Linear-algebraic list decoding of folded Reed-Solomon codes

Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal trade-off between rate and error-correction capability: specifically, for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)} time algorithm to list decode appropriate folded RS codes of rate $R$ from a fraction 1-R-\eps of errors. The algorithm is based on multivariate polynomial interpolation and root-finding over extension fields. It was noted by Vadhan that interpolating a linear polynomial suffices if one settles for a smaller decoding radius (but still enough for a statement of the above form). Here we give a simple linear-algebra based analysis of this variant that eliminates the need for the computationally expensive root-finding step over extension fields (and indeed any mention of extension fields). The entire list decoding algorithm is linear-algebraic, solving one linear system for the interpolation step, and another linear system to find a small subspace of candidate solutions. Except for the step of pruning this subspace, the algorithm can be implemented to run in {\em quadratic} time. The theoretical drawback of folded RS codes are that both the decoding complexity and proven worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above idea with a pseudorandom subset of all polynomials as messages, we get a Monte Carlo construction achieving a list size bound of O(1/\eps^2) which is quite close to the existential O(1/\eps) bound (however, the decoding complexity remains n^{\Omega(1/\eps)}). Our work highlights that constructing an explicit {\em subspace-evasive} subset that has small intersection with low-dimensional subspaces could lead to explicit codes with better list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on Computational Complexity (CCC), 201

Optimal rate list decoding via derivative codes

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

Decoding of Interleaved Reed-Solomon Codes Using Improved Power Decoding

We propose a new partial decoding algorithm for $m$-interleaved Reed--Solomon (IRS) codes that can decode, with high probability, a random error of relative weight $1-R^{\frac{m}{m+1}}$ at all code rates $R$, in time polynomial in the code length $n$. For $m>2$, this is an asymptotic improvement over the previous state-of-the-art for all rates, and the first improvement for $R>1/3$ in the last $20$ years. The method combines collaborative decoding of IRS codes with power decoding up to the Johnson radius.Comment: 5 pages, accepted at IEEE International Symposium on Information Theory 201