2024/02/12 by Somi Gupta, Francesco Pavese, Gupta, Somi +1 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2402.07882
An affine spread is a set of subspaces of AG(n, q) of the same dimension that partitions the points of AG(n, q). Equivalently, an \em affine spread is a set of projective subspaces of PG(n, q) of the same dimension which partitions the points of PG(n, q) ∖ H∞; here H∞ denotes the hyperplane at infinity of the projective closure of AG(n, q). Let Q be a non degenerate quadric of H_∞ and let Π be a generator of Q, where Π is a t-dimensional projective subspace. An affine spread P consisting of (t+1)-dimensional projective subspaces of PG(n, q) is called hyperbolic, parabolic or elliptic (according as Q is hyperbolic, parabolic or elliptic) if the following hold: each member of P meets H_∞ in a distinct generator of Q disjoint from Π; elements of P have at most one point in common; if S, T ∈ P, |S ∩ T| = 1, then ⟨ S, T ⟩ ∩ Q is a hyperbolic quadric of Q. In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of PG(n, q) is equivalent to a spread of Q+(n+1, q), Q(n+1, q) or Q-(n+1, q), respectively.