Folded Reed-Solomon (RS) codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error-correction capability: specifically, for any , Guruswami and Rudra presented an time algorithm to list decode appropriate folded RS codes of rate from a fraction 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 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 quadratic time. We also consider a closely related family of codes, called (order ) derivative codes and defined over fields of large characteristic, which consist of the evaluations of as well as its first formal derivatives at distinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achieve the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound are . 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 which is quite close to the existential bound (however, the decoding complexity remains ). Our work highlights that constructing an explicit subspace-evasive subset that has small intersection with low-dimensional subspaces-an interesting problem in pseudorandomness in its own right-could lead to explicit codes with better list-decoding guarantees.

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Abstract

Wireless Mesh Networks (WMNs) consist of radio nodes organized in a mesh topology for serving wireless mesh clients to communicate with one another or to connect to the Internet. Nodes in a mesh network can communicate with each other either directly or through one or more intermediate nodes, similar to social networks. WMNs share many common properties with social networks. We first identify the differences and similarities between social networks and WMNs and then use metrics that are typically used for social network analysis (SNA) to assess real WMNs. Analyzing real WMN data collected from the UCSB MeshNet and MIT Roofnet testbeds reveals that using SNA metrics are helpful in designing WMNs with better performance. We demonstrate the validity of our conclusions and this new approach by focusing on two sample applications of social networks: network reliability assessment and channel access scheduling.