Quadratic Forms, Rings and Codes

July 8, 2021

Laboratoire de Mathématiques de Lens, Unité de Recherche UR 2462

Organizing committee:   Ahmed Laghribi     André Leroy

This one-day meeting will be held online and it is the second part of the conference "NonCommutative Rings and their Applications, VII" on ring theory. The meeting is supported by the Université d'Artois in the framework of the BQR support (Bonus Qualité Recherche), and the department of mathematics in Lens. The aim of this meeting is to gather specialists of quadratic forms, ring theory and coding theory. The hope is to encourage exchange between these areas of research.

We are very happy to announce the speakers of this meeting:

  • 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.
      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).

Schedule of the talks

Videos Gomez and Chapman
Berhuy; Sivatski and Polcino

List of participants

  1. Mona Abdi, Shahrood University of Technology, Shahrood (Iran).
  2. Mohammad Ashraf, Alighar Muslim University (India).
  3. Shakir Ali, Aligarh Muslim University, Aligarh (India)
  4. Grégory Berhuy, Institut Fourier, Grenoble (France).
  5. Mhammed Boulagouaz Universite Sidi Mohammed ben Abdellah Fes (Morocco).
  6. Ranya Djihad Boulanouar, Algiers University USTHB, Alger (Algeria).
  7. Gary Birkenmeier, University of Louisiana at Lafayette (USA).
  8. Jérome Burési, Université d'Artois (France).
  9. Baptiste Calmes, Université d'Artois (France).
  10. Adam Chapman, Academic College of Tel-Aviv-Yaffo (Israel)
  11. Steven Dougherty, Scranton University (USA).
  12. Müge Diril, Izmir Institute of Technology (Turkey).
  13. Fatma Ebrahim, Al-Azhar University, Cairo, (Egypt).
  14. Gabriella d'Este, University of Milano (Italy).
  15. Sergio estrada, Unversidad de Murcia (Spain).
  16. Walter Ferrer Santos, University de la republica, Montevideo (Uruguay).
  17. Dipak Kumar Bhunia, Autonomous University of Barcelona (Spain)
  18. Jose Gomez-Torrecillas, University of Granada (Spain)
  19. Franco Guerriero, Ohio University (USA).
  20. Ashok Ji Gupta, Indian Institue of technology(BHU), Varanasi (India).
  21. Surender K. Jain, Ohio University, Athens (USA).
  22. Yeliz Kara, Bursa Uludag University (Turkey).
  23. Arda Kor, Gebze Technical University (Turkey).
  24. Ahmed Laghribi, Université d'Artois (France).
  25. André Leroy, Université d'Artois (France).
  26. Sergio Lopez-Permouth, Ohio University, Athens (USA).
  27. Najib Mahdou, University S.M. Ben Abdellah, Fez (Morocco).
  28. Xavier Mary, Univerité Paris Nanterre (France).
  29. Intan Muchtadi-Alamsyah, Institut Teknologi Bandung (Indonesia).
  30. Diksha Mukhija, Université d'Artois (France)
  31. Cesar Polcino Milies, Sao Paulo University (Brazil).
  32. Louis Rowen, Bar-Ilan University (Israel).
  33. Serap Sahinkaya, Tarsus University (Turkey).
  34. Anuradha Sharma, Department of Mathematics, IIIT-Delhi, New Delhi (India).
  35. Sandeep Sharma, IIIT-Delhi, New Delhi (India).
  36. Virgilio P. Sison, University of the Philippines Los Baños (Philippines).
  37. Alexander S. Sivatski, Universidade Federal do Rio Grande do Norte (Brazil)
  38. Daniel Smertnig, University of Graz (Austria).
  39. Patrick Solé, CNRS Aix-Marseille et centrale Marseille (France).
  40. Blas Torrecillas Universidad de Almería (Spain).
  41. Sarra Talbi, U.S.T.H.B., Algiers(Algeria).
  42. Jean Pierre Tignol, Université de Louvain la neuve (Belgique).
  43. Monika Yadav, IIIT-Delhi, New Delhi (India).
  44. Tulay Yildirim, Karabuk University (Turkey).

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