A Recursive Construction of Perfect $K^n\times K^n$ Arrays over $K$ Roots of Unity

Abstract

We give a uniform, explicit and elementary recursive construction of two-dimensional arrays with perfect (ideal) periodic autocorrelation over a \emph{fixed} alphabet of $K$ roots of unity and \emph{unbounded} size: for every integer $K\ge2$ and every $n\ge1$ a perfect $K^n\times K^n$ array whose entries are $K$-th roots of unity, given by a nested sum of floor terms. The construction works with \emph{twisted} (quasi-perfect) arrays indexed by a type $v\in\Z_K^2$, of which ordinary perfect arrays are the type $v=0$ members. For $K=2$ the types are Jedwab's generalised perfect binary arrays and the recursion specialises to the doubling constructions of Jedwab--Mitchell and Wild; for prime $K$ the arrays are the relative difference sets whose existence in abelian $p$-groups was settled by Ma and Schmidt; and the bare existence of perfect $K^n\times K^n$ arrays over $K$ roots of unity follows, for every $K$, from recent constructions of group-invariant Butson Hadamard matrices. The contribution of this note is therefore not existence but the calculus: types with arbitrary entries in $\Z_K^2$; three spectral lemmas (a twisted periodic extension of a type-$v$ array has a flat spectrum supported on one coset of $K\Z^2$; interleaving arrays whose spectra are flat on disjoint cosets gives a perfect array; and a shear of the index plane with a floor-function carry permutes types through the action of $SL_2(\Z_K)$ on $\Z_K^2$, whose orbits are the classes $\gcd(v_1,v_2,K)=g$); a one-directional lift that produces every type at side $KM$ from every type at side $M$ for arbitrary $K$; and the resulting closed index function, which is the recursive form of the floor-function arrays of \cite{Blake2017}.