Related papers: Coding over Core Models

A generalization of the polar coding scheme called mixed-kernels is introduced. This generalization exploits several homogeneous kernels over alphabets of different sizes. An asymptotic analysis of the proposed scheme shows that its…

We present a proof of Jensen's Coding Theorem (assumong -0#) which quotes the covering lemma, but otherwise makes no appeal to fine structure theory. The key idea is to use a modified definition of the coding at limit cardinals, using…

It is well known that there is a deep relationship between codes and lattices. Concepts from coding theory are related to concepts of lattice theory as, for example, weight enumerators to theta series, MacWilliams identity to Jacobi…

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings,…

Theorem: Let $n\ge 2.$ There is a CCC in $L$ forcing notion $P=P_n\in L$ such that $P$-generic extensions of $L$ are of the form $L[a],$ where $a\subseteq\omega$ and 1) $a$ is $\Delta^1_{n+1}$ in $L[a]$; and 2) if $b\in L[a],$…

Coding Theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. Such codes over rings had important applications and many interesting mathematical…

We obtain a new coding and decoding method using the generalized Pell $(p,i)$ -numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory. We give two new blocking…

Predictive coding offers a potentially unifying account of cortical function -- postulating that the core function of the brain is to minimize prediction errors with respect to a generative model of the world. The theory is closely related…

The Jensen-Steel core model is a canonical inner model which plays a fundamental role in the meta-mathematics of set theory. Its definition depends on exactly which hierarchy of fine-structural models of set theory, premice, one uses. Each…

The concept of Central sets, introduced by Furstenberg through the framework of topological dynamics, has played a pivotal role in combinatorial number theory. Furstenberg's Central Sets Theorem highlighted their rich combinatorial…

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…

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…

The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction…

Decoding error-correctiong codes by methods of mathematical optimization, most importantly linear programming, has become an important alternative approach to both algebraic and iterative decoding methods since its introduction by Feldman…

Coding, which targets compressing and reconstructing data, and intelligence, often regarded at an abstract computational level as being centered around model learning and prediction, interweave recently to give birth to a series of…

Model-based coding, described by John Pierce in 1961, has great potential to reduce the volume of information that needs to be transmitted in moving big data, without loss of information, from one place to another, or in lossless…

This paper deals with a universal coding problem for a certain kind of multiterminal source coding system that we call the complementary delivery coding system. In this system, messages from two correlated sources are jointly encoded, and…

A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…

Large models have achieved remarkable performance across various tasks, yet they incur significant computational costs and privacy concerns during both training and inference. Distributed deployment has emerged as a potential solution, but…

Large language models (LLMs) have achieved remarkable progress in code generation, yet their true programming competence remains underexplored. We introduce the Code Triangle framework, which systematically evaluates LLMs across three…