Families of multi-dimensional arrays with optimal correlations between all members

Anatol (Andrew) Zygmunt Tirkel, Benjamin Cavy, Imants Dzintars Svalbe

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6 Citations (Scopus)


Families of sequences with low off-peak autocorrelation and low rosscorrelation
are highly valued in spread-spectrum communication. Digital watermarking has an equal need for diverse families of orthogonal multi-dimensional (nD) arrays, where each array has optimal correlation properties. In this reported work, a 1D discrete projection method is used to construct new families of nD orthogonal arrays of size pn, with p a 4k − 1 prime. Finite field algebra and Hadamard atrices are applied to analyse these arrays. The periodic autocorrelation of each array is ‘perfect’ (p2 − 1 peak value, with −1 off-peak for p × p arrays). The cross-correlation between any pair of the p members of each 2D family has the lowest possible values, 0 or ±p. The arrays can be synthesised for arbitrarily large p and outperform Kasami sequences. The alphabet values for these optimal arrays can be roots of unity or signed integers. The aperiodic autocorrelation of the p × p arrays can attain a merit factor of above 3 at shift (p/4, p/4), consistent with Golay’s conjecture in 1D.
Original languageEnglish
Pages (from-to)1167-1168
Number of pages2
JournalElectronics Letters
Issue number15
Publication statusPublished - 2015

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