doi: 10.3934/amc.2021051
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Constructions of irredundant orthogonal arrays

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

Received  April 2021 Revised  August 2021 Early access November 2021

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

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.

Citation: Guangzhou Chen, Xiaotong Zhang. Constructions of irredundant orthogonal arrays. Advances in Mathematics of Communications, doi: 10.3934/amc.2021051
References:
[1]

C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phy. Rev. Lett., 68 (1992), 3121-3124.  doi: 10.1103/PhysRevLett.68.3121.

[2]

C. H. BennettG. BrassardC. CrépeauR. JozsaA. Peres and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett., 70 (1993), 1895-1899.  doi: 10.1103/PhysRevLett.70.1895.

[3]

D. BouwmeesterJ. W. PanK. MattleM. EiblH. Weinfurter and A. Zeilinger, Experimental quantum teleportation, Nature, 390 (1997), 575-579. 

[4]

G. ChenX. Zhang and Y. Guo, New results for 2-uniform states based on irredundant orthogonal arrays, Quantum Inf. Process., 20 (2021), 43.  doi: 10.1007/s11128-020-02978-x.

[5] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996.  doi: 10.1201/9781420049954.
[6]

A. Dey and R. Mukerjee, Fractional Factorial Plans, John Wiley & Sons, Inc, New York, NY, 1999. doi: 10.1002/9780470316986.

[7]

A. K. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett., 67 (1991), 661-663.  doi: 10.1103/PhysRevLett.67.661.

[8]

P. Facchi, Multipartite entanglement in qubit systems, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 20 (2009), 25-67.  doi: 10.4171/RLM/532.

[9]

P. FacchiG. FlorioG. Parisi and S. Pascazio, Maximally multipartite entangled states, Phys. Rev. A., 77 (2008), 060304.  doi: 10.1103/PhysRevA.77.060304.

[10]

K. Q. FengL. F. JinC. P. Xing and C. Yuan, Multipartite entangled states, symmetric matrices and error-correcting codes, IEEE Trans. Inform. Theory, 63 (2017), 5618-5627.  doi: 10.1109/tit.2017.2700866.

[11]

G. Ge, On (g, 4;1)-difference matrices, Discrete Math., 301 (2005), 164-174.  doi: 10.1016/j.disc.2005.07.004.

[12]

D. GoyenecheJ. Bielawski and K. Życzkowski, Multipartite entanglement in heterogeneous systems, Phys. Rev. A., 94 (2016), 012346.  doi: 10.1103/PhysRevA.94.012346.

[13]

D. GoyenecheZ. RaissiS. D. Martino and K. Życzkowski, Entanglement and quantum combinatorial designs, Phys. Rev. A., 97 (2018), 062326.  doi: 10.1103/PhysRevA.97.062326.

[14]

D. Goyeneche and K. Życzkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A., 90 (2014), 022316.  doi: 10.1103/PhysRevA.90.022316.

[15]

M. Grassl and M. Rötteler, Quantum MDS codes over small fields,, IEEE International Symposium on Information Theory (ISIT), (2015), 1104–1108. doi: 10.1109/ISIT.2015.7282626.

[16]

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, Springer Series in Statistics. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1478-6.

[17]

W. Helwig, Absolutely maximally entangled qudit graph states, preprint, arXiv: 1306.2879v1.

[18]

W. Helwig and W. Cui, Absolutely maximally entangled states: existence and applications, preprint, arXiv: 1306.2536.

[19]

W. HelwigW. CuiJ. I. LatorreA. Riera and H. K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A., 86 (2012), 052335.  doi: 10.1103/PhysRevA.86.052335.

[20]

A. Higuchi and A. Sudbery, How entangled can two couples get?, Phys. Lett. A, 273 (2000), 213-217.  doi: 10.1016/S0375-9601(00)00480-1.

[21]

R. HorodeckiP. HorodeckiM. Horodecki and K. Horodecki, Quantum entanglement, Rev. Modern Phys, 81 (2009), 865-942.  doi: 10.1103/RevModPhys.81.865.

[22]

P. Horodecki, Ł. Rudnicki and K. Życzkowski, Five open problems in quantum information, prepint, arXiv: 2002.03233v1.

[23]

F. HuberO. Gühne and J. Siewert, Absolutely maximally entangled states of seven qubits do not exist, Phys. Rev. Lett., 118 (2017), 200502.  doi: 10.1103/PhysRevLett.118.200502.

[24]

L. Ji and J. Yin, Constructions of new orthogonal arrays and covering arrays of strength three, J. Combi. Theory, Ser. A, 117 (2010), 236-247.  doi: 10.1016/j.jcta.2009.06.002.

[25]

R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2011-2032.  doi: 10.1098/rspa.2002.1097.

[26]

M. S. Li and Y. L. Wang, $k$-uniform quantum states arising from orthogonal arrays, Phy. Rev. A., 99 (2019), 042332. 

[27]

H. K. LoM. Curty and B. Qi, Measurement-device-independent quantum key distribution, Phys. Rev. Lett., 108 (2012), 130503.  doi: 10.1103/PhysRevLett.108.130503.

[28]

S. Q. PangX. ZhangJ. Du and T. Wang, Multipartite entanglement states of higher uniformity, J. Phys. A: Math. Theor., 54 (2021), 015305.  doi: 10.1088/1751-8121/abc9a4.

[29]

S. Q. PangX. ZhangS. M. Fei and Z. J. Zheng, Quantum $k$-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays, Quantum Inf. Process., 20 (2021), 156.  doi: 10.1007/s11128-021-03040-0.

[30]

S. Q. PangX. ZhangX. Lin and Q. J. Zhang, Two and three-uniform states from irredundant orthogonal arrays, npj Quantum Inf., 5 (2019), 1-10.  doi: 10.1038/s41534-019-0165-8.

[31]

F. PastawskiB. YoshidaD. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys., 6 (2015), 149.  doi: 10.1007/JHEP06(2015)149.

[32]

E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45 (1999), 1827-1832.  doi: 10.1109/18.782103.

[33]

Z. RaissiA. TeixidoC. Gogolin and A. Acín, Constructions of $k$-uniform and absolutely maximally entangled states beyond maximum distance codes, Physical Review Research, 2 (2020), 033411.  doi: 10.1103/PhysRevResearch.2.033411.

[34]

C. R. Rao, Factorial experiments derivable from combinational arrangements of arrays, Suppl. J. Roy. Statist. Soc., 9 (1947), 128-139.  doi: 10.2307/2983576.

[35]

S. A. Rather, A. Burchardt, W. Bruzda, G. R.-Mieldzioć, A. Lakshminarayan and K. Życzkowski, Thirty-six entangled officers of Euler, preprint, arXiv: 2104.05122v1.

[36]

M. RiebeH. HaffnerF. C. Roos and et al, Deterministic quantum teleportation with atoms, Nature, 429 (2004), 734-737.  doi: 10.1038/nature02570.

[37]

C. F. RoosM. RiebeH. Haffner and et al, Control and measurement of three-qubit entangled states, Science, 304 (2004), 1478-1480.  doi: 10.1126/science.1097522.

[38]

A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, Phys. Rev. A., 69 (2004), 052330.  doi: 10.1103/PhysRevA.69.052330.

[39]

F. Shi, Y. Shen, L. Chen and X. Zhang, Constructions of $k$-uniform states from mixed orthogonal arrays, preprint, arXiv: 2006.04086v1.

[40]

D. R. Stinson, Ideal ramp schemes and related combinatorial objects, Discrete Math., 341 (2018), 299-307.  doi: 10.1016/j.disc.2017.08.041.

[41]

C. SuenA. Das and A. Dey, On the construction of asymmetric orthogonal arrays, Statistica Sinica, 11 (2001), 241-260. 

[42]

Y. J. Zang, G. Z. Chen, K. J. Chen and Z. H. Tian, Further results on $2$-uniform states arising from irredundant orthogonal arrays, Advances in Mathematics of Communications, 2020. doi: 10.3934/amc.2020109.

[43]

Y. J. ZangH. J. Zuo and Z. H. Tian, 3-uniform states and orthogonal arrays of strength 3, Int. J. Quantum Inf., 17 (2019), 1950003.  doi: 10.1142/S0219749919500035.

[44]

X. W. ZhaI. Ahmed and Y. P. Zhang, 3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28. 

[45]

X. W. Zha, C. Z. Yuan and Y. P. Zhang, Generalized criterion for a maximally multi-qubit entangled states, Laser Phys. Lett., 10 (2013), 045201. doi: 10.1088/1612-2011/10/4/045201.

[46]

Z. ZhaoY. A. ChenA. N. ZhangT. YangH. J. Briegel and J. Pan, Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature, 430 (2004), 54-58.  doi: 10.1038/nature02643.

show all references

References:
[1]

C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phy. Rev. Lett., 68 (1992), 3121-3124.  doi: 10.1103/PhysRevLett.68.3121.

[2]

C. H. BennettG. BrassardC. CrépeauR. JozsaA. Peres and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett., 70 (1993), 1895-1899.  doi: 10.1103/PhysRevLett.70.1895.

[3]

D. BouwmeesterJ. W. PanK. MattleM. EiblH. Weinfurter and A. Zeilinger, Experimental quantum teleportation, Nature, 390 (1997), 575-579. 

[4]

G. ChenX. Zhang and Y. Guo, New results for 2-uniform states based on irredundant orthogonal arrays, Quantum Inf. Process., 20 (2021), 43.  doi: 10.1007/s11128-020-02978-x.

[5] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996.  doi: 10.1201/9781420049954.
[6]

A. Dey and R. Mukerjee, Fractional Factorial Plans, John Wiley & Sons, Inc, New York, NY, 1999. doi: 10.1002/9780470316986.

[7]

A. K. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett., 67 (1991), 661-663.  doi: 10.1103/PhysRevLett.67.661.

[8]

P. Facchi, Multipartite entanglement in qubit systems, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 20 (2009), 25-67.  doi: 10.4171/RLM/532.

[9]

P. FacchiG. FlorioG. Parisi and S. Pascazio, Maximally multipartite entangled states, Phys. Rev. A., 77 (2008), 060304.  doi: 10.1103/PhysRevA.77.060304.

[10]

K. Q. FengL. F. JinC. P. Xing and C. Yuan, Multipartite entangled states, symmetric matrices and error-correcting codes, IEEE Trans. Inform. Theory, 63 (2017), 5618-5627.  doi: 10.1109/tit.2017.2700866.

[11]

G. Ge, On (g, 4;1)-difference matrices, Discrete Math., 301 (2005), 164-174.  doi: 10.1016/j.disc.2005.07.004.

[12]

D. GoyenecheJ. Bielawski and K. Życzkowski, Multipartite entanglement in heterogeneous systems, Phys. Rev. A., 94 (2016), 012346.  doi: 10.1103/PhysRevA.94.012346.

[13]

D. GoyenecheZ. RaissiS. D. Martino and K. Życzkowski, Entanglement and quantum combinatorial designs, Phys. Rev. A., 97 (2018), 062326.  doi: 10.1103/PhysRevA.97.062326.

[14]

D. Goyeneche and K. Życzkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A., 90 (2014), 022316.  doi: 10.1103/PhysRevA.90.022316.

[15]

M. Grassl and M. Rötteler, Quantum MDS codes over small fields,, IEEE International Symposium on Information Theory (ISIT), (2015), 1104–1108. doi: 10.1109/ISIT.2015.7282626.

[16]

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, Springer Series in Statistics. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1478-6.

[17]

W. Helwig, Absolutely maximally entangled qudit graph states, preprint, arXiv: 1306.2879v1.

[18]

W. Helwig and W. Cui, Absolutely maximally entangled states: existence and applications, preprint, arXiv: 1306.2536.

[19]

W. HelwigW. CuiJ. I. LatorreA. Riera and H. K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A., 86 (2012), 052335.  doi: 10.1103/PhysRevA.86.052335.

[20]

A. Higuchi and A. Sudbery, How entangled can two couples get?, Phys. Lett. A, 273 (2000), 213-217.  doi: 10.1016/S0375-9601(00)00480-1.

[21]

R. HorodeckiP. HorodeckiM. Horodecki and K. Horodecki, Quantum entanglement, Rev. Modern Phys, 81 (2009), 865-942.  doi: 10.1103/RevModPhys.81.865.

[22]

P. Horodecki, Ł. Rudnicki and K. Życzkowski, Five open problems in quantum information, prepint, arXiv: 2002.03233v1.

[23]

F. HuberO. Gühne and J. Siewert, Absolutely maximally entangled states of seven qubits do not exist, Phys. Rev. Lett., 118 (2017), 200502.  doi: 10.1103/PhysRevLett.118.200502.

[24]

L. Ji and J. Yin, Constructions of new orthogonal arrays and covering arrays of strength three, J. Combi. Theory, Ser. A, 117 (2010), 236-247.  doi: 10.1016/j.jcta.2009.06.002.

[25]

R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2011-2032.  doi: 10.1098/rspa.2002.1097.

[26]

M. S. Li and Y. L. Wang, $k$-uniform quantum states arising from orthogonal arrays, Phy. Rev. A., 99 (2019), 042332. 

[27]

H. K. LoM. Curty and B. Qi, Measurement-device-independent quantum key distribution, Phys. Rev. Lett., 108 (2012), 130503.  doi: 10.1103/PhysRevLett.108.130503.

[28]

S. Q. PangX. ZhangJ. Du and T. Wang, Multipartite entanglement states of higher uniformity, J. Phys. A: Math. Theor., 54 (2021), 015305.  doi: 10.1088/1751-8121/abc9a4.

[29]

S. Q. PangX. ZhangS. M. Fei and Z. J. Zheng, Quantum $k$-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays, Quantum Inf. Process., 20 (2021), 156.  doi: 10.1007/s11128-021-03040-0.

[30]

S. Q. PangX. ZhangX. Lin and Q. J. Zhang, Two and three-uniform states from irredundant orthogonal arrays, npj Quantum Inf., 5 (2019), 1-10.  doi: 10.1038/s41534-019-0165-8.

[31]

F. PastawskiB. YoshidaD. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys., 6 (2015), 149.  doi: 10.1007/JHEP06(2015)149.

[32]

E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45 (1999), 1827-1832.  doi: 10.1109/18.782103.

[33]

Z. RaissiA. TeixidoC. Gogolin and A. Acín, Constructions of $k$-uniform and absolutely maximally entangled states beyond maximum distance codes, Physical Review Research, 2 (2020), 033411.  doi: 10.1103/PhysRevResearch.2.033411.

[34]

C. R. Rao, Factorial experiments derivable from combinational arrangements of arrays, Suppl. J. Roy. Statist. Soc., 9 (1947), 128-139.  doi: 10.2307/2983576.

[35]

S. A. Rather, A. Burchardt, W. Bruzda, G. R.-Mieldzioć, A. Lakshminarayan and K. Życzkowski, Thirty-six entangled officers of Euler, preprint, arXiv: 2104.05122v1.

[36]

M. RiebeH. HaffnerF. C. Roos and et al, Deterministic quantum teleportation with atoms, Nature, 429 (2004), 734-737.  doi: 10.1038/nature02570.

[37]

C. F. RoosM. RiebeH. Haffner and et al, Control and measurement of three-qubit entangled states, Science, 304 (2004), 1478-1480.  doi: 10.1126/science.1097522.

[38]

A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, Phys. Rev. A., 69 (2004), 052330.  doi: 10.1103/PhysRevA.69.052330.

[39]

F. Shi, Y. Shen, L. Chen and X. Zhang, Constructions of $k$-uniform states from mixed orthogonal arrays, preprint, arXiv: 2006.04086v1.

[40]

D. R. Stinson, Ideal ramp schemes and related combinatorial objects, Discrete Math., 341 (2018), 299-307.  doi: 10.1016/j.disc.2017.08.041.

[41]

C. SuenA. Das and A. Dey, On the construction of asymmetric orthogonal arrays, Statistica Sinica, 11 (2001), 241-260. 

[42]

Y. J. Zang, G. Z. Chen, K. J. Chen and Z. H. Tian, Further results on $2$-uniform states arising from irredundant orthogonal arrays, Advances in Mathematics of Communications, 2020. doi: 10.3934/amc.2020109.

[43]

Y. J. ZangH. J. Zuo and Z. H. Tian, 3-uniform states and orthogonal arrays of strength 3, Int. J. Quantum Inf., 17 (2019), 1950003.  doi: 10.1142/S0219749919500035.

[44]

X. W. ZhaI. Ahmed and Y. P. Zhang, 3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28. 

[45]

X. W. Zha, C. Z. Yuan and Y. P. Zhang, Generalized criterion for a maximally multi-qubit entangled states, Laser Phys. Lett., 10 (2013), 045201. doi: 10.1088/1612-2011/10/4/045201.

[46]

Z. ZhaoY. A. ChenA. N. ZhangT. YangH. J. Briegel and J. Pan, Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature, 430 (2004), 54-58.  doi: 10.1038/nature02643.

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)
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)
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
$(\mathbb{C}^s)^{\otimes k}$ 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
[1]

Yajuan Zang, Guangzhou Chen, Kejun Chen, Zihong Tian. Further results on 2-uniform states arising from irredundant orthogonal arrays. Advances in Mathematics of Communications, 2022, 16 (2) : 231-247. doi: 10.3934/amc.2020109

[2]

K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026

[3]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2022, 16 (3) : 597-619. doi: 10.3934/amc.2020125

[4]

Bingsheng Shen, Yang Yang, Ruibin Ren. Three constructions of Golay complementary array sets. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022019

[5]

Leonid Berlyand, Giuseppe Cardone, Yuliya Gorb, Gregory Panasenko. Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Networks and Heterogeneous Media, 2006, 1 (3) : 353-377. doi: 10.3934/nhm.2006.1.353

[6]

Masayuki Sato, Naoki Fujita, A. J. Sievers. Logic operations demonstrated with localized vibrations in a micromechanical cantilever array. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1287-1298. doi: 10.3934/dcdss.2011.4.1287

[7]

Samuel T. Blake, Andrew Z. Tirkel. A multi-dimensional block-circulant perfect array construction. Advances in Mathematics of Communications, 2017, 11 (2) : 367-371. doi: 10.3934/amc.2017030

[8]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[9]

Hui Ma, Dongxu Qi, Ruixia Song, Tianjun Wang. The complete orthogonal V-system and its applications. Communications on Pure and Applied Analysis, 2007, 6 (3) : 853-871. doi: 10.3934/cpaa.2007.6.853

[10]

Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260

[11]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial and Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[12]

Cuiling Fan, Koji Momihara. Unified combinatorial constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2014, 8 (1) : 53-66. doi: 10.3934/amc.2014.8.53

[13]

T. L. Alderson, K. E. Mellinger. Geometric constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2008, 2 (4) : 451-467. doi: 10.3934/amc.2008.2.451

[14]

Dario Corona. A multiplicity result for orthogonal geodesic chords in Finsler disks. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5329-5357. doi: 10.3934/dcds.2021079

[15]

Liqun Qi, Chen Ling, Jinjie Liu, Chen Ouyang. An orthogonal equivalence theorem for third order tensors. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021154

[16]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161

[17]

Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036

[18]

Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499

[19]

Liren Lin, Hongwei Liu, Bocong Chen. Existence conditions for self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (1) : 1-7. doi: 10.3934/amc.2015.9.1

[20]

Bertold Bongardt. Geometric characterization of the workspace of non-orthogonal rotation axes. Journal of Geometric Mechanics, 2014, 6 (2) : 141-166. doi: 10.3934/jgm.2014.6.141

2020 Impact Factor: 0.935

Metrics

  • PDF downloads (325)
  • HTML views (216)
  • Cited by (0)

Other articles
by authors

[Back to Top]