- Grégory Berhuy (Institut Fourier)
- Title:A quadratic form approach to Construction A
- Abstract:
We propose a construction of lattices from (skew-) polynomial codes,
by endowing quotients of some ideals in both number fields and cyclic
algebras with a suitable trace form. We give criteria for
unimodularity. This yields integral and unimodular lattices with a
multiplicative structure.
- Adam Chapman (Academic College of Tel-Aviv-Yaffo)
- Title: n-Linked fields of characteristic 2
- Abstract: We say that a field is n-linked if every n quaternion algebras over the field share a quadratic splitting field. 2-Linked fields with quaternion division algebras have u-invariant 4 or 8, and they are 3-linked if and only if their u-invariant is 4. The u-invariant is therefore insufficient for determining whether a field is 4-linked or not.
Global fields are n-linked for any n, and the question of the existence of 3-linked fields that are not 4-linked arose.
In a joint work with Tignol, the speaker showed that the function field in two variables over the complex numbers is such a field.
In a more recent paper, the speaker proved that the function field in two variables over an algebraically
closed field of characteristic 2 is another such field. The proof involves the lesser known
w-invariant, introduced by Tignol in the 90's.
Here are the Slides
of this talk
- José Gomez-Torrecillas (Universidad de Grenada)
- Title: Decoding Reed-Solomon Skew-differential codes
- Abstract: A large class of MDS linear codes is constructed. These codes are endowed
with an efficient decoding algorithm. Both the definition of the codes and the design of
their decoding algorithm only require from Linear Algebra methods, making them fully
accesible for everyone. Thus, the first part of the paper develops a direct presentation
of the codes by means of parity-check matrices, and the decoding algorithm rests upon
matrix and linear maps manipulations. The somewhat more sophisticated mathematical
context (non-commutative rings) needed for the proof of the correctness of the decoding
algorithm is postponed to the second part. A final section locates the Reed-Solomon skewdifferential codes introduced here within the general context of codes defined by means of
skew polynomial rings.
Here are the Slides
of this talk.
- César Polcino Milies (Universidade de São Paulo)
- Title: Essential idempotent in nilpotent group codes
- Abstract:We recall the definition of essential idempotents and its implications for cyclic and Abelian
codes. Then, we consider nilpotent group codes, i.e. codes that can be realized as ideals in the
finite (semisimple) group algebra of a nilpotent group. We discuss the existence of essential
idempotents in this context and study properties of minimal nilpotent codes.
bibliography
G.Chalom, R. Ferraz and C.Polcino Milies, Essential idempotents and simplex codes, (Algebra, Discrete Structures and Appl., 4, 2 (2017), 181-188.
G.Chalom, R. Ferraz and C.Polcino Milies, Essencial idempotents and codes of constant weight, Sâo Paulo J. of Math. Sci, 11 (2) (2018), 253-260
A. Duarte, On nilpotent and constacyclic codes, tese de doutoramento, Universidade Federal dO ABC, Santo André, Brazil, 2021.
Here are the Slides
of this talk.
- Alexander S. Sivatski (Universidade Federal do Rio Grande do Norte)
- Title: On the strong Hasse principle for rational field extensions:
- Abstract: Let F be a field of characteristic different from 2, let L/F be a field
extension. For a discrete F-valuation v on L denote by Lv the completion of L with
respect to v. We say that the strong Hasse principle (SHP) holds for the extension
L/F and anisotropic quadratic forms over L if for any anisotropic quadratic form φ
over L there exists a discrete F-valuation v on L such that the form φLv remains
anisotropic.
We investigate SHP for quadratic forms and the extension k(X)(t)/k(X), where
X is an arbitrary variety over an algebraically closed field k with dim X = 3. We
show that if dim X = 1, there are counterexamples to SHP for 4-dimensional forms
over k(X)(t). If dim X = 2, there are counterexamples in dimensions 5 and 6. If
dim X = 3, there are counterexamples in dimension 9.
Also we consider SHP for quadratic forms and the extension k0(x, t)/k0, where
k0 is an arbitrary not quadratically closed C1-field. We prove that in this case there
are counterexamples in dimensions 5 and 6.
Similarly to quadratic forms one can consider SHP for central division algebras.
We say that SHP holds for the extension L/F and central division algebras over L if
for any finite-dimensional central division algebra D over L there exists a discrete F-valuation v on L such that the algebra DLv remains a division one. For any n = 2 we
construct counterexamples to SHP for some extensions F(t)/F and central division
algebras of exponent two and index 2n over F(t).
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