28,764 research outputs found
Semidefinite programming, binary codes and a graph coloring problem
Experts in information theory have long been interested in the maximal size, A(n, d), of a binary error-correcting code of length n and minimum distance d, The problem of determining A(n, d) involves both the construction of good codes and the search for good upper bounds. For quite some time now, Delsarte\u27s linear programming approach has been the dominant approach to obtaining the strongest general purpose upper bounds on the efficiency of error-correcting codes. From 1973 forward, the linear programming bound found many applications, but there were few significant theoretical advances until Schrijver proposed a new code upper bound via semidefinite programming in 2003. Using the Terwilliger algebra, a recently introduced extension of the Bose-Mesner algebra, Schrijver formulated a new SDP strengthening of the LP approach. In this project we look at the dual solutions of the semidefinite programming bound for binary error-correcting codes. We explore the combinatorial meaning of these variables for small n and d, such as n = 4 and d = 2. To obtain information like this, we wrote a computer program with both Matlab and CVX modules to get solution of our primal SDP formulation. Our program efficiently generates the primal solutions with corresponding constraints for any n and d. We also wrote a program in C++ to parse the output of the primal SDP problem, and another Matlab script to generate the dual SDP problem, which could be used in assigning combinatorial meaning to the values given in the dual optimal solution. Our code not only computes both the primal and dual optimal variable values, but allows the researcher to display them in meaningful ways and to explore their relationship and dependence on arameters. These values are expected to be useful for later study of the combinatorial meaning of such solutions
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 -round interactive protocol
using rounds over an adversarial channel that corrupts up to
transmissions. Important performance measures for a coding scheme are its
maximum tolerable error rate , communication complexity , 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 with a linear 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 adaptively, if only one party is required to
decode, and 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 the design of an ECOC-compliant genetic algorithm
Genetic Algorithms (GA) have been previously applied to Error-Correcting Output Codes (ECOC) in state-of-the-art works in order to find a suitable coding matrix. Nevertheless, none of the presented techniques directly take into account the properties of the ECOC matrix. As a result the considered search space is unnecessarily large. In this paper, a novel Genetic strategy to optimize the ECOC coding step is presented. This novel strategy redefines the usual crossover and mutation operators in order to take into account the theoretical properties of the ECOC framework. Thus, it reduces the search space and lets the algorithm to converge faster. In addition, a novel operator that is able to enlarge the code in a smart way is introduced. The novel methodology is tested on several UCI datasets and four challenging computer vision problems. Furthermore, the analysis of the results done in terms of performance, code length and number of Support Vectors shows that the optimization process is able to find very efficient codes, in terms of the trade-off between classification performance and the number of classifiers. Finally, classification performance per dichotomizer results shows that the novel proposal is able to obtain similar or even better results while defining a more compact number of dichotomies and SVs compared to state-of-the-art approaches
Universal secure rank-metric coding schemes with optimal communication overheads
We study the problem of reducing the communication overhead from a noisy
wire-tap channel or storage system where data is encoded as a matrix, when more
columns (or their linear combinations) are available. We present its
applications to reducing communication overheads in universal secure linear
network coding and secure distributed storage with crisscross errors and
erasures and in the presence of a wire-tapper. Our main contribution is a
method to transform coding schemes based on linear rank-metric codes, with
certain properties, to schemes with lower communication overheads. By applying
this method to pairs of Gabidulin codes, we obtain coding schemes with optimal
information rate with respect to their security and rank error correction
capability, and with universally optimal communication overheads, when , being and the number of columns and number of rows,
respectively. Moreover, our method can be applied to other families of maximum
rank distance codes when . The downside of the method is generally
expanding the packet length, but some practical instances come at no cost.Comment: 21 pages, LaTeX; parts of this paper have been accepted for
presentation at the IEEE International Symposium on Information Theory,
Aachen, Germany, June 201
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