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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 subset Y of the general linear group $\text{GL}(n,q)$ is called t -intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$ , or equivalently x and y agree pointwise on a t -dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$. We show that, if n is sufficiently large compared to t , the size of every such t -intersecting set is at most that of the stabiliser of a basis of a t -dimensional subspace of $\mathbb{F}_q^n$. In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t -dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$. These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q -analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$. [ABSTRACT FROM AUTHOR]

The Pursley-Sarwate criterion of a pair of finite complex-valued sequences measures the collective smallness of the aperiodic autocorrelations and the aperiodic cross-correlations of the two sequences. It is known that this quantity is always at least 1 with equality if and only if the sequence pair is a Golay pair. We exhibit pairs of complex-valued sequences whose entries have unit magnitude for which the Pursley-Sarwate criterion tends to 1 as the sequence length tends to infinity. Our constructions use different carefully chosen Chu sequences. [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]

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from F q n to F q to the set of affine functions from F q n to F q. We prove the conjecture for each q such that the characteristic of F q lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when n tends to infinity. This also determines the asymptotic behaviour of the covering radius of the [ q n , n + 1 ] Reed-Muller code over F q and so answers a question raised by Leducq in 2013. Our results extend the case q = 2 , which was recently proved by the author and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory. [ABSTRACT FROM AUTHOR]