Further results on 2-uniform states arising from irredundant orthogonal arrays

Zang, Yajuan; Chen, Guangzhou; Chen, Kejun; Tian, Zihong

doi:10.3934/amc.2020109

Cited by 5 publications

(9 citation statements)

References 25 publications

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“…Goyeneche and Życzkowski [14] had established a link between the irredundant orthogonal arrays and t-uniform states since 2014, IrOA and t-uniform states has been paid great attention by many scholars, and then there are some results for uniform states [10,44,45] and irredundant orthogonal arrays [26,30,42,43].…”

Section: Discussionmentioning

confidence: 99%

“…For t = 3, s = 4, 5, although there exists a 3-uniform state of 7 qudits with local dimension s, it is clear that the existence of an IrOA(N, 7, s, 3) ⇐⇒ the existence of an IrOA(s 4 , 7, s, 3) ⇐⇒ the existence of an OA(s 4 , 7, s, 4), which is a very difficult problem. Recently, Zang et al [42] studied the existence of the IrOA with factor k = 5, 6 and almost solved the existence of these arrays from Lemma 1.1 and Lemma 1.2. Chen et al [4] showed that there exists an IrOA(v 3 , 12, v, 2) for any v ≥ 4 and v ≡ 2 (mod 4) and there exists an IrOA(v 3 , 3v, v, 2) for any prime or prime power v. Li et al [26] gave the existence of irredundant orthogonal arrays with parameters IrOA(q n , k, q, 2), where q is a prime power and q n−1 −1 q−1 + 3 ≤ k ≤ q n −1 q−1 + 3 for n ≥ 2.…”

Section: Discussionmentioning

confidence: 99%

“…Construction 2.1. [42] (Product Construction). If an IrOA(λ 1 v t 1 , k, v 1 , t) and an IrOA(λ 2 v t 2 , k, v 2 , t) both exist, then so does an IrOA(λ…”

Section: Preliminarymentioning

confidence: 99%

“…Let gcd(a, b) be the greatest common divisor of integers a and b. Recently, Zang et al [42] studied the existence of the IrOA with factor k = 5, 6 and obtained the following results.…”

Section: Introductionmentioning

confidence: 99%

“…For more results on irredundant orthogonal arrays, we refer the reader to [26,30,42,43,44] and references therein. In this article, we focus on the constructions of irredundant orthogonal arrays.…”

Section: Introductionmentioning

confidence: 99%

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Constructions of irredundant orthogonal arrays

Chen¹,

Zhang²

2023

AMC

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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} $\end{document}</tex-math></inline-formula> is said to be an orthogonal array with <inline-formula><tex-math id="M5">\begin{document}$ 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) $\end{document}</tex-math></inline-formula> is called irredundant, denoted by IrOA<inline-formula><tex-math id="M15">\begin{document}$ (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.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…Construction 2.1. [42] (Product Construction). If an IrOA(λ 1 v t 1 , k, v 1 , t) and an IrOA(λ 2 v t 2 , k, v 2 , t) both exist, then so does an IrOA(λ…”

Section: Preliminarymentioning

confidence: 99%

“…Let gcd(a, b) be the greatest common divisor of integers a and b. Recently, Zang et al [42] studied the existence of the IrOA with factor k = 5, 6 and obtained the following results.…”

Section: Introductionmentioning

confidence: 99%