Applications of ordered weights in information transmission

This dissertation is devoted to a study of a class of linear codes related to a particular metric space that generalizes the Hamming space in that the metric function is defined by a partial order on the set of coordinates of the vector. We begin with developing combinatorial and linear-algebraic aspects of linear ordered codes. In particular, we define multivariate rank enumerators for linear codes and show that they form a natural set of invariants in the study of the duality of linear codes. The rank enumerators are further shown to be connected to the shape distributions of linear codes, and enable us to give a simple proof of a MacWilliams-like theorem for the ordered case. We also pursue the connection between linear codes and matroids in the ordered case and show that the rank enumerator can be thought of as an instance of the classical matroid invariant called the Tutte polynomial. Finally, we consider the distributions of support weights of ordered codes and their expression via the rank enumerator. Altogether, these results generalize a group of well-known results for codes in the Hamming space to the ordered case. Extending the research in the first part, we define simple probabilistic channel models that are in a certain sense matched to the ordered distance, and prove several results related to performance of linear codes on such channels. In particular, we define ordered wire-tap channels and establish several results related to the use of linear codes for reliable and secure transmission in such channel models. In the third part of this dissertation we study polar coding schemes for channels with nonbinary input alphabets. We construct a family of linear codes that achieve the capacity of a nonbinary symmetric discrete memoryless channel with input alphabet of size q=2^r, r=2,3,.... A new feature of the coding scheme that arises in the nonbinary case is related to the emergence of several extremal configurations for the polarized data symbols. We establish monotonicity properties of the configurations and use them to show that total transmission rate approaches the symmetric capacity of the channel. We develop these results to include the case of ``controlled polarization'' under which the data symbols polarize to any predefined set of extremal configurations. We also outline an application of this construction to data encoding in video sequences of the MPEG-2 and H.264/MPEG-4 standards

Associative nn-categories

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold diagrams, which are stratifications of the n-cube such that strata are transversal to the flag foliation of the n-cube. The first part of this thesis develops a combinatorial model for manifold diagrams called singular n-cubes. In the second part we apply this model to build our notions of higher categories. Singular n-cubes are "directed triangulations" of space together with a decomposition into a collection of subspaces or strata. Singular n-cubes can be naturally organised into two categories. The first, whose morphisms are bundles themselves, is used for the inductive definition of singular (n+1)-cubes. The second, whose morphisms are "open" base changes, admits an (epi,mono) factorisation system. Monomorphisms will be called embeddings of cubes. Epimorphisms will be called collapses and describe how triangulations can be coarsened. Each cube has a unique coarsest triangulation called its normal form. The existence of normal forms makes the equality relation of (combinatorially represented) manifold diagrams decidable. As the main application of the resulting combinatorial framework for manifold diagrams, we give algebraic definitions of various notions of higher categories. Namely, we define associative n-categories, presented associative n-categories and presented associative n-groupoids. All three notions will have strict units and associators; the only weak coherences are homotopies, but we develop a mechanism for recovering the usual coherence data of weak n-categories, such as associators and pentagonators and their higher analogues. This will motivate the conjecture that the theory of associative higher categories is equivalent to its fully weak counterpart.Comment: 499 page

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum