List decoding of a class of affine variety codes

Consider a polynomial $F$ in $m$ variables and a finite point ensemble $S=S_1 \times ... \times S_m$. When given the leading monomial of $F$ with respect to a lexicographic ordering we derive improved information on the possible number of zeros of $F$ of multiplicity at least $r$ from $S$. We then use this information to design a list decoding algorithm for a large class of affine variety codes.Comment: 11 pages, 5 table

Explicit measurements with almost optimal thresholds for compressed sensing

We consider the deterministic construction of a measurement matrix and a recovery method for signals that are block sparse. A signal that has dimension N = nd, which consists of n blocks of size d, is called (s, d)-block sparse if only s blocks out of n are nonzero. We construct an explicit linear mapping Φ that maps the (s, d)-block sparse signal to a measurement vector of dimension M, where s•d <N(1-(1-M/N)^(d/(d+1))-o(1). We show that if the (s, d)- block sparse signal is chosen uniformly at random then the signal can almost surely be reconstructed from the measurement vector in O(N^3) computations

List decoding of repeated codes

Assuming that we have a soft-decision list decoding algorithm of a linear code, a new hard-decision list decoding algorithm of its repeated code is proposed in this article. Although repeated codes are not used for encoding data, due to their parameters, we show that they have a good performance with this algorithm. We compare, by computer simulations, our algorithm for the repeated code of a Reed-Solomon code against a decoding algorithm of a Reed-Solomon code. Finally, we estimate the decoding capability of the algorithm for Reed-Solomon codes and show that performance is somewhat better than our estimates

Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200