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We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled. [ABSTRACT FROM AUTHOR]

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double-exponentially growing arithmetic function T such that no 3-phase Barker array of size [ABSTRACT FROM AUTHOR]

Let $A_n=(a_0,a_1,\ldots ,a_{n-1})$ be drawn uniformly at random from ${\{ }-1,+1{\} }^n$ and define\[M(A_n)=\max _{01}{\mathrm {}}.\]It is proved that $M(A_n)/\sqrt {n\log n}$ converges in probability to $\sqrt {2}$. This settles a problem first studied by Moon and Moser in the 1960s and proves in the affirmative a recent conjecture due to Alon, Litsyn, and Shpunt. It is also shown that the expectation of $M(A_n)/\sqrt {n\log n}$ tends to $\sqrt {2}$. [ABSTRACT FROM AUTHOR]