The classical family of [n,k]q Reed-Solomon codes over a field \F_q
consist of the evaluations of polynomials f \in \F_q[X] of degree <k at
n distinct field elements. In this work, we consider a closely related family
of codes, called (order m) {\em derivative codes} and defined over fields of
large characteristic, which consist of the evaluations of f as well as its
first m−1 formal derivatives at n distinct field elements. For large enough
m, we show that these codes can be list-decoded in polynomial time from an
error fraction approaching 1−R, where R=k/(nm) is the rate of the code.
This gives an alternate construction to folded Reed-Solomon codes for achieving
the optimal trade-off between rate and list error-correction radius. Our
decoding algorithm is linear-algebraic, and involves solving a linear system to
interpolate a multivariate polynomial, and then solving another structured
linear system to retrieve the list of candidate polynomials f. The algorithm
for derivative codes offers some advantages compared to a similar one for
folded Reed-Solomon codes in terms of efficient unique decoding in the presence
of side information.Comment: 11 page
