Constructions of irredundant orthogonal arrays

Guangzhou Chen; Xiaotong Zhang

doi:10.3934/amc.2021051

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2021, Volume 0. Doi: 10.3934/amc.2021051

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Constructions of irredundant orthogonal arrays

Guangzhou Chen^1, , and
Xiaotong Zhang^2,

1.
Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
2.
School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

^* Corresponding author: chenguangzhou0808@163.com
^* Corresponding author: chenguangzhou0808@163.com

Received: March 31, 2021

Revised: July 31, 2021

Early access: November 2021

The first author is supported by National Natural Science Foundation of China (Grant Nos. 11871417 and 11501181)

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Abstract

An $ N \times k $ array $ A $ with entries from $ v $-set $ \mathcal{V} $ is said to be an orthogonal array with $ v $ levels, strength $ t $ and index $ \lambda $, denoted by OA$ (N,k,v,t) $, if every $ N\times t $ sub-array of $ A $ contains each $ t $-tuple based on $ \mathcal{V} $ exactly $ \lambda $ times as a row. An OA$ (N,k,v,t) $ is called irredundant, denoted by IrOA$ (N,k,v,t) $, if in any $ N\times (k-t ) $ sub-array, all of its rows are different. Goyeneche and $ \dot{Z} $yczkowski firstly introduced the definition of an IrOA and showed that an IrOA$ (N,k,v,t) $ corresponds to a $ t $-uniform state of $ k $ subsystems with local dimension $ v $ (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 $ t $-uniform states arise from these irredundant orthogonal arrays.

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Mathematics Subject Classification: Primary: 05B15, 62K15; Secondary: 81P99.

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Table 1. Correspondence between parameters of IrOAs and quantum states

Parameters	Irredundant Orthogonal array	Multipartite quantum state $\|\psi\rangle$
N	Runs	Number of linear terms in the state
k	Factors	Number of qudits
v	Levels	Dimension of the subsystem(v=2 for qubits)
t	Strength	Class of entanglement(t-uniform)

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Table 2. Existence of t-uniform states of k qudits (s ≥ 2)

$(\mathbb{C}^s)^{\otimes k}$	Existence	Nonexistence	Unknown
1-uniform	$s\geq 2, k\geq 2$	no	no
2-uniform	$s\geq 2, k\geq 4$ except (s,k) = (2,4)	s = 2 and k = 4	no
3-uniform	$s\geq 2, k\geq 6$ except s ≡ 2 (mod 4) and k = 7	s = 2 and k = 7	s ≥ 6, s ≡ 2 (mod 4) and k = 7

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