begin{document}$ A $\end{document}</tex-math></inline-formula> contains each <inline-formula><tex-math id="M11">\begin{document}$ t $\end{document}</tex-math></inline-formula>-tuple based on <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{V} $\end{document}</tex-math></inline-formula> exactly <inline-formula><tex-math id="M13">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> times as a row. An OA<inline-formula><tex-math id="M14">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> is called <i>irredundant</i>, denoted by IrOA<inline-formula><tex-math id="M15">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula>, if in any <inline-formula><tex-math id="M16">\begin{document}$ N\times (k-t ) $\end{document}</tex-math></inline-formula> sub-array, all of its rows are different. Goyeneche and <inline-formula><tex-math id="M17">\begin{document}$ \dot{Z} $\end{document}</tex-math></inline-formula>yczkowski firstly introduced the definition of an IrOA and showed that an IrOA<inline-formula><tex-math id="M18">\begin{document}$ (N,k,v,t) $\end{document}</tex-math></inline-formula> corresponds to a <inline-formula><tex-math id="M19">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform state of <inline-formula><tex-math id="M20">\begin{document}$ k $\end{document}</tex-math></inline-formula> subsystems with local dimension <inline-formula><tex-math id="M21">\begin{document}$ v $\end{document}</tex-math></inline-formula> (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of <inline-formula><tex-math id="M22">\begin{document}$ t $\end{document}</tex-math></inline-formula>-uniform states arise from these irredundant orthogonal arrays.</p>