  In a recent breakthrough [BGM23, GZ23, AGL23], it was shown that randomly
punctured Reed-Solomon codes are list decodable with optimal list size with
high probability, i.e., they attain the Singleton bound for list decoding
[ST20, Rot22, GST22]. We extend this result to the family of polynomial ideal
codes, a large class of error-correcting codes which includes several
well-studied families of codes such as Reed-Solomon, folded Reed-Solomon, and
multiplicity codes. More specifically, similarly to the Reed-Solomon setting,
we show that randomly punctured polynomial ideal codes over an exponentially
large alphabet exactly achieve the Singleton bound for list-decoding; while
such codes over a polynomially large alphabet approximately achieve it.
  Combining our results with the efficient list-decoding algorithm for a large
subclass of polynomial ideal codes of [BHKS21], implies as a corollary that a
large subclass of polynomial ideal codes (over random evaluation points) is
efficiently list decodable with optimal list size. To the best of our
knowledge, this gives the first family of codes that can be efficiently list
decoded with optimal list size (for all list sizes), as well as the first
family of linear codes of rate $R$ that can be efficiently list decoded up to a
radius of $1 -R-\epsilon$ with list size that is polynomial (and even linear)
in $1/\epsilon$. Our result applies to natural families of codes with algebraic
structure such as folded Reed-Solomon or multiplicity codes (over random
evaluation points).
  Our proof follows the general framework of [BGM23, GZ23, AGL23], but several
new ingredients are needed. The main two new ingredients are a polynomial-ideal
GM-MDS theorem (extending the algebraic GM-MDS theorem of [YH19, Lov21]), as
well as a duality theorem for polynomial ideal codes, both of which may be of
independent interest.
