  Lifted Reed-Solomon and multiplicity codes are classes of codes, constructed
from specific sets of $m$-variate polynomials. These codes allow for the design
of high-rate codes that can recover every codeword or information symbol from
many disjoint sets. Recently, the underlying approaches have been combined for
the bi-variate case to construct lifted multiplicity codes, a generalization of
lifted codes that can offer further rate improvements. We continue the study of
these codes by first establishing new lower bounds on the rate of lifted
Reed-Solomon codes for any number of variables $m$, which improve upon the
known bounds for any $m\ge 4$. Next, we use these results to provide lower
bounds on the rate and distance of lifted multiplicity codes obtained from
polynomials in an arbitrary number of variables, which improve upon the known
results for any $m\ge 3$.
  Specifically, we investigate a subcode of a lifted multiplicity code formed
by the linear span of $m$-variate monomials whose restriction to an arbitrary
line in $\mathbb{F}_q^m$ is equivalent to a low-degree univariate polynomial.
We find the tight asymptotic behavior of the fraction of such monomials when
the number of variables $m$ is fixed and the alphabet size $q=2^\ell$ is large.
Using these results, we give a new explicit construction of batch codes
utilizing lifted Reed-Solomon codes. For some parameter regimes, these codes
have a better trade-off between parameters than previously known batch codes.
Further, we show that lifted multiplicity codes have a better trade-off between
redundancy and the number of disjoint recovering sets for every codeword or
information symbol than previously known constructions, thereby providing the
best known PIR codes for some parameter regimes. Additionally, we present a new
local self-correction algorithm for lifted multiplicity codes.
