Related papers: Mathematical Programming Decoding of Binary Linear…

We consider communication over memoryless channels using low-density parity-check code ensembles above the iterative (belief propagation) threshold. What is the computational complexity of decoding (i.e., of reconstructing all the typical…

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…

The notion of programming paradigms, with associated programming languages and methodologies, is a well established tenet of Computer Science pedagogy, enshrined in international curricula. However, this notion sits ill with Kuhn's classic…

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable…

The problem of low complexity, close to optimal, channel decoding of linear codes with short to moderate block length is considered. It is shown that deep learning methods can be used to improve a standard belief propagation decoder,…

Probabilistic programming has emerged as a powerful paradigm in statistics, applied science, and machine learning: by decoupling modelling from inference, it promises to allow modellers to directly reason about the processes generating…

Mathematics has many useful properties for developing of complex software systems. One is that it can exactly describe a physical situation of the object or outcome of an action. Mathematics support abstraction and this is an excellent…

Probabilistic ideas and tools have recently begun to permeate into several fields where they had traditionally not played a major role, including fields such as numerical linear algebra and optimization. One of the key ways in which these…

Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on…

We overview our recently introduced theory of n-fold integer programming which enables the polynomial time solution of fundamental linear and nonlinear integer programming problems in variable dimension. We demonstrate its power by…

We introduce a new method for decoding short and moderate length linear block codes with dense parity-check matrix representations of cyclic form, termed multiple-bases belief-propagation (MBBP). The proposed iterative scheme makes use of…

We study the properties of the constructive linear programing problems. The parameters of linear functions in such problems are constructive real numbers. To solve such a problem is to find the optimal plan with the constructive real number…

We propose an approximation of maximum-likelihood detection in ISI channels based on linear programming or message passing. We convert the detection problem into a binary decoding problem, which can be easily combined with LDPC decoding. We…

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making…

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary…

This paper introduces and explores a new programming paradigm, Model-based Programming, designed to address the challenges inherent in applying deep learning models to real-world applications. Despite recent significant successes of deep…

Recently, it was shown that a binary linear code can be associated to a binomial ideal given as the sum of a toric ideal and a non-prime ideal. Since then two different generalizations have been provided which coincide for the binary case.…

We consider so-called squaring the square-puzzles where a given square (or rectangle) should be dissected into smaller squares. For a specific instance of such problems we demonstrate that a mathematically rigorous solution can be quite…

Schubert calculus provides algebraic tools to solve enumerative problems. There have been several applied problems in systems theory, linear algebra and physics which were studied by means of Schubert calculus. The method is most powerful…

Mathematical Selection is a method in which we select a particular choice from a set of such. It have always been an interesting field of study for mathematicians. Accordingly, Combinatorial Optimization is a sub field of this domain of…