Correcting a Fraction of Errors in Nonbinary Expander Codes with Linear Programming

A linear-programming decoder for \emph{nonbinary} expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties. It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where \delta_A and \delta_B are the relative minimum distances of the constituent codes).Comment: Part of this work was presented at the IEEE International Symposium on Information Theory 2009, Seoul, Kore

Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel

Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R is below 1-2p, for stabilizer codes: we also derive an improved upper bound of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant - low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings.Comment: 32 page

High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity $\exp(\tilde{O}(\sqrt{\log n}))$. Previously such codes were known to exist only with $\Omega(n^{\beta})$ query complexity (for constant $\beta > 0$), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby~\cite{AL96_codes}. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant relative distance}. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity $\exp(\tilde{O}(\sqrt{\log n}))$, which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with $\exp(\tilde{O}(\sqrt{\log n}))$ query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any $o(n)$ query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters

Homological Error Correction: Classical and Quantum Codes

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.Comment: Revtex4 fil

Improved Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: (i) the codes lie close to the Singleton bound, (ii) they can be encoded in time complexity that is linear in their code length, and (iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace MDS outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size.Comment: Part of this work was presented at the 2004 IEEE Int'l Symposium on Information Theory (ISIT'2004), Chicago, Illinois (June 2004). This work was submitted to IEEE Transactions on Information Theory on January 21, 2005. To appear in IEEE Transactions on Information Theory, August 2006. 12 page

Structured Error Recovery for Codeword-Stabilized Quantum Codes

Codeword stabilized (CWS) codes are, in general, non-additive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical non-linear codes. For an n-qubit quantum code correcting errors on up to t qubits, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. In this paper, we suggest an error grouping technique that allows to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about 3^t times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.Comment: 13 pages, 9 eps figure

Sparse Graph Codes for Quantum Error-Correction

We present sparse graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse graph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.Comment: Version 7.3e: 42 pages. Extended version, Feb 2004. A shortened version was resubmitted to IEEE Transactions on Information Theory Jan 20, 200