Linear codes and modular curves

Results of recent investigations at the juncture of coding theory, the theory of computability, and algebraic geometry over finite fields are presented. The basic problems of the asymptotic theory of codes and Goppa's construction of codes on the basis of algebraic curves are presented, and a detailed algorithmic analysis is given of the codes arising on the modular curves of elliptic modules of V. G. Drinfel'd.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 25, pp. 209–257, 1984.     

Algebraic geometric codes with applications

The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.      

Nonlinear codes with asymptotic parameters better than the Gilbert-Varshamov and the Xing Bounds

In this paper, we show that the Gilbert-Varshamov and the Xing bounds can be improved significantly around two points where these two bounds intersect by nonlinear codes from algebraic curves over finite fields.     

Generalized Hermitian codes

We investigate one-point algebraic geometry codes defined from curves related to the Hermitian curve. We obtain codes attaining new records on the parameters.     

Integrable structures of dispersionless systems and differential geometry

We develop the theory of Whitham-type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application, we construct Gibbons–Tsarev systems associated with the moduli space of algebraic curves of arbitrary genus and prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.     

The linear programming bound for codes over finite Frobenius rings

In traditional algebraic coding theory the linear-programming bound is one of the most powerful and restrictive bounds for the existence of both linear and non-linear codes. This article develops a linear-programming bound for block codes on finite Frobenius rings. An erratum to this article can be found at     

The order bound for general algebraic geometric codes

The order bound gives an in general very good lower bound for the minimum distance of one-point algebraic geometric codes coming from curves. This paper is about a generalization of the order bound to several-point algebraic geometric codes coming from curves.     

代数几何码概述

本文概述了有限域代数曲线上的码的一些最近结果．     

Nonbinary quantum error-correcting codes from algebraic curves

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia–Stichtenoth tower of function fields which are constructible in polynomial time.     

Sums of residues on algebraic surfaces and application to coding theory

In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results, we construct algebraic-geometric codes which are an extension to surfaces of the well-known differential codes on curves. We also study some properties of these codes and extend to them some known properties for codes on curves.     

The dual geometry of Hermitian two-point codes

In this paper, we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.     

Infinite families of 2‐designs from a class of cyclic codes

Combinatorial $t$‐designs have wide applications in coding theory, cryptography, communications, and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$‐design. In this paper, we first determine the weight distributions of a class of linear codes derived from the dual of some extended cyclic codes. We then obtain infinite families of 2‐designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By a simple counting argument, we obtain exponentially many 2‐designs.     

Parameters of AG codes from vector bundles

We investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples.     

Hasse–Weil bound for additive cyclic codes

We obtain a bound on the minimum distance of additive cyclic codes via the number of rational points on certain algebraic curves over finite fields. This is an extension of the analogous bound in the case of classical cyclic codes. Our result is the only general bound on such codes aside from Bierbrauer’s BCH bound. We compare our bounds’ performance against the BCH bound for additive cyclic codes in a special case and provide examples where it yields better results.     

A generalization of quasi-twisted codes: Multi-twisted codes

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their basic properties. Presenting several methods of constructing codes in this class and obtaining bounds on the minimum distances, we show that there exist codes with good parameters in this class that cannot be obtained as QT or constacyclic codes. This suggests that considering this larger class in computer searches is promising for constructing codes with better parameters than currently best-known linear codes. Working with this new class of codes motivated us to consider a problem about binomials over finite fields and to discover a result that is interesting in its own right.     

Translation planes and derivation sets

Using ideas from algebraic coding theory, a general notion of aderivation set for a projective plane is introduced. Certain geometric codes are used to locate such sets. These codes also lead to upper bounds for thep-ranks of incidence matrices of translation planes in terms of the dimensions of the associated codes.Dedicated to Professor Tallini on the occasion of his 60 ^th birthdayThis research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

A class of almost MDS codes

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed.     

多项式剩余类环上循环码新的表示

在编码理论中，多项式剩余类环是非常有意义的，它已经用来构造最优频率希望序列。本文，定义了多项式剩余类环上循环码的离散傅立叶变换及Mattson-Solomon（MS）多项式，证明了多项式剩余类环上的循环码同构于多项式剩余类环的Galois扩张的理想。