Generalization of the lee weight to Ζpᵏ

We introduce a new extension of the Lee weight to Ζpᵏ and later to Galois rings GR(pᵏ,m). The weight we define is a non-homogeneous weight and is different than the one that is generally labeled as "generalized Lee weight". Unlike the case of generalized Lee weight, we define a distance-preserving Gray map from (Ζpᵏ, extended Lee distance)to (Fppᵏ⁻¹, Hamming distance), thus making our extension practical for coding theory purposes.Publisher's Versio

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Kodierungstheorie

[no abstract available

Artin’s Conjecture on primes with prescribed primitive roots

La Conjectura d’Artin sobre la densitat del conjunt de nombres primers amb una arrel primitiva prescrita és un dels problemes matemàtics oberts més fàcils d’enunciar. En la seva versió més clàssica, es planteja la següent pregunta: hi ha infinits nombres primers p tal que 2 és una arrel primitiva mòdul p? El propósit d’aquest treball és introduir les tècniques més importants que s’han utilitzat per donar resultats parcials en aquesta àrea. En particular, fem una revisió detallada de [LT65, Artin’s Observation], [Bil37], [Hoo67], [W77] i [KR20; KM22]. A la Secció 4.2.3 i a la Secci´o 5.2 l’autor ha pogut fer dues contribucions modestes a problemes oberts a l’area.La Conjetura de Artin sobre la densidad del conjunto de números primos con una raíz primitiva prescrita es uno de los problemas matemáticos abiertos más fáciles de enunciar. En su versión más clásica, plantea la siguiente pregunta: hay infinitos números primos p tal que 2 es una raíz primitiva módulo p? El propósito de este trabajo es introducir las técnicas más importantes que se han utilizado para dar resultados parciales en este área. En particular, hacemos una revisión detallada de [LT65, Artin’s Observation], [Bil37], [Hoo67], [W77] y [KR20; KM22]. En la Sección 4.2.3 y en la Sección 5.2 el autor ha podido hacer dos contribuciones modestas a problemas abiertos en el áreaArtin’s Conjecture about primes with a prescribed primitive root is one of the simplest to state open questions in mathematics. In its most classical form it asks the following question: are there infinitely many primes p such that 2 is a primitive root modulo p? The purpose of this work is to introduce some of the most important results towards answering this and related questions. In particular, we give an in depth review of [LT65, Artin’s Observation], [Bil37], [Hoo67], [W77] and [KR20; KM22]. In Section 4.2.3 and Section 5.2 the author has been able to make modest contributions about some open questions in the area.Outgoin

Codes Over Rings from Curves of Higher Genus

We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing “lifts” of these curves and then estimating the size of certain exponential sums. THE purpose of this paper is to construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools; notably, an estimate for the dimension of trace codes over rings (generalizing work of van der Vlugt over fields and some results on lifts of affin curves from field of characteristic p to Witt vectors of length two. This work partly generalizes our previous work on elliptic curves, although there are some differences which we will point out below

On a conjecture for ℓ\ell-torsion in class groups of number fields: from the perspective of moments

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments.