Related papers: Principles and Parameters: a coding theory perspec…
One of the oldest problems in coding theory is to match the Gilbert-Varshamov bound with explicit binary codes. Over larger-yet still constant-sized-fields, algebraic-geometry codes are known to beat the GV bound. In this work, we leverage…
We describe a new parameterized family of symmetric error-correcting codes with low-density parity-check matrices (LDPC). Our codes can be described in two seemingly different ways. First, in relation to Reed-Muller codes: our codes are…
In this thesis we present several results in coding theory, concerning error-correcting codes and the Shannon capacity. 1. We give a general symmetry reduction of matrices occuring in semidefinite programs in coding theory. 2. We apply the…
Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable…
Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is…
In this work we investigate codes in $\mathbb{Z}_{2^m}^n$ that can correct errors that occur in just one coordinate of the codeword, with a magnitude of up to a given parameter $t$. We will show upper bounds on these cross codes, derive…
Neural Language Models of Code, or Neural Code Models (NCMs), are rapidly progressing from research prototypes to commercial developer tools. As such, understanding the capabilities and limitations of such models is becoming critical.…
We study parameterized Constraint Satisfaction Problem for infinite constraint languages. The parameters that we study are weight of the satisfying assignment, number of constraints, maximum number of occurrences of a variable in the…
We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the…
Minimum distance is an important parameter of a linear error correcting code. For improved performance of binary Low Density Parity Check (LDPC) codes, we need to have the minimum distance grow fast with n, the codelength. However, the best…
We describe the theory of quantum convolutional error correcting codes. These codes are aimed at protecting a flow of quantum information over long distance communication. They are largely inspired by their classical analogs which are used…
The PCP Theorem is one of the most stunning results in computational complexity theory, a culmination of a series of results regarding proof checking it exposes some deep structure of computational problems. As a surprising side-effect, it…
Consider an ensemble of regular generalized LDPC (GLDPC) codes and assume that the same component code is associated with each parity check node. To decode a GLDPC code from the ensemble, we use the bit flipping bounded distance decoding…
We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory…
An error correcting code using a tree-like multilayer perceptron is proposed. An original message $\mbi{s}^0$ is encoded into a codeword $\boldmath{y}_0$ using a tree-like committee machine (committee tree) or a tree-like parity machine…
We extend the correspondence between two-stage coding procedures in data compression and penalized likelihood procedures in statistical estimation. Traditionally, this had required restriction to countable parameter spaces. We show how to…
Traditional Linear Genetic Programming (LGP) algorithms are based only on the selection mechanism to guide the search. Genetic operators combine or mutate random portions of the individuals, without knowing if the result will lead to a…
Encodings or the proof of their absence are the main way to compare process calculi. To analyse the quality of encodings and to rule out trivial or meaningless encodings, they are augmented with quality criteria. There exists a bunch of…
We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of…
Delsarte's method and its extensions allow to consider the upper bound problem for codes in 2-point-homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that…