SEMINAR SERIES

WORKING SEMINAR

COLLOQUIA

INVITED PEOPLE

UNDERGRADUATE TALKS - COFFE TALKS

PHOTOS

Welcome

During the Spring Semester we will have a series of Colloquia, Seminars and Working Seminar on Coding Theory and Related Topics at Eastern Kentucky University. The seminar will be in charge of Dr. Edgar Martinez-Moro, Vernon Wilson Endowed Chair in EKU's Department of Mathematics and Statistics and Assistant Professor at Valladolid University (Spain) and Steve Szabo EKU's Department of Mathematics and Statistics.

SEMINAR SERIES

Seminar: Canonical structures on coding theory

Edgar Martinez and Steve Szabo -- Wed. 15:30 at Wallace 434.

During the first two sessions we will cover the basic facts on coding theory as well as some aspects on the esay and hard problems sections in Barg's paper Complexity issues in Coding Theory. After we will review how the hard problems can be seen as a reduction process in an algebraic setting. A complete study of this last part can be seen in Irene Marquez's PhD thesis.

Slides 1     Slides 2   Slides 3    Slides 4    Slides 5

Extra material:

• What is a Groebner basis?, By Bernd Sturmfels. Notices of the AMS, Vol. 52, 10 (Nov. 2005)
• What is a Syzygy?, By Roger Wiegand. Notices of the AMS, Vol. 53, 4 (Apr. 2006)

Seminar talk: Interesting alphabets and weights for algebraic coding theory

Steven Dougherty (University of Scranton) -- Feb. 27th, 15:35 at Wallace 434.

We shall describe various alphabets and weights used in coding theory and explain the connections between these spaces. Various families of codes will be described in these spaces including self-dual codes and cyclic codes and we shall show how these types of codes have interesting connections to other objects.

Slides 1     Slides 2

Seminar talk series: Linear codes over finite rings and modules

Jay Wood (Western Michigan University) -- Weeek of March the 4th.

The 1962 doctoral dissertation of Jessie MacWilliams contained two theorems, the MacWilliams extension theorem and the MacWilliams identities, which have proved valuable for the mathematical understanding of linear codes defined over finite fields. I will describe the generalizations of these theorems in the context of linear codes defined over finite rings and modules.

Slides 1     Slides 2   Slides 3

Talks can also be find here.

1. Seminar 1: The MacWilliams identities
   -- Mon. 15:30 at Wallace 434.
   
   The MacWilliams identities relate the weight enumerator of a linear code to the weight enumerator of its dual code. For self-dual codes, the MacWilliams identities place strong restrictions on possible weight enumerators. A form of the identities will first be proved for additive codes using character theory. Then, by using certain identifications, the identities will be derived for linear codes over finite Frobenius rings.
   
   
   
2. Seminar 2: The MacWilliams extension theorem over Frobenius rings
   -- Tue. 15:30 at Wallace 345.
   
   The original MacWilliams extension theorem says that a linear transformation between linear codes that preserves the Hamming weight must extend to a monomial transformation of the ambient space. This theorem generalizes to linear codes over finite Frobenius rings and, in fact, provides a characterization of those rings. The proof requires a generalization to linear codes with module alphabets.
   
   
   
3. Seminar 3: The MacWilliams extension theorem for general weights
   -- Wed. 15:30 at Wallace 434.
   
   While we understand the extension theorem for the Hamming weight, our knowledge of what happens for other weights is surprisingly small. I will describe some of what is known about the extension theorem for different choices of weights.

Seminar talk series: Duality and MacWilliams Identities in Coding Theory

Heide Gluesing-Luerssen (University of Kentucky) -- Th. March 21 3:30-4:45 and Tue. March 26 3:30-4:45. Place Wallace 345.

In this talk we will consider the vast field of MacWilliams identities for codes over groups and rings and present an approach based on partitions that enjoy a certain invariance property under the Fourier transform. This will allow us to recover MacWilliams identities for many well-known cases, where we can also give shorter and sometimes more insightful proofs. We will also encounter a few more cases for which a MacWilliams identity can be derived. Special attention will be paid to a) the homogeneous weight, b) poset structures, and, if time permits, c) the rank metric.

Seminar talk series: Bent Functions, Their Generalizations and Relatives

Xiang-dong Hou (University of South Florida) -- Wed April 3, 3:30-4:45. Place Wallace 345.

Abstract.

Seminar talk series: Ordered linear codes

Alexander Barg (University of Maryland) -- Wed. April 10 3:30-4:45. Place Wallace 424.

Slides.

Seminar talk series: On trace codes, duality and Galois invariance over finite commutative chain rings

Edgar Martinez-Moro (University of Valladolid) --Wed. April 17 3:30-4:45. Place Wallace 424.

Codes over finite commutative chain rings have been introduced as a generalization of codes over finite fields. Let S | R be a Galois extension of finite commutative chain rings. If C ⊆ S^n is an S-code, it is possible to define, starting from C , two different R-codes: Res(C ) = C ∩ R^n and Tr( C ), where Tr is the trace function. We will analyze the relationships between these R-codes and the duality operator.

Slides.

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WORKING SEMINAR

Mon. 15:30 at Wallace 111 (Library).

It will be mostly devoted to study and postgraduate student presentations of the following papers:

Paper 1     Paper 2   Paper 3

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COLLOQUIA

• Feb. 11 th, 14:30, Wallace 426
  
  Edgar Martinez-Moro Code based cryptography.
  
  Abstract: All Public Key Cryptosystems (PKCs) are based on the (computational) hardness of some mathematical problem. RSA or hyperelliptic curve cryptosystem are based on the hardness of factorisation and of the discrete log respectively. P. Shor showed how, employing a quantum computer, this problems can be solved. This thread has reinitialised the research on some alternative PKCs that are not broken with Shor's approach.
  This is the case of McEliece PKC based on the hardness of decoding a "random code". Unfortunately the "good guys" need to know how to decode in order to implement the PKC and therefore the code is not so "random". We will see at the end of the talk how some extensions of some classical results on geometry as "the number of points defining a conic" can help us in breaking McEliece PKC in the case that the code is an Algebraic Geometry code.
  
  Slides. Related papers: Paper 1, Paper 2.



• Feb. 26th, 15:30, Wallace 344
  
  Steven Dougherty (University of Scranton) Foundations of Coding Theory as Pure Mathematics.
  
  Abstract: We describe the origins of coding theory and show how it evolves into a branch of pure mathematics with roots in abstract algebra and combinatorics.

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INVITED PEOPLE

The following people will be visitting us during the semester. They will collaborate in the seminars and colloquia as well as lecture some undergraduate talks.

Steven Dougherty (University of Scranton)	Week of Feb. 25th
Jay Wood (Western Michigan University)	Week of Mar. 4th
Heide Gluesing-Luerssen (University of Kentucky)	Week of Mar. 25th
Xiang-Dong Hou (University of South Florida)	Week of Apr. 1st
Alexander Barg (University of Maryland)	Week of Apr. 18th
Sergio Lopez-Permouth (Ohio University)	Week of Mar. 4th

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UNDERGRADUATE TALKS - COFFEE TALKS

• Feb. 28th, 17:00, Wallace 328.
  
  Steven Dougherty (University of Scranton) Japanese ladders and games.
  
  Abstract: We shall describe a visual representation of permutations as Japanese ladders and use this representation to make a series of interesting mathematical games. These games have interesting mathematical aspects but can be played by anyone. The ladders have applications to the Braid group in Topology.
  
  
• April 2nd, 17:00, Wallace 328.
  
  Xiang-dong Hou (University of South Florida) Sums of Reciprocals of Polynomials Over Finite Fields .
  
  Abstract.

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