Related papers: State spaces of convolutional codes, codings and e…
Quantum convolutional coding is a technique for encoding a stream of quantum information before transmitting it over a noisy quantum channel. Two important goals in the design of quantum convolutional encoders are to minimize the memory…
This paper investigates the existence of minimal $p$-encoders for convolutional codes over $\mathbb{Z}_{p^r}$, where $p$ is a prime. This addresses a conjecture from \cite{k}, which posits that every such code admits a minimal $p$-encoder,…
Higher-dimensional analogs of the predictable degree property and column reducedness are defined, and it is proved that the two properties are equivalent. It is shown that every multidimensional convolutional code has, what is called, a…
We will study convolutional codes with the help of state space realizations. It will be shown that two such minimal realizations belong to the same code if and only if they are equivalent under the full state feedback group. This result…
We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum…
Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized are being polynomial left prime…
This paper explores the design of convolutional codes for varying constraint lengths, focusing on their role in error correction in digital communication systems. Convolutional codes are essential in achieving reliable data transmission…
Discrete dynamical systems defined by the iteration of a polynomial map of the unit simplex to itself appear in the context of population genetic systems evolving under mutation, recombination and weak selection. Although exceptional…
This paper investigates the concept of self-dual convolutional code. We derive the basic properties of this interesting class of codes and we show how some of the techniques to construct self-dual linear block codes generalize to self-dual…
Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace…
A lower bound on minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of generalized cosets of the codes generated by bottom rows of the polarizing matrix. Moreover, a construction of…
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…
The article provides a survey on convolutional codes stressing the connections to module theory and systems theory. Constructions of codes with maximal possible distance and distance profile are provided. The article will appear as book…
Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous…
One of the most important open questions in the theory of quantum convolutional coding is to determine a minimal-memory, non-catastrophic, polynomial-depth convolutional encoder for an arbitrary quantum convolutional code. Here, we present…
We present a general theory of entanglement-assisted quantum convolutional coding. The codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to quantum communication, and…
Function encoders are a recent technique that learn neural network basis functions to form compact, adaptive representations of Hilbert spaces of functions. We show that function encoders provide a principled connection to feature learning…
This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…
Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is…
A spread code is a set of vector spaces of a fixed dimension over a finite field Fq with certain properties used for random network coding. It can be constructed in different ways which lead to different decoding algorithms. In this work we…