On vote-taking and complete decoding of certain error-correcting codes

It is shown how complete decoding of maximum distance separable codes can be accomplished by a vote-taking algorithm or an equivalent distance correlation method. It is also indicated where this method of decoding might find application

Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding

We study coding schemes for error correction in interactive communications. Such interactive coding schemes simulate any $n$-round interactive protocol using $N$ rounds over an adversarial channel that corrupts up to $\rho N$ transmissions. Important performance measures for a coding scheme are its maximum tolerable error rate $\rho$, communication complexity $N$, and computational complexity. We give the first coding scheme for the standard setting which performs optimally in all three measures: Our randomized non-adaptive coding scheme has a near-linear computational complexity and tolerates any error rate $\delta < 1/4$ with a linear $N = \Theta(n)$ communication complexity. This improves over prior results which each performed well in two of these measures. We also give results for other settings of interest, namely, the first computationally and communication efficient schemes that tolerate $\rho < \frac{2}{7}$ adaptively, $\rho < \frac{1}{3}$ if only one party is required to decode, and $\rho < \frac{1}{2}$ if list decoding is allowed. These are the optimal tolerable error rates for the respective settings. These coding schemes also have near linear computational and communication complexity. These results are obtained via two techniques: We give a general black-box reduction which reduces unique decoding, in various settings, to list decoding. We also show how to boost the computational and communication efficiency of any list decoder to become near linear.Comment: preliminary versio

On Coding over Sliced Information

The interest in channel models in which the data is sent as an unordered set of binary strings has increased lately, due to emerging applications in DNA storage, among others. In this paper we analyze the minimal redundancy of binary codes for this channel under substitution errors, and provide several constructions, some of which are shown to be asymptotically optimal up to constants. The surprising result in this paper is that while the information vector is sliced into a set of unordered strings, the amount of redundant bits that are required to correct errors is order-wise equivalent to the amount required in the classical error correcting paradigm

Design of a fault tolerant airborne digital computer. Volume 1: Architecture

This volume is concerned with the architecture of a fault tolerant digital computer for an advanced commercial aircraft. All of the computations of the aircraft, including those presently carried out by analogue techniques, are to be carried out in this digital computer. Among the important qualities of the computer are the following: (1) The capacity is to be matched to the aircraft environment. (2) The reliability is to be selectively matched to the criticality and deadline requirements of each of the computations. (3) The system is to be readily expandable. contractible, and (4) The design is to appropriate to post 1975 technology. Three candidate architectures are discussed and assessed in terms of the above qualities. Of the three candidates, a newly conceived architecture, Software Implemented Fault Tolerance (SIFT), provides the best match to the above qualities. In addition SIFT is particularly simple and believable. The other candidates, Bus Checker System (BUCS), also newly conceived in this project, and the Hopkins multiprocessor are potentially more efficient than SIFT in the use of redundancy, but otherwise are not as attractive

How to correct small quantum errors

The theory of quantum error correction is a cornerstone of quantum information processing. It shows that quantum data can be protected against decoherence effects, which otherwise would render many of the new quantum applications practically impossible. In this paper we give a self contained introduction to this theory and to the closely related concept of quantum channel capacities. We show, in particular, that it is possible (using appropriate error correcting schemes) to send a non-vanishing amount of quantum data undisturbed (in a certain asymptotic sense) through a noisy quantum channel T, provided the errors produced by T are small enough.Comment: LaTeX2e, 23 pages, 6 figures (3 eps, 3 pstricks

Limitations of Passive Protection of Quantum Information

The ability to protect quantum information from the effect of noise is one of the major goals of quantum information processing. In this article, we study limitations on the asymptotic stability of quantum information stored in passive N-qubit systems. We consider the effect of small imperfections in the implementation of the protecting Hamiltonian in the form of perturbations or weak coupling to a ground state environment. We prove that, regardless of the protecting Hamiltonian, there exists a perturbed evolution that necessitates a final error correcting step when the state of the memory is read. Such an error correction step is shown to require a finite error threshold, the lack thereof being exemplified by the 3D compass model. We go on to present explicit weak Hamiltonian perturbations which destroy the logical information stored in the 2D toric code in a time O(log(N)).Comment: 17 pages and appendice