Abstract In this paper, we give a geometric construction of the three strong non-lifted $$(3\mod 5)$$ ( 3 mod 5 ) -arcs in $${{\,\textrm{PG}\,}}(3,5)$$ PG ( 3 , 5 ) of respective sizes 128, 143, and 168, and construct an infinite family of non-lifted, strong $$(t\mod q)$$ ( t mod q ) -arcs in $${{\,\textrm{PG}\,}}(r,q)$$ PG ( r , q ) with $$t=(q+1)/2$$ t = ( q + 1 ) / 2 for all $$r\ge 3$$ r ≥ 3 and all odd prime powers q.

Abstract An affine spread is a set of subspaces of $$\textrm{AG}(n, q)$$ AG ( n , q ) of the same dimension that partitions the points of $$\textrm{AG}(n, q)$$ AG ( n , q ) . Equivalently, an affine spread is a set of projective subspaces of $$\textrm{PG}(n, q)$$ PG ( n , q ) of the same dimension which partitions the points of $$\textrm{PG}(n, q) \setminus H_{\infty }$$ PG ( n , q ) \ H ∞ ; here $$H_{\infty }$$ H ∞ denotes the hyperplane at infinity of the projective closure of $$\textrm{AG}(n, q)$$ AG ( n , q ) . Let $$\mathcal {Q}$$ Q be a non-degenerate quadric of $$H_\infty $$ H ∞ and let $$\Pi $$ Π be a generator of $$\mathcal {Q}$$ Q , where $$\Pi $$ Π is a t-dimensional projective subspace. An affine spread $$\mathcal {P}$$ P consisting of $$(t+1)$$ ( t + 1 ) -dimensional projective subspaces of $$\textrm{PG}(n, q)$$ PG ( n , q ) is called hyperbolic, parabolic or elliptic (according as $$\mathcal {Q}$$ Q is hyperbolic, parabolic or elliptic) if the following hold: Each member of $$\mathcal {P}$$ P meets $$H_\infty $$ H ∞ in a distinct generator of $$\mathcal {Q}$$ Q disjoint from $$\Pi $$ Π ; Elements of $$\mathcal {P}$$ P have at most one point in common; If $$S, T \in \mathcal {P}$$ S , T ∈ P , $$|S \cap T| = 1$$ | S ∩ T | = 1 , then $$\langle S, T \rangle \cap \mathcal {Q}$$ ⟨