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

On Algebraic Decoding of qq-ary Reed-Muller and Product-Reed-Solomon Codes

We consider a list decoding algorithm recently proposed by Pellikaan-Wu \cite{PW2005} for $q$-ary Reed-Muller codes $\mathcal{RM}_q(\ell, m, n)$ of length $n \leq q^m$ when $\ell \leq q$. A simple and easily accessible correctness proof is given which shows that this algorithm achieves a relative error-correction radius of $\tau \leq (1 - \sqrt{{\ell q^{m-1}}/{n}})$. This is an improvement over the proof using one-point Algebraic-Geometric codes given in \cite{PW2005}. The described algorithm can be adapted to decode Product-Reed-Solomon codes. We then propose a new low complexity recursive algebraic decoding algorithm for Reed-Muller and Product-Reed-Solomon codes. Our algorithm achieves a relative error correction radius of $\tau \leq \prod_{i=1}^m (1 - \sqrt{k_i/q})$. This technique is then proved to outperform the Pellikaan-Wu method in both complexity and error correction radius over a wide range of code rates.Comment: 5 pages, 5 figures, to be presented at 2007 IEEE International Symposium on Information Theory, Nice, France (ISIT 2007

Iterative Soft Input Soft Output Decoding of Reed-Solomon Codes by Adapting the Parity Check Matrix

An iterative algorithm is presented for soft-input-soft-output (SISO) decoding of Reed-Solomon (RS) codes. The proposed iterative algorithm uses the sum product algorithm (SPA) in conjunction with a binary parity check matrix of the RS code. The novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the SPA is applied at each iteration. The proposed algorithm can be geometrically interpreted as a two-stage gradient descent with an adaptive potential function. This adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and, therefore, significantly improves the performance. Simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding (HDD) and compare favorably with other popular soft decision decoding methods.Comment: 10 pages, 10 figures, final version accepted by IEEE Trans. on Information Theor

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

A Combinatorial Bound on the List Size

In this paper we study the scenario in which a server sends dynamic data over a single broadcast channel to a number of passive clients. We consider the data to consist of discrete packets, where each update is sent in a separate packet. On demand, each client listens to the channel in order to obtain the most recent data packet. Such scenarios arise in many practical applications such as the distribution of weather and traffic updates to wireless mobile devices and broadcasting stock price information over the Internet. To satisfy a request, a client must listen to at least one packet from beginning to end. We thus consider the design of a broadcast schedule which minimizes the time that passes between a clients request and the time that it hears a new data packet, i.e., the waiting time of the client. Previous studies have addressed this objective, assuming that client requests are distributed uniformly over time. However, in the general setting, the clients behavior is difficult to predict and might not be known to the server. In this work we consider the design of universal schedules that guarantee a short waiting time for any possible client behavior. We define the model of dynamic broadcasting in the universal setting, and prove various results regarding the waiting time achievable in this framework