Abstract:
Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric $n$-vertices colourable graph and a group of operations (colouring rules) on the graph: find the minimum sequence of operations that maps between two given graph colourings. We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code. Computing the distance of a quantum code is a highly non-trivial problem and our method may be of use in the construction of better codes.

Abstract:
We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes.

Abstract:
Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where C_1 and C_2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C_1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.

Abstract:
The connection between index coding and matroid theory have been well studied in the recent past. El Rouayheb et al. established a connection between multilinear representation of matroids and wireless index coding. Muralidharan and Rajan showed that a vector linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Recently index coding with erroneous transmission was considered by Dau et al.. Error correcting index codes in which all receivers are able to correct a fixed number of errors was studied. In this paper we consider a more general scenario in which each receiver is able to correct a desired number of errors, calling such index codes differential error correcting index codes. A link between differential error correcting index codes and certain matroids is established. We define matroidal differential error correcting index codes and we show that a scalar linear differential error correcting index code exists if and only if it is matroidal differential error correcting index code associated with a representable matroid.

Abstract:
A general theory of quantum error avoiding codes is established, and new light is shed on the relation between quantum error avoiding and correcting codes. Quantum error avoiding codes are found to be a special type of highly degenerate quantum error correcting codes. A combination of the ideas of correcting and avoiding quantum errors may yield better codes. We give a practical example.

Abstract:
This paper computationally obtains optimal bounded-weight, binary, error-correcting codes for a variety of distance bounds and dimensions. We compare the sizes of our codes to the sizes of optimal constant-weight, binary, error-correcting codes, and evaluate the differences.

Abstract:
Error correcting codes are defined and important parameters for a code are explained. Parameters of new codes constructed on algebraic surfaces are studied. In particular, codes resulting from blowing up points in $\proj^2$ are briefly studied, then codes resulting from ruled surfaces are covered. Codes resulting from ruled surfaces over curves of genus 0 are completely analyzed, and some codes are discovered that are better than direct product Reed Solomon codes of similar length. Ruled surfaces over genus 1 curves are also studied, but not all classes are completely analyzed. However, in this case a family of codes are found that are comparable in performance to the direct product code of a Reed Solomon code and a Goppa code. Some further work is done on surfaces from higher genus curves, but there remains much work to be done in this direction to understand fully the resulting codes. Codes resulting from blowing points on surfaces are also studied, obtaining necessary parameters for constructing infinite families of such codes. Also included is a paper giving explicit formulas for curves with more \field{q}-rational points than were previously known for certain combinations of field size and genus. Some upper bounds are now known to be optimal from these examples.

Abstract:
The possibility of using the two-fold topological degeneracy of spin-1/2 chiral spin liquid states on the torus to construct quantum error correcting codes is investigated. It is shown that codes constructed using these states on finite periodic lattices do not meet the necessary and sufficient conditions for correcting even a single qubit error with perfect fidelity. However, for large enough lattice sizes these conditions are approximately satisfied, and the resulting codes may therefore be viewed as approximate quantum error correcting codes.

Abstract:
We present a universal framework for quantum error-correcting codes, i.e., the one that applies for the most general quantum error-correcting codes. This framework is established on the group algebra, an algebraic notation for the nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.

Abstract:
We consider network coding for networks experiencing worst-case bit-flip errors, and argue that this is a reasonable model for highly dynamic wireless network transmissions. We demonstrate that in this setup prior network error-correcting schemes can be arbitrarily far from achieving the optimal network throughput. We propose a new metric for errors under this model. Using this metric, we prove a new Hamming-type upper bound on the network capacity. We also show a commensurate lower bound based on GV-type codes that can be used for error-correction. The codes used to attain the lower bound are non-coherent (do not require prior knowledge of network topology). The end-to-end nature of our design enables our codes to be overlaid on classical distributed random linear network codes. Further, we free internal nodes from having to implement potentially computationally intensive link-by-link error-correction.