On the automorphisms of generalized algebraic geometry codes

Designs, Codes and Cryptography - We consider the class of generalized algebraic geometry codes (GAG codes) formed by two collections of places, with places of the same degree in each collection....     

On the order bound of one-point algebraic geometry codes

Let S={s_i}_i∈N⊆N be a numerical semigroup. For each i∈N, let ν(s_i) denote the number of pairs (s_i−s_j,s_j)∈S²: it is well-known that there exists an integer m such that the sequence {ν(s_i)}_i∈N is non-decreasing for i>m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitly. In particular we give the value of m when the Cohen-Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When S is the Weierstrass semigroup of a family {C_i}_i∈N of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {C_i}.     

On the algebraic geometry ofS-duality

This paper deals with algebro-geometric questions arising in the verification of theS-duality conjecture for supersymmetric Yang-Mills quantum field theories in the four-dimensional case. We describe all the cases for the gauge groups of rank 1 and 2, where the Gell-Man-Law beta-function is either zero or negative, and point out some series of such cases for gauge groups of arbitrary rank. Realization of one of these series on the complex projective plane demonstrates a relationship with exceptional bundles. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 163–178, February, 1997. Translated by S. K. Lando     

Further improvements on the designed minimum distance of algebraic geometry codes

In the literature about algebraic geometry codes one finds a lot of results improving Goppa’s minimum distance bound. These improvements often use the idea of “shrinking” or “growing” the defining divisors of the codes under certain technical conditions. The main contribution of this article is to show that most of these improvements can be obtained in a unified way from one (rather simple) theorem. Our result does not only simplify previous results but it also improves them further.     

On row-cyclic codes with algebraic structure

In this article, some row-cyclic error-correcting codes are shown to be ideals in group rings in which the underlying group is metacyclic. For a given underlying group, several nonequivalent codes with this structure may be generated. Each is related to a cyclic code generated in response, to the metrics associated with the underlying metacyclic group. Such codes in the same group ring are isomorphic as vector spaces but may vary greatly in weight distributions and so are nonequivalent. If the associated cyclic code is irreducible, examining the structure of its isomorphic finite field yields all nonequivalent codes with the desired structure. Several such codes have been found to have minimum distances equalling those of the best known linear codes of the same length and dimension.This work was presented in part at the First International Conference on Finite Fields, Coding Theory, and Advances in Communications and Computing, Las Vegas, Nevada, August, 1991.     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

On the algebraic representation of projectively embeddable affine geometries

The main result of this article is an application of [1] and [2] which yields that an at least 2-dimensional affine geometry is module-induced if and only if it is projectively embeddable into an Arguesian projective lattice geometry.     

On some very strong compactness conditions

The aim of this paper is to consider compactness notions by utilizing ??-sets, V-sets, locally closed sets, locally open sets, ??-closed sets and ??-open sets. We completely characterize these variations of compactness, and also provide various interesting examples that support our results.     

Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences

This paper is concerned with two applications of bases of Riemann-Roch spaces. In the first application, we define the floor of a divisor and obtain improved bounds on the parameters of algebraic geometry codes. These bounds apply to a larger class of codes than that of Homma and Kim (J. Pure Appl. Algebra 162 (2001) 273). Then we determine explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. These bases give better estimates on the parameters of a large class of m-point Hermitian codes. In the second application, these bases are used for fast implementation of Xing and Niederreiter's method (Acta. Arith. 72 (1995) 281) for the construction of low-discrepancy sequences.     

On degenerate diffusion with very strong absorption

We prove the existence of weak global solutions to the degenerate diffusion equation (1) with singular absorption term. Moreover we investigate the regularity up to the quenching time and we show by means of explicit solutions that our regularity results are optimal.     

关于代数多项式的一个积分表示式

研究了代数多项式导数的Bernstein不等式和Markov不等式．通过代数多项式导数的一个积分表示式,给出这两个著名不等式以及它们的离散形式的证明．     

代数函数域上的局部恢复码

假设C是有限域Fq上的[n,k]线性码,如果码字的每个坐标是其它至多r个坐标的函数,称C是(n,k,r)线性码,这里r是较小的数.本文在代数函数域上构造出了局部恢复码,它的码长不受字符集大小的限制,实际上,它的码长可以远远大于字符集的大小;并将此方法应用于广义Hermite函数域,得到了一类广义Hermite函数域上的...