Algebraic coding theory over finite commutative rings pdf

Cylinders, homotopies, and approximation theorems 121 3. All i know is a result that every finite commutative ring is a direct product of local commutative rings this is correct, right. Classification of finite commutative rings mathoverflow. Pdf commutative ring theory and applications download. We primarily investigate the the existence the lcd cyclic code of various lengths over. Some recent works on codes over such rings are 3, 7, 9, 22. Mceliece, finite fields for computer scientists and engineers, kluwer, 1987, m. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. Mathematics math ory, this volume is dedicated to codes constructed using algebraic geometry. Coding theory as pure mathematics page web dandre leroy. Hammons, kumar, calderbank, sloane, and sol e showed that important duality properties of certain.

Quasigalois rings 107 7 basic notions on codes over finite fields 121. Introduction in this paper we give a geometric version of the satake isomorphism sat. Moreover, frobenius rings are the largest class of finite rings over which. Is there a classification of finite commutative rings available. Pdf commutative ring theory and applications download full. 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 complexities and algebraic curves over finite fields d.

Slides for my talk in the special session on algebraic coding theory at the joint mathematics meetings, san diego, california, january 7, 2002. Finite commutative rings and their applications request pdf. The object of this chapter is to combine these areas of coding theory by introducing and studying algebraic geometric codes over rings. 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. This book introduces algebraic coding theory over finite frobenius rings. Bif a finite commutative ring r satisfies that all of its hamming isometries.

Hence we assume that the students have some basic background on ring theory and module theory. Dougherty department of mathematics university of scranton scranton, pa 18510, usa. Finite chain rings, besides their practical importance, are quite rich mathematical objects and so they have also theoretical attraction. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra and coding theory. 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. Finite commutative rings and their applications adds to the readers knowledge. Does a finite commutative ring necessarily have a unity. Noncommutative separable algebras have been studied in 2. Geometric langlands duality and representations of algebraic groups over commutative rings by i. Algebraic coding theory over finite commutative rings steven t. This text is suitable for courses in commutative algebra, finite commutative algebra, and coding theory. 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. The authors then introduce some applications of finite rings, in particular galois rings, to coding theory, using a solid algebraic and geometric theoretical background.

Algebraic coding theory over finite commutative rings. Constructions of selfdual codes over finite commutative chain rings. Constructions of selfdual codes over finite commutative. For nonnoetherian rings, and also nonlocal rings, the dimension may be infinite, but noetherian local rings have finite dimension. 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. 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. Constacyclic codes over finite commutative semisimple rings. Finite commutative rings and their applications is the first to address both theoretical and practical aspects of. Many areas of mathematics are used in coding theory, and we focus on the interplay between algebra. Matsumura covers the basic material, including dimension theory, depth, cohenmacaulay rings, gorenstein rings, krull rings and valuation rings. Geometric langlands duality and representations of algebraic.

Computational methods in commutative algebra and algebraic geometry. Some standard terminology from algebraic coding theory is discussed in. They have connections in both geometry pappian hjelmslev planes and algebraic number theory quotient rings of algebraic integers. Pdf finite commutative rings and their applications. Mathematics math algebraic coding theory is an area of discrete applied mathematics that is concerned in part with developing errorcontrol codes and encodingdecoding procedures.

Geometric langlands duality and representations of. We provide a class of cyclic groups and a class of. This book provides a selfcontained introduction to algebraic coding theory over finite frobenius 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. 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. Published for the conference board of the mathematical sciences by the american mathematical society, 2002.

An important partial con verse of this which we use here is that. Duality for modules over finite rings and applications to coding. This construction carries over to the wider context of monoidal monads, see tensor product of algebras over a commutative monad. 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. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Lamprechtcalculation of general gauss sums and quadratic gauss sums in finite rings j.

In a finite commutative ring every nonzerodivisor is a unit. Algebraic coding theory is an area of discrete applied mathematics that is concerned in part with developing errorcontrol codes and encodingdecoding procedures. 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. Finite frobenius rings as a setting for algebraic coding theory. Chudnovsky department of mathematics, columbia university, new york, new york 100. The notion of commutative algebraic theory was formulated in terms of monads by anders kock. 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. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is. Algebraic coding theory revised 1984 edition by elwyn r. We define alternant codes over a commutative ring r and a corresponding key equation. Codes over chain rings are of special interest theoretically and practically since most known good codes are from chain rings.

The remainder of this chapter is structured as follows. Hence, in addition to being applicable to systems over the real or complex numbers, kalmans theory can be applied to systems over finite. Coding theory and applications, cim series in mathematical sciences, vol. This text is intended for a one or twosemester undergraduate course in abstract algebra. Algebraic coding theory over finite commutative rings request. 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 frobenius rings. It analyzes the following three properties of a code mainly. 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.

Linear codes over finite chain ringsalgebraic theory. Algebraic complexities and algebraic curves over finite fields. The exercises at the ends of the chapters are mostly routine, with a few more. Among others, necessary and sufficient conditions for the existence of selfdual, lcd, and hermitian dualcontaining. Taylorthe arithmetic theory of local galois gauss sums for tame characters. Using this framework we construct an equivalence between the category of geometric linear codes over spec k and linear codes over k. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. Quadratic gauss sums over finite commutative rings. Finite commutative semisimple rings are direct sum of finite fields. The section on quasigalois rings presents new and unpublished results as well. Algebraic numbers, wileyinterscience, new york 1972. Idempotent is one of central notions in semigroup theory and algebra. A but we are particularly concerned with a when it has an algebraic structure.

I ask because of the following theorem given in my lecture notes. On linear complimentary dual codes over finite commutative. On the key equation over a commutative ring springerlink. It is the first to offer a comprehensive account on the subject. I there were some results on codes over rings in the 1970s, but the real breakthrough came in 1994. Abstract algebra course notes for rings and fields pdf. On linear complimentary dual codes over finite commutative frobenius local rings ramakrishna bandi1. If not, what are the best structure theorem that are known at present. Introduction to algebraic coding theory with gap fall 2006 sarah spence adams. Schroeder, number theory in science and communication, springer, 1986, or indeed any book on. Request pdf algebraic coding theory over finite commutative rings this book provides a selfcontained introduction to algebraic coding theory over finite. Constructions of selfdual codes over finite commutative chain rings steven t. Homotopy invariance and quillens algebraic ktheory of rings 128 5.