The book is devoted to the theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics On one side there are such classical areas as algebraic geometry and number theory; on the other, information transmission theory, combinatorics, finite geometries, dense packings, etc The authors give a unique perspective on the subject Whereas most books on coding theory build up coding theory from within, starting from elementary concepts and almost always finishing without reaching a certain depth, this book constantly looks for interpretations that connect coding theory to algebraic geometry and number theory There are no prerequisites other than a standard algebra graduate course The first two chapters of the book can serve as an introduction to coding theory and algebraic geometry respectively Special attention is given to the geometry of curves over finite fields in the third chapter Finally, in the last chapter the authors explain relations between all of these: the theory of algebraic geometric codes

Algebraic Geometric Codes: Basic Notions

The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the purpose of the paper is to review this work. Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced. Code constructions from particular classes of curves, including the Klein quartic, elliptic, and hyperelliptic curves, and Hermitian curves, are presented. Decoding algorithms for these classes of codes, and others, are considered. The construction of classes of asymptotically good codes using modular curves is also discussed.

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Algebraic-geometry codes

The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover, it is explained in terms of linear algebra and the theory of semigroups only.

https://www.win.tue.nl/~ruudp/paper/20.pdf

The minimum distance of codes in an array coming from telescopic semigroups

In this paper we use coding theory to give simple explanations of some recent results on universal hashing. We first apply our approach to give a precise and elegant analysis of the Wegman-Carter construction for authentication codes. Using Reed-Solomon codes and the well known concept of concatenated codes we can then give some new constructions, which require much less key size than previously known constructions. The relation to coding theory allows the use of codes from algebraic curves for the construction of hash functions. Particularly, we show how codes derived from Artin-Schreier curves, Hermitian curves and Suzuki curves yield good classes of universal hash functions.

https://link.springer.com/content/pdf/10.1007%2F3-540-48329-2_28.pdf

On families of hash functions via geometric codes and concatenation

The q-ary Reed-Muller (RM) codes RM/sub q/(u,m) of length n=q/sup m/ are a generalization of Reed-Solomon (RS) codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for RM codes were given in and . The algorithm in Sudan et al. (1999) is an improvement of the algorithm in , it is applicable to codes RM/sub q/(u,m) with u<q/2 and works for up to E<n(1-/spl radic/2u/q) errors. In this correspondence, following , we show that q-ary RM codes are subfield subcodes of RS codes over F/sub q//sup m/. Then, using the list- decoding algorithm in Guruswami and Sudan (1999) for RS codes over F/sub q//sup m/, we present a list-decoding algorithm for q-ary RM codes. This algorithm is applicable to codes of any rates, and achieves an error-correction bound n(1-/spl radic/(n-d)/n). The algorithm achieves a better error-correction bound than the algorithm in , since when u is small. The implementation of the algorithm requires O(n) field operations in F/sub q/ and O(n/sup 3/) field operations in F/sub q//sup m/ under some assumption.

/pdf/list-decoding-of-q-ary-reed-muller-codes-3r81funm13.pdf

List decoding of q-ary Reed-Muller codes

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If conditions are imposed on the degree of the divisor use, then criteria are derived for linear codes to be algebraic-geometric. In particular. the family of q-ary Hamming codes is investigated, and it is proven that only those with redundancy one or two and the binary (7,4,3) code are algebraic-geometric in this sense. For these codes. the authors explicitly give a curve, rational points, and a divisor. It is proven that this triple is in a certain sense unique in the case of the (7,4,3) code. >

/pdf/which-linear-codes-are-algebraic-geometric-2616xvovcy.pdf

R. Pellikan

Papers

Which linear codes are algebraic-geometric?