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The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then further researched. [citation needed] Algebraic coding theory is basically divided into two major types of codes: [citation needed] Linear block codes; Convolutional codes
This book is mainly centered around algebraic and combinatorial techniques for designing and using error-correcting linear block codes. [1] [3] [9] It differs from previous works in this area in its reduction of each result to its mathematical foundations, and its clear exposition of the results follow from these foundations.
Algebraic geometry codes are a generalization of Reed–Solomon codes. Constructed by Irving Reed and Gustave Solomon in 1960, Reed–Solomon codes use univariate polynomials to form codewords, by evaluating polynomials of sufficiently small degree at the points in a finite field . [8]
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Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .
Robert J. McEliece (May 21, 1942 – May 8, 2019) [1] was the Allen E. Puckett Professor and a professor of electrical engineering at the California Institute of Technology (Caltech) best known for his work in error-correcting coding and information theory. He was the 2004 recipient of the Claude E. Shannon Award and the 2009 recipient of the ...
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the ...
The Fano matroid, derived from the Fano plane.Matroids are one of many kinds of objects studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.