On line covers of finite projective and polar spaces

@article{Cossidente2019OnLC,
  title={On line covers of finite projective and polar spaces},
  author={Antonio Cossidente and Francesco Pavese},
  journal={Designs, Codes and Cryptography},
  year={2019},
  pages={1-18}
}
An m-cover of lines of a finite projective space $$\mathrm{PG}(r,q)$$PG(r,q) (of a finite polar space $${\mathcal {P}}$$P) is a set of lines $${\mathcal {L}}$$L of $$\mathrm{PG}(r,q)$$PG(r,q) (of $${\mathcal {P}}$$P) such that every point of $$\mathrm{PG}(r,q)$$PG(r,q) (of $${\mathcal {P}}$$P) contains m lines of $${\mathcal {L}}$$L, for some m. Embed $$\mathrm{PG}(r,q)$$PG(r,q) in $$\mathrm{PG}(r,q^2)$$PG(r,q2). Let $${{\bar{{\mathcal {L}}}}}$$L¯ denote the set of points of $$\mathrm{PG}(r,q^2… 
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