Reweighted LP Decoding for LDPC Codes

We introduce a novel algorithm for decoding binary linear codes by linear programming (LP). We build on the LP decoding algorithm of Feldman and introduce a postprocessing step that solves a second linear program that reweights the objective function based on the outcome of the original LP decoder output. Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. We also show significant empirical performance gains for the reweighted LP decoding algorithm with very small additional computational complexity

A Lower Bound for Relaxed Locally Decodable Codes

A locally decodable code (LDC) C \colon \bitset^k \to \bitset^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to distributed storage. However, despite nearly two decades of extensive study, the best known constructions of $O(1)$-query LDCs have super-polynomial blocklength. The notion of relaxed LDCs is a natural relaxation of LDCs, which aims to bypass the foregoing barrier by requiring local decoding of nearly all individual message bits, yet allowing decoding failure (but not error) on the rest. State of the art constructions of $O(1)$-query relaxed LDCs achieve blocklength $n = O\left(k^{1+ \gamma}\right)$ for an arbitrarily small constant $\gamma$. We prove a lower bound which shows that $O(1)$-query relaxed LDCs cannot achieve blocklength $n = k^{1+ o(1)}$. This resolves an open problem raised by Goldreich in 2004

Interior Point Decoding for Linear Vector Channels

In this paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, called interior point decoding, is designed for linear vector channels. The linear vector channels include many practically important channels such as inter symbol interference channels and partial response channels. It is shown that the maximum likelihood decoding (MLD) rule for a linear vector channel can be relaxed to a convex optimization problem, which is called a relaxed MLD problem. The proposed decoding algorithm is based on a numerical optimization technique so called interior point method with barrier function. Approximate variations of the gradient descent and the Newton methods are used to solve the convex optimization problem. In a decoding process of the proposed algorithm, a search point always lies in the fundamental polytope defined based on a low-density parity-check matrix. Compared with a convectional joint message passing decoder, the proposed decoding algorithm achieves better BER performance with less complexity in the case of partial response channels in many cases.Comment: 18 pages, 17 figures, The paper has been submitted to IEEE Transaction on Information Theor

A Noise Resilient Transformation for Streaming Algorithms

In a streaming algorithm, Bob receives an input $x \in \{0,1\}^n$ via a stream and must compute a function $f$ in low space. However, this function may be fragile to errors in the input stream. In this work, we investigate what happens when the input stream is corrupted. Our main result is an encoding of the incoming stream so that Bob is still able to compute any such function $f$ in low space when a constant fraction of the stream is corrupted. More precisely, we describe an encoding function $\mathsf{enc}(x)$ of length $\text{poly}(n)$ so that for any streaming algorithm $A$ that on input $x$ computes $f(x)$ in space $s$, there is an explicit streaming algorithm $B$ that computes $f(x)$ in space $s \cdot \text{polylog}(n)$ as long as there were not more than $\frac14 - \varepsilon$ fraction of (adversarial) errors in the input stream $\mathsf{enc}(x)$