extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

irreducibility criterion for germs of analytic functions of two complex variables let s x y be the germ of an analytic function of two complex variables near the origin to decide whether f is irreducible or not we can apply a succession of blowing ups to desingularize f and see if the proper transform off ever bifurcates and then use the fact that f is irreducible iff such a bifurcation never takes place alternatively we can use newton s theorem also called puiseux s theorem to completely factor s into linear factors in y by allowing fractional power series in x and then collate these factors into conjugacy classes and now use the fact that fis irreducible iff there is only one conjugacy class in a recent conversation t c kuo of the university of sydney in australia asked me the following question

systematic encoding via grobner bases for a class of algebraic geometric goppa codes any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring the theory of grobner bases for modules gives a compact description and implementation of a systematic encoder we present examples of algebraic geometric goppa codes that can be encoded by these methods including the one point hermitian codes

fast decoding of algebraic geometric codes up to the designed minimum distance we present a decoding algorithm for algebraic geometric codes from regular plane curves in particular the hermitian curve which corrects all error patterns of weight less than d 2 with low complexity the algorithm is based on the majority scheme of feng and rao 1993 and uses a modified version of sakata s 1988 generalization of the berlekamp massey algorithm

algebraic function fields over finite fields with many rational places algebraic function fields or equivalently algebraic curves provide a useful tool for coding theory for instance algebraic geometric codes and trace codes but also for other branches of information theory in these applications the number of rational places of a function field plays a crucial role one is particularly interested in function fields having a large number of rational places after a short introduction into the mathematical theory of algebraic functions the paper gives a survey of old and new results on the number of rational places of function fields

decoding binary 2 d cyclic codes by the 2 d berlekamp massey algorithm a method of decoding two dimensional 2 d cyclic codes by applying the 2 d berlekamp massey algorithm is proposed to explain this decoding method the author introduces a subclass of 2 d cyclic codes which are called 2 d bch codes due to their similarity with bch codes it is shown that there are some short 2 d cyclic codes with a better cost parameter value the merit of the approach is verified by showing several simple examples of 2 d cyclic codes

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

a generalized forney formula for algebraic geometric codes this correspondence contains a straightforward generalization of decoding of bch codes to the decoding of algebraic geometric codes couched in terms of varieties ideals and grobner bases this consists of 1 a berlekamp massey type lattice shifting row reduction algorithm with majority voting similar to algorithms in the current literature 2 a realization that it produces a minimal grobner basis b for the error locator ideal i v relative to a particular weighted total degree monomial ordering 3 a factoring of that basis into several minimal plex bases that facilitates finding the variety v of error positions and 4 a direct generalization of forney s formula to calculate error magnitudes using functions spl sigma p which are by products of this factoring

the complete list of genera of quotients of the fq2 maximal hermitian curve for q 1 mod 4 abstract let f q 2 be the finite field with q 2 elements most of the known f q 2 maximal curves arise as quotient curves of the f q 2 maximal hermitian curve h q after a seminal paper by garcia stichtenoth and xing many papers have provided genera of quotients of h q but their complete determination is a challenging open problem in this paper we determine completely the spectrum of genera of quotients of h q for any q 1 mod 4

two point ag codes on the gk maximal curves we determine the weierstrass semigroup of a pair of certain rational points on the giulietti korchmaros maximal curves we use this semigroup to obtain two point algebraic geometric ag codes with better parameters than comparable one point ag codes arising from these curves these parameters are new records in the mint s tables

a new family of maximal curves over a finite field it has been known for a long time that the deligne lusztig curves associated to the algebraic groups of type 2a 2 2b 2 and 2g 2 defined over the finite field mathbb f n all have the maximum number of mathbb f n rational points allowed by the weil explicit formulas and that these curves are mathbb f q 2 maximal curves over infinitely many algebraic extensions mathbb f q 2 of mathbb f n serre showed that an mathbb f q 2 rational curve which is mathbb f q 2 covered by an mathbb f q 2 maximal curve is also mathbb f q 2 maximal this has posed the problem of the existence of mathbb f q 2 maximal curves other than the deligne lusztig curves and their mathbb f q 2 subcovers see for instance garcia on curves with many rational points over finite fields in finite fields with applications to coding theory cryptography and related areas pp 152 163 springer berlin 2002 and garcia and stichtenoth a maximal curve which is not a galois subcover of the hermitan curve bull braz math soc n s 37 139 152 2006 in this paper a positive answer to this problem is obtained for every q n 3 with n p r 2 p 2 prime we give a simple explicit construction of an mathbb f q 2 maximal curve mathcal x that is not mathbb f q 2 covered by any mathbb f q 2 maximal deligne lusztig curve furthermore the mathbb f q 2 automorphism group aut mathcal x has size n 3 n 3 1 n 2 1 n 2 n 1 interestingly mathcal x has a very large mathbb f q 2 automorphism group with respect to its genus g frac 1 2 n 3 1 n 2 2 1

codes on the klein quartic ideals and decoding corresp a sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field gf 2 3 in the sequence there is for instance a code of length 21 and dimension 10 with minimal distance 9 and a code of length 21 and dimension 16 with minimal distance 3 the codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by goppa the curve used in the present work is the klein quartic this curve has the maximal number of rational points over gf 2 3 allowed by serre s improvement of the hasse weil bound which together with the low genus accounts for the good code parameters the klein quartic has the frobenius group g of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes in fact the codes are given alternative descriptions as left ideals in the group algebra gf 2 3 g this description allows for easy decoding for instance in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 decoding is obtained by multiplication with an idempotent in the group algebra

fast decoding of multipoint codes from algebraic curves multipoint codes are a broad class of algebraic geometry codes derived from algebraic functions which have multiple poles and or zeros on an algebraic curve thus they are more general than one point codes which are an important class of algebraic geometry codes in the sense that they can be decoded efficiently using the berlekamp massey sakata algorithm we present a fast method for decoding multipoint codes from a plane curve particularly a hermitian curve our method with some adaptation can be applied to decode multipoint codes from a general algebraic curve embedded in the n dimensional affine space fqn over a finite field fq so that those algebraic geometry codes can be decoded efficiently if the dimension n of the affine space including the defining curve is small

two point codes for the generalized gk curve we improve previously known lower bounds for the minimum distance of certain two point ag codes constructed using a generalized giulietti korchmaros curve ggk castellanos and tizziotti recently described such bounds for two point codes coming from the giulietti korchmaros curve our results completely cover and in many cases improve on their results using different techniques while also supporting any ggk curve our method builds on the order bound for ag codes to enable this we study certain weierstrass semigroups this allows an efficient algorithm for computing our improved bounds we find several new improvements upon the mint minimum distance tables

unique decoding of general ag codes a unique decoding algorithm for general ag codes namely multipoint evaluation codes on algebraic curves is presented it is a natural generalization of the previous decoding algorithm which was only for one point ag codes as such it retains the same advantages of fast speed regular structure and direct message recovery upon this generalization we add a technique from the guruswami sudan list decoding that boosts the decoding speed significantly compared with other known decoding algorithms for general ag codes it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure

an explicit construction of a sequence of codes attaining the tsfasman vladut zink bound the first steps we present a sequence of codes attaining the tsfasman vladut zink bound the construction is based on the tower of artin schreier extensions described by garcia and stichtenoth 1995 we also determine the dual codes the first steps of the constructions are explicitly given as generator matrices

three superposition principles currents continuity equations and curves of measures abstract we establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure representing them as marginals of measures concentrated on the solutions of the associated ode defined by some algebra of observables we relate this result with decomposition of acyclic normal currents in metric spaces as an application a slightly extended version of a probabilistic representation for absolutely continuous curves in kantorovich wasserstein spaces originally due to s lisini is provided in the metric framework this gives a hierarchy of implications between superposition principles for curves of measures and for metric currents

one point ag codes on the gk maximal curves in this paper algebraic geometric ag codes associated to a recently discovered class of maximal curves are investigated as a result some linear codes with better parameters with respect to the previously known ones are discovered and 70 improvements on mint s tables tables of optimal parameters for linear codes university of salzburg salzburg austria are obtained

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

automorphisms of harbater katz gabber curves let k be a perfect field of characteristic p 0 and let g be a finite group we consider the pointed g curves over k associated by harbater katz and gabber to faithful actions of g on k t over k we use such hkg g curves to classify the automorphisms of k t of p power order that can be expressed by particularly explicit formulas namely those mapping t to a power series lying in a mathbb z p mathbb z artin schreier extension of k t in addition we give necessary and sufficient criteria to decide when an hkg g curve with an action of a larger finite group j is also an hkg j curve

majority coset decoding a majority coset decoding mcd procedure that can be applied to an arbitrary geometric code is discussed in general the basic algorithm for decoding of algebraic geometric codes does not correct up to the designed minimum distance in mcd a reduction step is added to the basic algorithm in case the basic algorithm fails a majority scheme is used to obtain an additional syndrome for the error vector thus a strictly smaller cost containing the error vector is obtained in this way the basic algorithm is applied to a decreasing chain of cosets and after finitely many steps the coset will be small enough for successful application of the basic algorithm

improved two point codes on hermitian curves one point codes on the hermitian curve produce long codes with excellent parameters feng and rao introduced a modified construction that improves the parameters while still using one point divisors a separate improvement of the parameters was introduced by matthews considering the classical construction but with two point divisors those two approaches are combined to describe an elementary construction of two point improved codes upon analysis of their minimum distance and redundancy it is observed that they improve on the previous constructions for a large range of designed distances

the poincare metric and isoperimetric inequalities for hyperbolic polygons we prove several isoperimetric inequalities for the conformal radius or equivalently for the poincare density of polygons on the hyperbolic plane our results include as limit cases the isoperimetric inequality for the conformal radius of euclidean n gons conjectured by g polya and g szego in 1951 and a similar inequality for the hyperbolic n gons of the maximal hyperbolic area conjectured by j hersch both conjectures have been proved in previous papers by the third author our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by a yu solynin we also employ the dissymmetrization transformation of v n dubinin as an important part of our proofs we obtain monotonicity and convexity results for special combinations of the euler gamma functions which appear to have a significant interest in their own right

generalized hamming weights of trace codes linear codes over f sub p often admit a natural representation as trace codes of codes that are defined over an extension field f sub p m in the paper the authors obtain estimates for the weights of subcodes of such trace codes their main result is a far reaching generalization of the carlitz uchiyama bound for the duals of binary bch codes in particular they prove sharp bounds for the generalized hamming weights of a large class of codes including duals of bch codes classical goppa codes melas codes and arbitrary cyclic codes of length n p sup m 1 the main tool is the theory of algebraic functions over finite fields in particular the hasse weil bound for the number of places of degree one

algebraic geometry codes 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

algebraic curves with many automorphisms abstract let x be a projective geometrically irreducible non singular algebraic curve of genus g 2 defined over an algebraically closed field k of odd characteristic p let aut x be the group of all automorphisms of x which fix k element wise it is known that if aut x 8 g 3 then the p rank equivalently the hasse witt invariant of x is zero this raises the problem of determining the minimum value function f g such that whenever aut x f g then x has zero p rank for even g we prove that f g 900 g 2 the odd genus case appears to be much more difficult although for any genus g 2 if aut x has a solvable subgroup g such that g 252 g 2 then x has zero p rank and g fixes a point of x our proofs use the hurwitz genus formula and the deuring shafarevich formula together with a few deep results from group theory characterizing finite simple groups whose sylow 2 subgroups have a cyclic subgroup of index 2 we also point out some connections with the abhyankar conjecture and the katz gabber covers

generalized algebraic geometric codes from maximal curves some new results on generalized algebraic geometric gag codes are obtained first we provide some constructions which significantly improve the general lower bounds on the minimum distance of a gag code gag codes associated to specific maximal curves over finite fields are then investigated as a result 2895 improvements on mint s tables are obtained finally we construct asymptotically good gag codes with better parameters with respect to those constructed by spera in 2005 maximal curves play a role in this context as well

modular curves and codes with a polynomial construction in this paper we study q ary codes arising from modular curves x 0 11l over gf p 2 and some binary codes attached to them all these codes have a polynomial construction and have good asymptotic parameters for certain values of the parameters the q ary codes with q p 2 geq 49 are better than the varshamov gilbert bound and the binary codes are better than the ziablov bounds

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

algebraic decoding using special divisors the basic algorithm for decoding of algebraic geometric codes corrects up to d sub c 1 2 g 2 errors where d sub c denotes the designed minimum distance of a code and g denotes the genus of a curve the modified algorithm improves on this but applies to a restricted class of codes an extended modified algorithm that applies to all codes is formulated it will correct up to d sub c 1 2 s errors s is called the clifford defect of a curve for curves with g or 1 this defect satisfies 0

flows of measures generated by vector fields the scope of the paper is twofold we show that for a large class of measurable vector fields in the sense of weaver i e derivations over the algebra of lipschitz functions called in the paper laminated the notion of integral curves may be naturally defined and characterized when appropriate by an ordinary differential equation we further show that for such vector fields the notion of a flow of the given positive borel measure similar to the classical one generated by a smooth vector field in a space with smooth structure may be defined in a reasonable way so that the measure flows along the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

on the main conjecture on geometric mds codes a new way to attack the main conjecture on mds codes for geometric codes is proposed in particular the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough is proven

codes from the suzuki function field we construct algebraic geometry ag codes from the function field f 2 sup 2n 1 x y f 2 sup 2n 1 defined by y 2 sup 2n 1 y x 2 sup 2n x where n is a positive integer these codes are supported by two places and many have parameters that are better than those of any comparable code supported by one place of the same function field to define such codes we determine and exploit the structure of the weierstrass gap set of an arbitrary pair of rational places of f 2 sup 2n 1 x y f 2 sup 2n 1 moreover we find some codes over f sub 8 with parameters that are better than any known code

the weight hierarchy of higher dimensional hermitian codes studies a class of projective systems and linear codes corresponding to hermitian varieties over finite fields the weight hierarchy also known as the set of generalized hamming weights of the code is calculated the higher weight distribution is also found

minimum distance decoding of general algebraic geometry codes via lists algebraic geometry codes are defined by divisors d and g on a curve over a finite field f often g is supported by a single f rational point and the resulting code is called a one point code recently there has been interest in allowing the divisor g to be more general as this can result in superior codes in particular one may obtain a code with better parameters by allowing g to be supported by m distinct f rational points where m 1 in this paper we demonstrate that a multipoint algebraic geometry code c may be embedded in a one point code c exploiting this fact we obtain a minimum distance decoding algorithm for the multipoint code c this is accomplished via list decoding in the one point code c

simplified understanding and efficient decoding of a class of algebraic geometric codes an efficient decoding algorithm for algebraic geometric codes is presented for codes from a large class of irreducible plane curves including hermitian curves it can correct up to d 1 2 errors where d is the designed minimum distance with it we also obtain a proof of d sub min spl ges d without directly using the riemann roch theorem the algorithm consists of gaussian elimination on a specially arranged syndrome matrix followed by a novel majority voting scheme a fast implementation incorporating block hankel matrix techniques is obtained whose worst case running time is o mn sup 2 where m is the degree of the curve applications of our techniques to decoding other algebraic geometric codes to decoding bch codes to actual minimum distance and to two dimensional shift register synthesis are also presented

structure of metric cycles and normal one dimensional currents abstract we prove that every one dimensional real ambrosio kirchheim normal current in a polish i e complete separable metric space can be naturally represented as an integral of simpler currents associated to lipschitz curves as a consequence a representation of every such current with zero boundary i e a cycle as an integral of so called elementary solenoids which are very roughly speaking more or less the same as asymptotic cycles introduced by s schwartzman is obtained the latter result on cycles is in fact a generalization of the analogous result proven by s smirnov for classical whitney currents in a euclidean space the same results are true for every complete metric space under suitable set theoretic assumptions

on curves covered by the hermitian curve abstract for each proper divisordof q q 1 withqan even power of a prime maximal curves of genus 1 2 q q 1 d 1 that are f q covered by the hermitian curve are constructed

decomposition of acyclic normal currents in a metric space abstract we prove that every acyclic normal one dimensional real ambrosio kirchheim current in a polish i e complete separable metric space can be decomposed in curves thus generalizing the analogous classical result proven by s smirnov in euclidean space setting the same assertion is true for every complete metric space under a suitable set theoretic assumption

on the geometry of hermitian one point codes abstract here we describe the algebraic geometry of one point codes from the hermitian curve in particular we employ zero dimensional schemes in the plane to characterize the minimum weight codewords of their dual codes providing explicit formulas for their number we discuss also some natural improvements of the duals of hermitian one point codes by means of geometric arguments finally some cohomological tools are developed to characterize the small weight codewords of such codes

the automorphism groups of a family of maximal curves abstract the hasse weil bound restricts the number of points of a curve which are defined over a finite field if the number of points meets this bound the curve is called maximal giulietti and korchmaros introduced a curve c 3 which is maximal over f q 6 and determined its automorphism group garcia guneri and stichtenoth generalized this construction to a family of curves c n indexed by an odd integer n 3 such that c n is maximal over f q 2 n in this paper we determine the automorphism group aut c n when n 3 in contrast with the case n 3 it fixes the point at infinity on c n the proof requires a new structural result about automorphism groups of curves in characteristic p such that each sylow p subgroup has exactly one fixed point

improved geometric goppa codes i basic theory in this paper we present a construction of improved geometric goppa codes which for the case of r 2g are often more efficient than the current geometric goppa codes derived from some varieties which include algebraic curves hyperplanes surfaces and other varieties for the special case of a plane in a three dimensional projective space the improved geometric goppa codes are reduced to linear multilevel codes for these improved geometric goppa codes a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given

an isoperimetric inequality for logarithmic capacity of polygons we verify an old conjecture of g polya and g szeg o saying that the regular n gon minimizes the logarithmic capacity among all n gons with a fixed area

two point coordinate rings for gk curves giulietti and korchmaros presented new curves with the maximal number of points over a field of size q6 garcia guneri and stichtenoth extended the construction to curves that are maximal over fields of size q2n for odd n 3 the generalized gk curves have affine equations xq x yq 1 and yq2 y zr for r qn 1 q 1 we give a new proof for the maximality of the generalized gk curves and we outline methods to efficiently obtain their two point coordinate ring

geometric reed solomon codes of length 64 and 65 over f sub 8 we determine the actual parameters for a class of one point codes of length 64 and 65 over f sub 8 defined by hansen and stichtenoth 1990 several codes have a minimum distance that exceeds the feng rao bound the codes with parameters 64 5 51 64 10 42 64 11 42 64 12 40 65 5 52 65 10 43 65 11 42 65 12 41 and 65 13 40 are better than any known code

vardohus codes polar codes based on castle curves kernels in this paper we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w r t the field operations we will also study the minimum distance of the polar codes proposed their duals and the exponents of the matrices used for defining them all the restrictions that we make to our curves will be accomplished by the so called castle curves

a tower of artin schreier extensions of function fields attaining the drinfeld vladut bound for an algebraic function fieldf having a finite constant field letg f resp n f denote the genus off resp the number of places off of degree one we construct a tower of function fields f 1 subseteq f 2 subseteq f 3 subseteq ldots over mathbb f q 2 such that the ration fi g fi tends to the drinfeld vladut boundq 1

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

projective reed muller codes a class of codes in the reed muller family the projective reed muller codes prm codes is studied the author defines the prm codes of all orders and discusses their relation to polynomial codes the exact parameters of prm codes are given the duals are characterized and in parallel to the classical works on generalized reed muller codes the cyclic properties are studied tables over parameters of the codes are given

on automorphisms of geometric goppa codes the algebraic geometric codes which were introduced by v d goppa in 1977 7 8 we call them geometric goppa codes can be used to prove the existence of long linear codes which are better than the gilbert varshamov bound there are many good expositions of this important result ll 13 15 181 in that context one has to consider curves of high genus which admit many rational points on the other hand new interesting codes were found among the geometric goppa codes by looking at special curves e g elliptic curves hermite curves or artin schreier curves 3 5 8 13 16 171 in coding theory one is often interested in codes admitting many automorphisms e g cyclic codes goppa ls already observed that automorphisms of the underlying algebraic curve induce automorphisms of the associated geometric codes since goppa s paper is not quite correct we shall give a detailed exposition of this connexion in section 2 section 3 is the central part of our paper it is devoted to the case of geometric goppa codes associated with a curve of genus 0 so called rational goppa codes these codes are particularly interesting since many well known classes of codes e g bch codes classical goppa codes reed solomon codes cauchy codes 6 can be represented as rational goppa codes or subfield subcodes of them cl our main result is theorem 3 1 it states that in general all automorphisms of a rational goppa code are induced by automorphisms of the rational curve i e by projective transformations of the projective line

algebraic curves with a large non tame automorphism group fixing no point let k be an algebraically closed field of characteristic p 0 and let x be a curve over k of genus g 2 assume that the automorphism group aut x of x over k fixes no point of x the following result is proven if there is a point p on x whose stabilizer in aut x contains a p subgroup of order greater than gp p 1 then x is birationally equivalent over k to one of the irreducible plane curves ii iii iv v listed in the introduction

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

improvements on parameters of one point ag codes from hermitian curves by choosing specific divisors we show that some algebraic geometry ag codes from the hermitian curves have better parameters than one point hermitian codes the general lower bound is discussed

a class of linear codes with good parameters from algebraic curves a class of linear codes with good parameters is constructed in this correspondence it turns out that linear codes of this class are subcodes of the subfield subcodes of goppa s geometry codes in particular we find 61 improvements on brouwer s table based on our codes

decoding geometric goppa codes using an extra place decoding geometric goppa codes can be reduced to solving the key congruence of a received word in an affine ring if the codelength is smaller than the number of rational points on the curve then this method can correct up to 1 2 d l 2 s errors where d is the designed minimum distance of the code and s is the clifford defect the affine ring with respect to a place p is the set of all rational functions which have no poles except at p and it is somehow similar to a polynomial ring for a special kind of geometric goppa code namely c sub omega d mp the decoding algorithm is reduced to solving the key equation in the affine ring which can be carried out by the subresultant sequence in the affine ring with complexity o n sup 3 where n is the length of codewords

on semigroups generated by two consecutive integers and improved hermitian codes analysis of the berlekamp massey sakata algorithm for decoding one point codes leads to two methods for improving code rate one method due to feng and rao removes parity checks that may be recovered by their majority voting algorithm the second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic in this work formulae are given for the redundancies of hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance the results proceed from an analysis of numerical semigroups generated by two consecutive integers

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

on the decoding of algebraic geometric codes over f sub q for q or 16 it is proved that for algebraic geometric codes on a curve over f sub q for q or 37 or on a curve of sufficiently large genus over f sub q for q or 16 there exists a polynomial decoding algorithm up to d 1 2 errors d being the designed minimum distance

near optimal depth constrained codes this note considers an n letter alphabet in which the ith letter is accessed with probability p sub i the problem is to design efficient algorithms for constructing near optimal depth constrained huffman and alphabetic codes we recast the problem as one of determining a probability vector q sup q sup sub 1 q sup sub n in an appropriate convex set s so as to minimize the relative entropy d p spl par q over all q spl isin s methods from convex optimization give an explicit solution for q sup in terms of p we show that the huffman and alphabetic codes so constructed are within 1 and 2 bits of the corresponding optimal depth constrained codes

fast decoding of codes from algebraic plane curves improvement to an earlier decoding algorithm for codes from algebraic geometry is presented for codes from an arbitrary regular plane curve the authors correct up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance of the code and m is the degree of the curve the complexity of finding the error locator is o n sup 7 3 where n is the length of the code for codes from hermitian curves the complexity of finding the error values given the error locator is o n sup 2 and the same complexity can be obtained in the general case if only d 2 m sup 2 2 errors are corrected