Families of multi-level Legendre-like arrays

Abstract

Families of new, multi-level integer 2D arrays are introduced here as an extension of the well-known binary Legendre sequences that are derived from quadratic residues. We present a construction, based on Fourier and Finite Radon Transforms, for families of periodic perfect arrays, each of size \(p\times p\) for many prime values p. Previously delta functions were used as the discrete projections which, when back-projected, build 2D perfect arrays. Here we employ perfect sequences as the discrete projected views. The base family size is \(p+1\). All members of these multi-level array families have perfect autocorrelation and constant, minimal cross-correlation. Proofs are given for four useful and general properties of these new arrays. 1) They are comprised of odd integers, with values between at most \(-p\) and \(+p\), with a zero value at just one location. 2) They have the property of ‘conjugate’ spatial symmetry, where the value at location (i, j) is always the negative of the value at location \((p-i, p-j)\). 3) Any change in the value assigned to the array’s origin leaves all of its off-peak autocorrelation values unchanged. 4) A family of \(p+1\), \(p\times p\) arrays can be compressed to size \((p+1)^2\) and each family member can be exactly and rapidly unpacked in a single \(p\times p\) decompression pass.