Related papers: Linear Programming Bounds
This chapter is written for the forthcoming book "A Concise Encyclopedia of Coding Theory" (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Sol\'e. This book will collect short but foundational articles, emphasizing…
Finding the largest code with a given minimum distance is one of the most basic problems in coding theory. In this paper, we study the linear programming bound for codes in the Lee metric. We introduce refinements on the linear programming…
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…
Whereas many results are known about thresholds for ensembles of low-density parity-check codes under message-passing iterative decoding, this is not the case for linear programming decoding. Towards closing this knowledge gap, this paper…
The article surveys quantum error control, focusing on quantum stabilizer codes, stressing on the how to use classical codes to design good quantum codes. It is to appear as a book chapter in "A Concise Encyclopedia of Coding Theory,"…
We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…
The linear programming method is applied to the space $\U_n(\C)$ of unitary matrices in order to obtain bounds for codes relative to the diversity sum and the diversity product. Theoretical and numerical results improving previously known…
This is a chapter of the upcoming "A Concise Encyclopedia of Coding Theory", W.C. Huffman, J.-L. Kim, and P. Sole' Eds., CRC Press. The chapter gives an introduction to the mathematical theory of rank-metric codes. Treated topics include:…
This is a book about large language models. As indicated by the title, it primarily focuses on foundational concepts rather than comprehensive coverage of all cutting-edge technologies. The book is structured into five main chapters, each…
This paper provides a primer on Large Language Models (LLMs) and identifies their strengths, limitations, applications and research directions. It is intended to be useful to those in academia and industry who are interested in gaining an…
We survey recent generalizations and improvements of the linear programming method that involve semidefinite programming. A general framework using group representations and tools from graph theory is provided.
I developed the lecture notes based on my ``Linear Model'' course at the University of California, Berkeley over the past ten years. This book provides an intermediate-level introduction to the linear model. It balances rigorous proofs and…
We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory…
In his thesis, Wiberg showed the existence of thresholds for families of regular low-density parity-check codes under min-sum algorithm decoding. He also derived analytic bounds on these thresholds. In this paper, we formulate similar…
Qualification has been recently introduced as a generalization of uncertainty in the field of Logic Programming. In this report we investigate a more expressive language for First-Order Functional Logic Programming with Constraints and…
We discuss here constraint programming (CP) by using a proof-theoretic perspective. To this end we identify three levels of abstraction. Each level sheds light on the essence of CP. In particular, the highest level allows us to bring CP…
We define a metric on $\mathbb{F}_q^n$ using the linear complexity of finite sequences. We will then develop a coding theory for this metric. We will give a Singleton-like bound and we will give constructions of subspaces of…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
Given the increasing demands in computer programming education and the rapid advancement of large language models (LLMs), LLMs play a critical role in programming education. This study provides a systematic review of selected empirical…
Although large language models (LLMs) store vast amount of knowledge in their parameters, they still have limitations in the memorization and utilization of certain knowledge, leading to undesired behaviors such as generating untruthful and…