Open AccessFamilies of multi‐dimensional arrays with optimal correlations between all members
Author(s) -
Tirkel A.,
Cavy B.,
Svalbe I.
Publication year - 2015
Publication title -
electronics letters
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.375
H-Index - 146
ISSN - 1350-911X
DOI - 10.1049/el.2015.1046
Subject(s) - binary golay code , mathematics , hadamard transform , autocorrelation , complementary sequences , aperiodic graph , conjecture , combinatorics , finite field , hadamard matrix , orthogonal array , pseudorandom number generator , prime (order theory) , discrete mathematics , algorithm , mathematical analysis , statistics , taguchi methods
Families of sequences with low off‐peak autocorrelation and low cross‐correlation are highly valued in spread‐spectrum communication. Digital watermarking has an equal need for diverse families of orthogonal multi‐dimensional ( n D) arrays, where each array has optimal correlation properties. In this reported work, a 1D discrete projection method is used to construct new families of n D orthogonal arrays of size p n , with p a 4 k − 1 prime. Finite field algebra and Hadamard matrices are applied to analyse these arrays. The periodic autocorrelation of each array is ‘perfect’ ( p 2 − 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.