Related papers: Endomorphisms of Linear Block Codes
Recent years have seen rapid development in the subject of quantum coding theory, with breakthroughs on many exciting classes of codes, including quantum LDPC codes, quantum locally testable codes, and quantum codes with interesting…
The paper is organized as a self-contained literate Prolog program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The complete Prolog code is available at…
We introduce a general framework for end-to-end optimization of the rate--distortion performance of nonlinear transform codes assuming scalar quantization. The framework can be used to optimize any differentiable pair of analysis and…
Quantum low-density parity-check codes are promising candidates for low-overhead fault-tolerant quantum computing, but degeneracy is known to impair the convergence of belief-propagation (BP) decoding of these codes. In this work, we show…
We analyze the list-decodability, and related notions, of random linear codes. This has been studied extensively before: there are many different parameter regimes and many different variants. Previous works have used complementary styles…
Molecular "fingerprints" encoding structural information are the workhorse of cheminformatics and machine learning in drug discovery applications. However, fingerprint representations necessarily emphasize particular aspects of the…
Large Language Models trained on code corpora (code-LLMs) have demonstrated impressive performance in various coding assistance tasks. However, despite their increased size and training dataset, code-LLMs still have limitations such as…
Block-encoding operators are one of the essential components in quantum algorithms based on Quantum Signal Processing. Their gate complexity largely determines the overall gate complexity of the full algorithm. Using variational methods, we…
Deep learning on graphs has become a popular research topic with many applications. However, past work has concentrated on learning graph embedding tasks, which is in contrast with advances in generative models for images and text. Is it…
We study the automorphisms of binary stabilizer codes and states. We prove that they almost always form a solvable group, and thereby shed new light on the fact that there is no universal set of transversal gates. We also determine the…
We describe and present a new construction method for codes using encodings from group rings. They consist primarily of two types: zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; for example cyclic…
In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincar\'e theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms…
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…
In this paper, we focus on the problem of identifying semantic factors of variation in large image datasets. By training a convolutional Autoencoder on the image data, we create encodings, which describe each datapoint at a higher level of…
Self-attention mechanisms have revolutionised deep learning architectures, yet their core mathematical structures remain incompletely understood. In this work, we develop a category-theoretic framework focusing on the linear components of…
Convolutional codes are error-correcting linear codes that utilize shift registers to encode. These codes have an arbitrary block size and they can incorporate both past and current information bits. DNA codes represent DNA sequences and…
For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the M\"obius transformations that commute with $f$, and these form a finite subgroup of…
In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that generally consist of duplicate entries. We first introduce a class of binary matrices called…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
Using the EXIT chart approach, a necessary and sufficient condition is developed for the local stability of iterative decoding of multi-edge type (MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In such code…