CAAFRTA
About ACA
Talks
Talk 1: Exponents of Skew Polynomials ( Ahmed Cherchem, André Leroy )
Abstract: We generalize the classical definition of the exponent, also known as order or period.
Talk 2: The Module Isomorphism Problem for Finite Rings and Related Results ( Iuliana Ciocanea Teodorescu)
Abstract: Let R be a finite ring and let M;N be two finite left R-modules. We present two distinct deterministic algorithms that decide in polynomial time whether or not M and N are isomorphic, and if they are, exhibit an isomorphism.
Talk 3: Construction of codes for DNA computing by the Greedy Algorithm ( Nabil Bennenni, Kenza Guenda, T. Aaron Gulliver )
Abstract: In this paper we construct codes for DNA computing using the greedy algorithm over Z4. We obtain linear codes over Z4 with bounded GC content. We also consider the edit distance, we gave upper bounds for the edit distance and construct codes with bounded edit distance.
Talk 4: Multivariable Codes in Principal Ideal Polynomial Quotient Rings ( Edgar Martínez-Moro, Alejandro P. Nicolas, Ignacio F. Rua )
Abstract: Multivariable codes over a finite field are a natural generalization of several classes of codes, including cyclic, negacyclic, constancyclic, polycyclic and abelian codes. Since these particular families have been also considered in the context of codes over a finite chain ring, we proposed constructions of multivariable codes over such a class of finite rings. As in the case of traditional cyclic codes over finite fields the modular case (i.e., codes with repeated roots) is much more difficult to handle than the semisimple case (i.e., codes with non-repeated roots). In this sense, different authors have dedicated their efforts to provide a better understanding of the properties of cyclic, negacyclic, constancylic and polycyclic modular codes over a finite chain ring. Among these codes, those contained in an ambient space which is a principal ideal ring admit a relatively simple description, quite close to that of semisimple. This feature has been recently used in the description of abelian codes over a finite field, and in the description of modular additive cyclic codes over F4. As a natural continuation of these works, in this paper we consider the structure of multivariable modular codes in an ambient space which is a principal ideal ring.
Talk 5: Linear Codes over Z4[x]/(x^2-2x). Dual Preserving Maps and Images as Codes over Z4 (Edgar Martinez-Moro, Steve Szabo, Bahattin Yildiz)
Abstract: The most general class of rings to considered working on coding theory over are Frobenius rings. Since finite commutative Frobenius rings are isomorphic to a direct sum of local Frobenius rings, it is important to understand local Frobenius rings. Chain rings have been extensively studied which are examples of local Frobenius rings. There are however non-chain examples as well. These local Frobenius non-chain rings have not garnered much attention until recently. We consider linear codes over Z4[x]/(x^2-2x), which is one of the seven local Frobenius non-chain rings of order 16. Order 16 is of importance since there are no local Frobenius rings of smaller order that are not chain rings. A dual preserving map is presented along with a characterization of Z4 linear codes that are images of a codes overZ4[x]/(x^2-2x).
Talk 6: Codes Over Rings of Order 16 (Steven Dougherty, Esengül Saltürk, Steve Szabo)
Abstract: We study codes over finite commutative local Frobenius rings of order 16. We define a standard form for the generator matrix for linear codes over these rings. Finally, we describe the generating characters for each ring which produce MacWilliams relations for codes over these rings.
Talk 7: Codes over local rings of order 16 and their Gray maps (Steven Dougherty, Esengül Saltürk)
Abstract: Weight preserving Gray maps are defined from any non-chain local ring of order 16 to the binary Hamming space. This is used to define the Lee weight for codes in this setting. MacWilliams relations for the weight enumerator with respect to the Lee weight are given. Self-dual codes are studied over these rings and they are used to study binary codes whose weight enumerators are held invariant by the action of the MacWilliams relations.
Talk 8: A New Non-Associative Cryptosystem Based on NTOW Public Key Cryptosystem and Octonions Algebra (Kadijeh Bagheri, Mohammad-Reza Sadeghi)
Abstract: In this work, we present a public key cryptosystem, called OTWO, based on octonions algebra and NTWO cryptosystem which is a multivariate version of NTRU. Inherent security of this system relies on the difficulty of the shortest vector problem (SVP) in a certain type of lattices with a hybrid norm.
Talk 9: Network Coding via Skew Polynomials (Felice Manganiello)
Abstract: In 2003 it was proven independently by Kötter and Mérdard, and Li et al. that linear network coding over a suitable finite field can be used to achieve the capacity of multicast networks. After this result, networks were connected to matroids. In 2007 Dougherty et al. showed how matroids can be deployed to construct matroidal networks. Gadouleau and Goupil in 2011 proved the achievability of the capacity of a multicast network by means of matroids instead of linear spaces. As a consequence of this result together with the fact that matroids can be perceived as a generalization of linear spaces, one can obtain an increase in the size of the codebooks used for communication. The ring of skew polynomials is a non commutative generalization of the classic univariate polynomial ring. The multiplication of the former obeys a non commutative multiplication rule between the variable and a scalar defined by an automorphism of the underlined field and a derivation map. The ring maintains the structure of a right Euclidean ring without zero divisors. At this point, one obtains a natural evaluation map for skew polynomials as the remainder of the right division by a monomial. It can be proven that the set of the zero locus of all of the skew polynomials using the aforementioned evaluation map, forms a matroid. Focusing on the case of skew polynomial rings over finite fields with trivial derivation map, it is possible to characterize the flats of this matroid and their sets of generators using minimal skew polynomials. We are going to explore this matroid structure and connect it to multicast communication.
Talk 10: Computer Algebra Challenges for Constructing Skew Cyclic Codes (Noah Aydin)
Abstract: One of the challenging problems of coding theory is to construct codes with best possible parameters. Computers and computer algebra systems are often used in achieving this goal. Since the computation of the minimum distance of a linear code is computationally intractable (NP-hard), it is necessary to focus on certain promising classes of codes with rich algebraic structures. Cyclic codes and their various generalizations, such as constacyclic, quasi-cyclic, and quasi-twisted codes have been subject to much research, both theoretical and computational, for decades. As a result, a large number of best-known codes come from these families. More recently, a new generalization of cyclic codes, called theta-cyclic codes or skew cyclic codes, have been introduced. The algebraic study of skew cyclic codes requires one to work in a non-commutative ring called skew polynomial ring. This introduces new computational challenges for computer algebra systems when it comes to implementing search algorithms for constructing skew cyclic and related codes over rings or fields. In this talk, we will describe some of these challenges.
Talk 11: On Gröbner Bases over Rings and Residue Class Polynomials Rings with Torsion (Ambedkar Dukkipati, Nithish Pai, Maria Francis, Abhishek Dubey)
Abstract: In this work we generalize the Macaulay-Buchberger basis theorem for residue class ring A[x1 , . . . , xn ]/a, in the case when it is finitely generated as an A-module but need not necessarily a free module, where A is a Noetherian ring and a is an ideal.