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We consider a wireless relay network, where a transmitter node communicates with a receiver node with the help of relay nodes. Most coding strategies considered so far assume that the relay nodes are used for one hop. We address the problem of code design when relay nodes may be used for more than one hop. We consider as a protocol a more elaborated version of amplify-and-forward, called distributed space-time coding, where the relay nodes multiply their received signal with a unitary matrix, in such a way that the receiver senses a space-time code. We first show that in this scenario, as expected, the so-called full-diversity condition holds, namely, the codebook of distributed space-time codewords has to be designed such that the difference of any two distinct codewords is full rank. We then compute the diversity of the channel, and show that it is given by the minimum number of relay nodes among the hops. We finally give a systematic way of building fully diverse codebooks and provide simulation results for their performance.

5th International Conference Noncommutative Rings and their Applications -- JUN 12-15, 2017 -- Univ Artois, Lens, FRANCE WOS: 000473290900019 In this paper, we deal with some versions of reversibility and symmetricity on amalgamated rings along an ideal. Ahi Evran University Scientific Research Projects Coordination UnitAhi Evran University [FEF.A4.16.001] The first author was supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: FEF.A4.16.001.

This thesis is dedicated to the constructions of modular lattices with algebraic methods. The goal is to develop new methods as well as constructing new lattices. There are three methods considered: construction from number fields, construction from totally definite quaternion algebras over number fields and construction from linear codes via generalized Construction A. The construction of Arakelov-modular lattices, which result in modular lattices, was first introduced in [6] for ideal lattices from cyclotomic fields. We generalize this construction to other number fields and also to totally definite quaternion algebras over number fields. We give the characterization of Arakelov-modular lattices over the maximal real subfield of a cyclotomic field with prime power degree and totally real Galois fields with odd degrees. Characterizations of Arakelov-modular lattices of trace type, which are special cases of Arakelov- modular lattices, are given for quadratic fields and maximal real subfields of cyclotomic fields with non-prime power degrees. Furthermore, we give the classification of Arakelov-modular lattices of level l for l a prime over totally definite quaternion algebras with base field the field of rationals. Construction A is a well studied method to obtain lattices from codes via quotient of different rings, such as rings of integers, in which case mostly cyclotomic number fields have been considered. In this thesis, we will study Construction A over all totally real and CM fields. Using Construction A, the intersection between a lattice constructed from a linear complementary dual (LCD) code and its dual lattice is investigated. This is an attempt to find an equivalent definition to LCD codes for lattices. Several new constructions of existing extremal lattices as well as a new extremal lattice are obtained from the above mentioned methods. The mathematical concepts used in this thesis mainly involve algebraic number theory, class field theory, non commutative algebra and coding theory. ​Doctor of Philosophy (SPMS)

This thesis is dedicated to the study of information inequalities and quasi-uniform codes using group theory. Understanding the region of entropic vectors for dimension n ≥ 4 is an open problem in network information theory. It can be studied using information inequalities and their violations. The connection between entropic vectors and finite groups, known as 'group representability', is a useful tool to compute these violations. In the first part of this thesis we address the problem of extracting 'abelian group representable' vectors out of the whole set of group representable vectors. We prove that certain classes of non-abelian groups are abelian group representable and non-nilpotent groups are not abelian group representable. We then address the question of finding linear inequality violators for n = 5 and obtain the smallest group violators of two linear inequalities. Random variables which are uniformly distributed over their support are known as quasi-uniform. One way of getting quasi-uniform random variables is by using finite groups and subgroups. Codes are constructed in such a way that the associated random variables are quasi-uniform and group theory is used to construct such codes. In the second part of this thesis, we consider the construction of quasi-uniform codes coming from groups and their algebraic properties. We compute some coding parameters and bounds in terms of groups. Finally we propose some applications of quasi-uniform codes especially to distributed storage. DOCTOR OF PHILOSOPHY (SPMS)

This thesis contributes to the study of designing erasure codes for distributed storage systems. Erasure codes have traditionally been studied in the context of communication theory, but have also increasingly been used to ensure the reliability of distributed storage systems, since they provide a good trade-off between storage overhead (or rate) and reliability in case of storage node failures. There is however a critical difference between erasure codes for communication and for storage application: in the context of storage, reliability needs to be ensured over time. Failures of storage nodes keep happening, and the lost data must be rebuilt, to guarantee resiliency over time. This process is often called maintenance, or repair. Over the past years, there has been a vibrant research activity around the problem of designing good storage erasure codes, amenable to repair. A network coding inspired line of works was introduced by Dimakis et al. [13], who used a max-flow min-cut analysis to characterize the communication cost needed for repairing one failed node as a function of the code storage overhead. This setting has been generalized independently in [4, 5] to allow the simultaneous repair of several nodes failures thanks to node collaboration during the repair process, resulting in a better trade-off in terms of repair communication cost versus storage overhead than in the case of one failure repair at a time. The main contribution of this thesis is to introduce a family of storage codes that generalizes the framework of collaborative repairs, which subsumes the previously known constructions with either no collaboration (one failure at a time) or full collaboration (several failures at a time, involving all repair nodes in the collaboration as proposed in [4,5]). The motivation is twofold: to allow more degrees of freedom in the code design to best suit possible storage system requirements, and to trade gain in repair bandwidth with security impairments that typically appear in the presence of collaboration. More precisely, this thesis contains a trade-off between storage overhead and repair bandwidth as a function of the collaboration degree of the repair nodes involved, and code constructions: codes for the special cases where either the storage overhead or the repair bandwidth is minimized, and codes derived from group actions on Grassmannian spaces, known as orbit codes. Finally, the degree of collaboration is studied as a function of the confidentiality one would like to obtain in the presence of eavesdroppers. DOCTOR OF PHILOSOPHY (SPMS)

International audience; We extend the classical Reed Muller codes by using non commu-tative iterated skew polynomial rings instead of classical commutative polynomial rings. This involves the construction of iterated skew polynomial rings and the definition of the notion of points and evaluation at those points for iterated skew polynomials. Our approach is based on the notion of a left module Gröbner basis in iterated skew polynomial rings.

We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property. This condition applies to bi-invariant weights on a finite Frobenius ring as a special case. The complex-valued functions on a Frobenius bimodule are viewed as a module over the semigroup ring of the multiplicative semigroup of the coefficient ring., 15 pages

This thesis is dedicated to the Gaussian wiretap channel coding problem. In particular, lattice Gaussian wiretap codes are considered with respect to a new lattice coding invariant called secrecy gain defined as the maximum of the secrecy function, which characterizes the amount of confusion a lattice can cause at the eavesdropper. The weak secrecy gain is the value of the secrecy function achieved at its symmetry point, conjectured to be maximum, namely, the secrecy gain. The secrecy gain of unimodular lattices, i.e. lattices that are eqeal to their duals, in dimensions less than 24 are computed and the best lattice in each dimension is classified. The computation relies on the study of modular forms, a classical object in analytic number theory. Contemporary subject such as coding theory is also exploited in the computation. The weak secrecy gain of 2- and 3-modular lattices, lattices that are similar to their duals, in small dimensions are computed to compare with the best unimodular lattices. The data shows that, at least in small dimensions, 2- and 3-modular lattices typically have a bigger secrecy gain and hence give more secure lattice Gaussian wiretap codes. Modular lattices are constructed from the ring of integers of algebraic number fields. The computation in this part involves basic algebraic number theory, on top of heavy modular form theory and coding theory. Bounds on the weak secrecy gain of unimodular lattices are studied and the result shows that to maximize the weak secrecy gain of a unimodular lattice of dimension n, one has only to minimize the number of vectors with norm no bigger than ⌊n/ 8⌋ This facilitates a way to bound the weak secrecy gain of unimodular lattices by counting the number of their small norm vectors. ​Doctor of Philosophy (SPMS)

56 p. In this master thesis we study a special kind of central simple algebras called crossed product algebras. We will show how to construct them and study some of their properties, mainly their representation theory and divisibility. In particular, Brauer groups are employed to study the divisibility of crossed product algebras. A special kind of crossed product algebra called cyclic algebra has relatively simple structure and is particularly interesting. We compute the representation and Brauer group, of cyclic algebras and construct a new cyclic algebra with higher degree by tensoring two cyclic algebras with lower degree. Motivated by a coding theory problem, we next discuss the existence of unitary involution over crossed product algebras. We will show that the Hermitian conjugation of the representation matrix can be translated into the assumption of this involution. These generalize the results in [51. Finally we discuss how to find unitary elements in crossed product algebras. ​Master of Science