Related papers: Codes and Noncommutative Stochastic Matrices
Matrix mutation of skew-symmetrizable matrices is foundational in cluster algebra theory. Effective mutation invariants are essential for determining whether two matrices lie in the same mutation class. Casals~\cite{Casals} introduced a…
The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matriz is superregular if all of its minors that are not trivially zero are nonzero. Given a a times b, a larger than or equal to b,…
We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x;\theta] modulo the left ideal generated by a polynomial of the type x^n-a. This allows us to develop an approach to skew-constacyclic codes based on…
We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook…
Compression of integer sets and sequences has been extensively studied for settings where elements follow a uniform probability distribution. In addition, methods exist that exploit clustering of elements in order to achieve higher…
Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that…
In this paper we define contractive and nonexpansive properties for adapted stochastic processes $X_1, X_2, \ldots $ which can be used to deduce limiting properties. In general, nonexpansive processes possess finite limits while contractive…
In this paper, on one hand, a class of linear codes with one or two weights is obtained. Based on these linear codes, we construct two classes of constant composition codes, which includes optimal constant composition codes depending on…
A class of linear codes that extends classic Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these…
We provide a general formulation for the code-based test compression problem with fixed-length input blocks and propose a solution approach based on Evolutionary Algorithms. In contrast to existing code-based methods, we allow unspecified…
We introduce the new concept of computation coding. Similar to how rate-distortion theory is concerned with the lossy compression of data, computation coding deals with the lossy computation of functions. Particularizing to linear…
A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex…
Let $X_1, ..., X_m$ be a set of $m$ statistically dependent sources over the common alphabet $\mathbb{F}_q$, that are linearly independent when considered as functions over the sample space. We consider a distributed function computation…
We extend coded distributed computing over finite fields to allow the number of workers to be larger than the field size. We give codes that work for fully general matrix multiplication and show that in this case we serendipitously have…
Pattern matching is a powerful tool which is part of many functional programming languages as well as computer algebra systems such as Mathematica. Among the existing systems, Mathematica offers the most expressive pattern matching.…
Inspired from modern out-of-equilibrium statistical physics models, a matrix product based framework permits the formal definition of random vectors (and random time series) whose desired joint distributions are a priori prescribed. Its key…
We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of…
Non-asymptotic theory of random matrices strives to investigate the spectral properties of random matrices, which are valid with high probability for matrices of a large fixed size. Results obtained in this framework find their applications…
We extend our techniques developed in our earlier paper appeared in Computational Complexity, 2017 (preprint: arXiv:1508.00690) to obtain a deterministic polynomial time algorithm for computing the non-commutative rank together with…
Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous…