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.

Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a
Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can
grow with n. We consider such polynomials for odd square-free n, where \phi(n)
coefficients are determined by the Jacobi symbol, but the remaining
coefficients can be freely chosen. When n is prime, these polynomials have the
smallest known asymptotic value of the normalised L_4 norm ||f||_4/||f||_2
among all Littlewood polynomials, namely (7/6)^{1/4}.