Geometric langlands duality and representations of algebraic. Idempotent is one of central notions in semigroup theory and algebra. I ask because of the following theorem given in my lecture notes. Published for the conference board of the mathematical sciences by the american mathematical society, 2002. Algebraic coding theory over finite commutative rings. In this thesis, we develop a geometric foundation for classical coding over fields and commutative rings using modern algebraic geometry and the language of schemes. Geometric langlands duality and representations of algebraic groups over commutative rings by i. I there were some results on codes over rings in the 1970s, but the real breakthrough came in 1994.
Pdf commutative ring theory and applications download. Moreover, frobenius rings are the largest class of finite rings over which. Finite frobenius rings as a setting for algebraic coding. We provide a class of cyclic groups and a class of. Pdf finite commutative rings and their applications. Constructions of selfdual codes over finite commutative chain rings. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra. The object of this chapter is to combine these areas of coding theory by introducing and studying algebraic geometric codes over rings. The main objects of study in this paper are the commutative separable algebras over a commutative ring. Quasigalois rings 107 7 basic notions on codes over finite fields 121.
This text is suitable for courses in commutative algebra, finite commutative algebra, and coding theory. We define alternant codes over a commutative ring r and a corresponding key equation. It is the first to offer a comprehensive account on the subject. Some standard terminology from algebraic coding theory is discussed in. Commutative separable algebras have been studied in 1 and in 2, 6 where the main ideas are. In this study, we investigate the algebraic structure of. Mathematics math ory, this volume is dedicated to codes constructed using algebraic geometry. Classification of finite commutative rings mathoverflow. Is there a classification of finite commutative rings available.
Coding theory and applications, cim series in mathematical sciences, vol. Using this framework we construct an equivalence between the category of geometric linear codes over spec k and linear codes over k. If not, what are the best structure theorem that are known at present. Coding theory as pure mathematics page web dandre leroy. Computational methods in commutative algebra and algebraic geometry. Dougherty department of mathematics university of scranton scranton, pa 18510, usa. Lamprechtcalculation of general gauss sums and quadratic gauss sums in finite rings j. Finite chain rings, besides their practical importance, are quite rich mathematical objects and so they have also theoretical attraction. The exercises at the ends of the chapters are mostly routine, with a few more. Slides for my talk in the special session on algebraic coding theory at the joint mathematics meetings, san diego, california, january 7, 2002.
This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. Taylorthe arithmetic theory of local galois gauss sums for tame characters. Algebraic numbers, wileyinterscience, new york 1972. Mathematics math algebraic coding theory is an area of discrete applied mathematics that is concerned in part with developing errorcontrol codes and encodingdecoding procedures. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is. Finite commutative rings and their applications request pdf. Algebraic coding theory revised 1984 edition by elwyn r. On linear complimentary dual codes over finite commutative. Cylinders, homotopies, and approximation theorems 121 3. For nonnoetherian rings, and also nonlocal rings, the dimension may be infinite, but noetherian local rings have finite dimension.
The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings in particular, galois rings as well as to their applications to coding theory. On linear complimentary dual codes over finite commutative frobenius local rings ramakrishna bandi1. A but we are particularly concerned with a when it has an algebraic structure. The integers, groups, cyclic groups, permutation groups, cosets and lagranges theorem, algebraic coding theory, isomorphisms, normal subgroups and factor groups, matrix groups and symmetry, the sylow theorems, rings, polynomials. Finite frobenius rings as a setting for algebraic coding theory. This book introduces algebraic coding theory over finite frobenius rings.
Our definition of a code is motivated by the fact that every nonsingular algebraic variety x over the complex numbers c has a natural structure as a. Galois theory, commutative algebra, with applications to. Request pdf algebraic coding theory over finite commutative rings this book provides a selfcontained introduction to algebraic coding theory over finite. The term algebraic coding theory denotes the subfield of coding theory where the properties of codes are expressed in algebraic terms and then further researched. Geometric langlands duality and representations of. Introduction in this paper we give a geometric version of the satake isomorphism sat. Mceliece, finite fields for computer scientists and engineers, kluwer, 1987, m. Algebraic complexities and algebraic curves over finite fields. Finite commutative rings 1 introduction let s be a nonempty commutative semigroup, endowed with a binary associative operation let es be the set of idempotents of s, where e.
Noncommutative separable algebras have been studied in 2. Hence, in addition to being applicable to systems over the real or complex numbers, kalmans theory can be applied to systems over finite. Linear codes over finite chain ringsalgebraic theory. Finite commutative rings and their applications adds to the readers knowledge. An important partial con verse of this which we use here is that. Pdf commutative ring theory and applications download full. Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in commutative algebra, the goingup theorem and krulls principal. Constructions of selfdual codes over finite commutative. Finite commutative rings and their applications is the first to address both theoretical and practical aspects of. On the key equation over a commutative ring springerlink. In a finite commutative ring every nonzerodivisor is a unit. Duality for modules over finite rings and applications to coding. This text is intended for a one or twosemester undergraduate course in abstract algebra.
Constructions of selfdual codes over finite commutative chain rings steven t. Matsumura covers the basic material, including dimension theory, depth, cohenmacaulay rings, gorenstein rings, krull rings and valuation rings. Dedicated with respect and admiration to vera pless on the occasion of her retirement. A characterization of finite commutati ve rings roger a lperin and e loise h amann w e give a partial con verse to the w ellkno wn result.
Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Finite commutative rings and their applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Codes over chain rings are of special interest theoretically and practically since most known good codes are from chain rings. His areas of interest include algebraic coding theory and its applications to quantum coding, network coding, and biological. All i know is a result that every finite commutative ring is a direct product of local commutative rings this is correct, right. Commutative separable algebras have been studied in 1 and in 2, 6 where the main ideas are based on the classical galois theory of fields. Abstract algebra course notes for rings and fields pdf. This construction carries over to the wider context of monoidal monads, see tensor product of algebras over a commutative monad. The section on quasigalois rings presents new and unpublished results as well. They have connections in both geometry pappian hjelmslev planes and algebraic number theory quotient rings of algebraic integers.
Homotopy invariance and quillens algebraic ktheory of rings 128 5. Request pdf algebraic coding theory over finite commutative rings this book provides a selfcontained introduction to algebraic coding theory over finite frobenius rings. Constacyclic codes over finite commutative semisimple rings. Fulfillment by amazon fba is a service we offer sellers that lets them store their products in amazons fulfillment centers, and we directly pack, ship, and provide customer service for these products. Among others, necessary and sufficient conditions for the existence of selfdual, lcd, and hermitian dualcontaining. Some recent works on codes over such rings are 3, 7, 9, 22. Finite commutative semisimple rings are direct sum of finite fields. Algebraic coding theory over finite commutative rings request.
Quadratic gauss sums over finite commutative rings. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. Does a finite commutative ring necessarily have a unity. Hence we assume that the students have some basic background on ring theory and module theory. The remainder of this chapter is structured as follows. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Bif a finite commutative ring r satisfies that all of its hamming isometries. This book provides a selfcontained introduction to algebraic coding theory over finite frobenius rings. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra and coding theory. Hammons, kumar, calderbank, sloane, and sol e showed that important duality properties of certain. The notion of commutative algebraic theory was formulated in terms of monads by anders kock. It analyzes the following three properties of a code mainly. Finite commutative rings and their applications gilberto.
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