Theory Lunch

Optimal rate list decoding via derivative codes

Carol Wang

Sep 21, 2011

ABSTRACT: The classical family of $[n,k{]}_q$ Reed-Solomon codes over a field $F_q$ consist of the evaluations of polynomials $f\in F_q\left[X\right]$ of degree 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 $R=k/\left(nm\right)$ 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.