  In this work, we present an abstract framework for some algebraic
error-correcting codes with the aim of capturing codes that are list-decodable
to capacity, along with their decoding algorithm. In the polynomial ideal
framework, a code is specified by some ideals in a polynomial ring, messages
are polynomials and their encoding is the residue modulo the ideals. We present
an alternate way of viewing this class of codes in terms of linear operators,
and show that this alternate view makes their algorithmic list-decodability
amenable to analysis. Our framework leads to a new class of codes that we call
affine Folded Reed-Solomon codes (which are themselves a special case of the
broader class we explore). These codes are common generalizations of the
well-studied Folded Reed-Solomon codes and Multiplicity codes, while also
capturing the less-studied Additive Folded Reed-Solomon codes as well as a
large family of codes that were not previously known/studied.
  More significantly our framework also captures the algorithmic
list-decodability of the constituent codes. Specifically, we present a unified
view of the decoding algorithm for ideal theoretic codes and show that the
decodability reduces to the analysis of the distance of some related codes. We
show that good bounds on this distance lead to capacity-achieving performance
of the underlying code, providing a unifying explanation of known
capacity-achieving results. In the specific case of affine Folded Reed-Solomon
codes, our framework shows that they are list-decodable up to capacity (for
appropriate setting of the parameters), thereby unifying the previous results
for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.
