 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 TelAvivYaffo)
 Title: nLinked fields of characteristic 2
 Abstract: We say that a field is nlinked if every n quaternion algebras over the field share a quadratic splitting field. 2Linked fields with quaternion division algebras have uinvariant 4 or 8, and they are 3linked if and only if their uinvariant is 4. The uinvariant is therefore insufficient for determining whether a field is 4linked or not.
Global fields are nlinked for any n, and the question of the existence of 3linked fields that are not 4linked 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
winvariant, introduced by Tignol in the 90's.
Here are the Slides
of this talk
 José GomezTorrecillas (Universidad de Grenada)
 Title: Decoding ReedSolomon Skewdifferential 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 paritycheck matrices, and the decoding algorithm rests upon
matrix and linear maps manipulations. The somewhat more sophisticated mathematical
context (noncommutative rings) needed for the proof of the correctness of the decoding
algorithm is postponed to the second part. A final section locates the ReedSolomon 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), 181188.
G.Chalom, R. Ferraz and C.Polcino Milies, Essencial idempotents and codes of constant weight, Sâo Paulo J. of Math. Sci, 11 (2) (2018), 253260
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 Fvaluation v on L denote by L_{v} 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 Fvaluation v on L such that the form φ_{Lv} 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 4dimensional 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 k_{0}(x, t)/k_{0}, where
k_{0} is an arbitrary not quadratically closed C_{1}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 finitedimensional central division algebra D over L there exists a discrete Fvaluation v on L such that the algebra D_{Lv} 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 2^{n} over F(t).

