Families of multi-level Legendre-like arrays

• Published in: Annals of mathematics and artificial intelligence Vol. 92; no. 1; pp. 169 - 182
• Main Authors: Petersen, Timothy, Cavy, Benjamin, Paganin, David, Svalbe, Imants
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Arrays; Artificial Intelligence; Autocorrelation; Complex Systems; Computer Science; Cross correlation; Delta function; Mathematics; Radon transformation; Legendre sequences; Families of low cross-correlation and perfect autocorrelation arrays; Discrete Radon transform; Conjugate symmetry
• ISSN: 1012-2443; 1573-7470
• DOI: 10.1007/s10472-023-09903-9

Families of new, multi-level integer 2 D 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 × p for many prime values p . Previously delta functions were used as the discrete projections which, when back-projected, build 2 D 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 × p arrays can be compressed to size ( p + 1 ) 2 and each family member can be exactly and rapidly unpacked in a single p × p decompression pass.