The merit factor of binary arrays derived from the quadratic character

TL;DR

This paper computes the asymptotic merit factor of two families of binary arrays derived from quadratic characters, generalizing Legendre sequences to two dimensions and analyzing their rotational invariance.

Contribution

It provides the first non-trivial theoretical results on the asymptotic merit factor of two-dimensional binary arrays derived from quadratic characters.

Findings

Maximum asymptotic merit factor is 36/13 for both families.

Arrays are generalizations of Legendre sequences to two dimensions.

Asymptotic merit factors differ between the two families.

Abstract

We calculate the asymptotic merit factor, under all cyclic rotations of rows and columns, of two families of binary two-dimensional arrays derived from the quadratic character. The arrays in these families have size p x q, where p and q are not necessarily distinct odd primes, and can be considered as two-dimensional generalisations of a Legendre sequence. The asymptotic values of the merit factor of the two families are generally different, although the maximum asymptotic merit factor, taken over all cyclic rotations of rows and columns, equals 36/13 for both families. These are the first non-trivial theoretical results for the asymptotic merit factor of families of truly two-dimensional binary arrays.