  Lifted Reed-Solomon codes, introduced by Guo, Kopparty and Sudan in 2013, are
known as one of the few families of high-rate locally correctable codes. They
are built through the evaluation over the affine space of multivariate
polynomials whose restriction along any affine line can be interpolated as a
low degree univariate polynomial.
  In this work, we give a formal definition of their analogues over projective
spaces, and we study some of their parameters and features. Local correcting
algorithms are first derived from the very nature of these codes, generalizing
the well-known local correcting algorithms for Reed-Muller codes. We also prove
that the lifting of both Reed-Solomon and projective Reed-Solomon codes are
deeply linked through shortening and puncturing operations. It leads to
recursive formulae on their dimension and their monomial bases. We finally
emphasize the practicality of lifted projective Reed-Solomon codes by computing
their information sets and by providing an implementation of the codes and
their local correcting algorithms.
