623 research outputs found
On the complete weight enumerators of some linear codes with a few weights
Linear codes with a few weights have important applications in authentication
codes, secret sharing, consumer electronics, etc.. The determination of the
parameters such as Hamming weight distributions and complete weight enumerators
of linear codes are important research topics. In this paper, we consider some
classes of linear codes with a few weights and determine the complete weight
enumerators from which the corresponding Hamming weight distributions are
derived with help of some sums involving Legendre symbol
Explicit Reciprocity Laws in Iwasawa Theory -- A survey with some focus on the Lubin-Tate setting
Starting from Gau{\ss}' and Legendre's quadratic reciprocity law we want to
sketch how it gave rise to the development of higher and generalized
reciprocity laws and over all explicit reciprocity formulas in Iwasawa theory
Parallelisms & Lie Connections
The aim of this article is to study rational parallelisms of algebraic
varieties by means of the transcendence of their symmetries. The nature of this
transcendence is measured by a Galois group built from the Picard-Vessiot
theory of principal connections
Quadratic Residues and Non-Residues: Selected Topics
Number theory as a coherent mathematical subject started with the work of
Fermat in the decade from 1630 to 1640, but modern number theory, that is, the
systematic and mathematically rigorous development of the subject from
fundamental properties of the integers, began in 1801 with the appearance of
the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the
Disquisitiones deals with quadratic residues and nonresidues. Beginning with
these fundamental contributions of Gauss, the study of quadratic residues and
nonresidues has subsequently led directly to many of the key ideas and
techniques that are used everywhere in number theory today, and the primary
goal of these lectures is to use this study as a window through which to view
the development of some of those ideas and techniques. In pursuit of that goal,
we will employ methods from elementary, analytic, and combinatorial number
theory, as well as methods from the theory of algebraic numbers.Comment: xi+265 pp., 4 tables, 20 figures in Lecture Notes in Mathematics no.
2171, Springer, New York, 201
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