Related papers: State spaces of convolutional codes, codings and e…

Fully convolutional neural networks can process input of arbitrary size by applying a combination of downsampling and pooling. However, we find that fully convolutional image classifiers are not agnostic to the input size but rather show…

In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result…

Block encodings are a fundamental primitive in quantum algorithms, but can often have large ancilla overhead. In this work, we introduce novel techniques for reducing this overhead in two distinct ways. In Part I, we prove the existence of…

In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then…

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. Subsystem codes generalize all major quantum error protection schemes, and therefore are especially versatile. This…

Subspace codes are the $q$-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In…

An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional…

Recently, we have developed an efficient generic partition refinement algorithm, which computes behavioural equivalence on a state-based system given as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend this to a…

In this study, we present an investigation into the anisotropy dynamics and intrinsic dimension of embeddings in transformer architectures, focusing on the dichotomy between encoders and decoders. Our findings reveal that the anisotropy…

Motivated by applications to noncoherent network coding, we study subspace codes defined by sets of linear cellular automata (CA). As a first remark, we show that a family of linear CA where the local rules have the same diameter -- and…

We study the compact embedding between smoothness Morrey spaces on bounded domains and characterise its entropy numbers. Here we discover a new phenomenon when the difference of smoothness parameters in the source and target spaces is…

The rapidly improving performance of modern hardware renders convolutional codes obsolete, and allows for the practical implementation of more sophisticated correction codes such as low density parity check (LDPC) and turbo codes (TC). Both…

Additive codes and some nonadditive codes use the single and multiple invariant subspaces of the stabilizer G, respectively, to construct quantum codes, so the selection of the invariant subspaces is a key problem. In this paper, I provide…

In this paper, we generalize some halfspace type theorems for self-shrinkers of codimension 1 to the case of arbitrary codimension.

Polarization-adjusted convolutional (PAC) codes are a new family of linear block codes that can perform close to the theoretical bounds in the short block-length regime. These codes combine polar coding and convolutional coding. In this…

For most algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise,…

We define a subset of the closure of the upper half plane associated with an endomorphism on a real inner product space, which is called the leaf. When the dimension of the space is at least 3, the leaf is a convex with respect to the…

In 1991, Roth introduced a natural generalization of rank metric codes, namely tensor codes. The latter are defined to be subspaces of $r$-tensors where the ambient space is endowed with the tensor rank as a distance function. In this work,…

Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely…

We propose an efficient Adaptive Random Convolutional Network Coding (ARCNC) algorithm to address the issue of field size in random network coding. ARCNC operates as a convolutional code, with the coefficients of local encoding kernels…