Constructions of maximum rank distance codes, cyclic constant dimension codes, and subspace packings

Download
2018
Otal, Kamil
In this thesis, we aim to introduce main contributions to solve three main problems in coding theory. The first problem investigates the construction of inequivalent maximum rank distance (MRD) codes. Namely, we look for the constructions of the largest possible sets of $mtimes n$ matrices over a finite field $F_q$, such that the rank of the subtraction of any two different matrices in the set cannot be smaller than a certain number. Constructions of such codes under a suitable equivalence notion have taken a worthwhile attention in the last decade due to their applications in many areas, and most of the constructions including also our works have been discovered in last few years. We introduce these outcomes classifying them considering the main and most general equivalence idea. We basically use the language of linearized polynomials in this direction as usual in many works in the literature. The second problem, which is originated from an application related to the efficiency in random network coding, concerns the construction of large cyclic subspace codes of constant dimension. In this set up, we aim to construct large sets of $k$-dimensional subspaces of $F_q^n$ in a way that any two distinct subspaces cannot be close to each other more than a certain number in terms of the subspace distance, and the cyclic shifts of each subspace must be included in the set. We give the only systematic construction of such sets in the literature utilizing linearized polynomials again but in a slightly different way. We note that the basic structure of this construction was proposed in our another work. Additionally, we summarize the history of the solution and some further remarks. In the last problem, we focus on the constructions of subspace packings, which are the $q$-analogue of packing designs. This notion is a natural generalization of constant dimension codes, and has applications in the analysis of different network codes. We give a recursive construction of such codes using a generalization of the linkage construction rather than linearized polynomials. In particular, we make use of the matrix version of MRD codes together with some facts from linear algebra. This result is one of the main outcomes of our recent work. We express that these problems are different from but substantially related to each other. Connections among them are also expressed in related places. Furthermore, we remark that various areas of mathematics are used to solve these problems in general, e.g. finite geometry, algebraic geometry, algebra, and linear algebra. Therefore, it is not easy to introduce all advances properly with their complete preliminary information here. Moreover, we try to keep our language as simple as possible, and follow the historical journey of the advances. In that way, we target that this thesis can addresses to a more general reader group.

Suggestions

Contributions on plateaued (vectorial) functions for symmetric cryptography and coding theory
Sınak, Ahmet; Özbudak, Ferruh; Mesnager, Sihem; Department of Cryptography (2017)
Plateaued functions, used to construct nonlinear functions and linear codes, play a significant role in cryptography and coding theory. They can possess various desirable cryptographic properties such as high nonlinearity, low autocorrelation, resiliency, propagation criteria, balanced-ness and correlation immunity. In fact, they provide the best possible compromise between resiliency order and nonlinearity. Besides they resist against linear cryptanalysis and fast correlation attacks due to their low Walsh...
Computing cryptographic properties of Boolean functions from the algebraic normal orm representation
Çalık, Çağdaş; Doğanaksoy, Ali; Department of Cryptography (2013)
Boolean functions play an important role in the design and analysis of symmetric-key cryptosystems, as well as having applications in other fields such as coding theory. Boolean functions acting on large number of inputs introduces the problem of computing the cryptographic properties. Traditional methods of computing these properties involve transformations which require computation and memory resources exponential in the number of input variables. When the number of inputs is large, Boolean functions are ...
Characterizations of Partially Bent and Plateaued Functions over Finite Fields
Mesnager, Sihem; Özbudak, Ferruh; SINAK, AHMET (2018-12-30)
Partially bent and plateaued functions over finite fields have significant applications in cryptography, sequence theory, coding theory, design theory and combinatorics. They have been extensively studied due to their various desirable cryptographic properties. In this paper, we study on characterizations of partially bent and plateaued functions over finite fields, with the aim of clarifying their structure. We first redefine the notion of partially bent functions over any finite field Fq , with q a prim...
Aspects of coding theory with two recent applications
Bodur, Şeyma; Özbudak, Ferruh; Department of Cryptography (2019)
Coding Theory is a deep subject having a lot of applications in different areas. In this thesis we explain some background for two recent applications: Code base Cryptography, Entanglement Assisted Quantum Error-Correcting Codes (EAQECC).
Non-linear programming models for sector and policy analysis
Bauer, Siegfried; Kasnakoglu, Haluk (Elsevier BV, 1990-7)
This paper examines the basic problems of the mathematical programming models used for agricultural sector and policy analysis. Experience with traditional programming models shows that a considerable improvement in performance is possible by adequately incorporating non-linear relationships. Particular emphasis will be given to the calibration and validation problems involved in this type of model. With the help of the Turkish agricultural sector model it will be demonstrated that an empirical specificatio...
Citation Formats
K. Otal, “Constructions of maximum rank distance codes, cyclic constant dimension codes, and subspace packings,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.