The classical family of [n, k] q Reed-Solomon codes over a field F q consist of the evaluations of polynomials f ∈ 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) 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.
In REF , similar results were also shown for derivative codes, where the encoding of a polynomial f consists of the evaluations of f and its first m − 1 derivates at distinct field elements.
Optimal rate list decoding via derivative codes
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Mathematics Computer Science