  In this work, we show new and improved error-correcting properties of folded
Reed-Solomon codes and multiplicity codes. Both of these families of codes are
based on polynomials over finite fields, and both have been the sources of
recent advances in coding theory. Folded Reed-Solomon codes were the first
explicit constructions of codes known to achieve list-decoding capacity;
multivariate multiplicity codes were the first constructions of high-rate
locally correctable codes; and univariate multiplicity codes are also known to
achieve list-decoding capacity.
  However, previous analyses of the error-correction properties of these codes
did not yield optimal results. In particular, in the list-decoding setting, the
guarantees on the list-sizes were polynomial in the block length, rather than
constant; and for multivariate multiplicity codes, local list-decoding
algorithms could not go beyond the Johnson bound.
  In this paper, we show that Folded Reed-Solomon codes and multiplicity codes
are in fact better than previously known in the context of list-decoding and
local list-decoding. More precisely, we first show that Folded RS codes achieve
list-decoding capacity with constant list sizes, independent of the block
length; and that high-rate univariate multiplicity codes can also be
list-recovered with constant list sizes. Using our result on univariate
multiplicity codes, we show that multivariate multiplicity codes are high-rate,
locally list-recoverable codes. Finally, we show how to combine the above
results with standard tools to obtain capacity achieving locally list decodable
codes with query complexity significantly lower than was known before.
