Open Access. Powered by Scholars. Published by Universities.®
- Publication Year
- Publication Type
Articles 1 - 14 of 14
Full-Text Articles in Entire DC Network
Partitioning Groups With Difference Sets, Rebecca Funke
Partitioning Groups With Difference Sets, Rebecca Funke
Honors Theses
This thesis explores the use of difference sets to partition algebraic groups. Difference sets are a tool belonging to both group theory and combinatorics that provide symmetric properties that can be map into over mathematical fields such as design theory or coding theory. In my work, I will be taking algebraic groups and partitioning them into a subgroup and multiple McFarland difference sets. This partitioning can then be mapped to an association scheme. This bridge between difference sets and association schemes have important contributions to coding theory.
Cameron-Liebler Line Classes And Partial Difference Sets, Uthaipon Tantipongipat
Cameron-Liebler Line Classes And Partial Difference Sets, Uthaipon Tantipongipat
Honors Theses
The work consists of three parts. The first is a study of Cameron-Liebler line classes which receive much attention recently. We studied a new construction of infinite family of Cameron-Liebler line classes presented in the paper by Tao Feng, Koji Momihara, and Qing Xiang (rst introduced in 2014), and summarized our attempts to generalize this construction to discover any new Cameron-Liebler line classes or partial difference sets (PDSs) resulting from the Cameron-Liebler line classes. The second is our approach to finding PDS in non-elementary abelian groups. Our attempt eventually led to the same general construction of PDS presented in John …
Difference Sets In Non-Abelian Groups Of Order 256, Taylor Applebaum
Difference Sets In Non-Abelian Groups Of Order 256, Taylor Applebaum
Honors Theses
This paper considers the problem of determining which of the 56092 groups of order 256 contain (256; 120; 56; 64) difference sets. John Dillon at the National Security Agency communicated 724 groups which were still open as of August 2012. In this paper, we present a construction method for groups containing a normal subgroup isomorphic to Z4 Z4 Z2 . This construction method was able to produce difference sets in 643 of the 649 unsolved groups with the correct normal subgroup. These constructions elimated approximately 90% of the open cases, leaving 81 remaining unsolved groups.
G-Perfect Nonlinear Functions, James A. Davis, Laurent Poinsot
G-Perfect Nonlinear Functions, James A. Davis, Laurent Poinsot
Department of Math & Statistics Faculty Publications
Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both Z2 -valued and Za …
Some Recent Developments In Difference Sets, James A. Davis, Jonathan Jedwab
Some Recent Developments In Difference Sets, James A. Davis, Jonathan Jedwab
Department of Math & Statistics Faculty Publications
There are five known parameter families for (v, k, λ, n)- difference sets satisfying gcd(v, n)>1: the Hadamard, McFarland, Spence, Davis-Jedwab, and Chen families. The authors recently gave a recursive unifying construction for difference sets from the first four families which relies on relative difference sets. We give an overview of this construction and show that, by modifying it to use divisible difference sets in place of relative difference sets, the recent difference set discoveries of Chen can be brought within the unifying framework. We also demonstrate the recursive use of an auxiliary construction for …
Hadamard Difference Sets In Nonabelian 2-Groups With High Exponent, James A. Davis, Joel E. Iiams
Hadamard Difference Sets In Nonabelian 2-Groups With High Exponent, James A. Davis, Joel E. Iiams
Department of Math & Statistics Faculty Publications
Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22t + 2 has a difference set if and only if the exponent of the group is less than or equal to 2t + 2. In a previous work (R. A. Liebler and K. W. Smith, in “Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order …
On Some New Constructions Of Difference Sets, Sarah Agnes Spence
On Some New Constructions Of Difference Sets, Sarah Agnes Spence
Honors Theses
Difference sets are mathematical structures which arise in algebra and combinatorics, with applications in coding theory. The fundamental question is when and how one can construct difference sets. This largely expository paper looks at standard construction methods and describes recent findings that resulted in new families of difference sets. This paper provides explicit examples of difference sets that arise from the recent constructions. By gaining a thorough understanding of these new techniques, it may be possible to generalize the results to find additional new families of difference sets. The paper also introduces partial and relative difference sets and discusses how …
A Nonexistence Result For Abelian Menon Difference Sets Using Perfect Binary Arrays, K. T. Arasu, James A. Davis, Jonathan Jedwab
A Nonexistence Result For Abelian Menon Difference Sets Using Perfect Binary Arrays, K. T. Arasu, James A. Davis, Jonathan Jedwab
Department of Math & Statistics Faculty Publications
A Menon difference set has the parameters (4N2, 2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group H×K×Zpα contains a Menon difference set, where p is an odd prime, |K|=pα, and pj≡−1 (mod exp (H)) for some j. Using the viewpoint of perfect binary arrays we prove that K must be cyclic. A …
Research Announcement: Recursive Construction For Families Of Difference Sets, James A. Davis, Jonathan Jedwab
Research Announcement: Recursive Construction For Families Of Difference Sets, James A. Davis, Jonathan Jedwab
Department of Math & Statistics Faculty Publications
A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d1d2-1 : d1, d2, ∈ D} contains each nonzero element of G exactly λ times; n = k-λ.
New Constructions Of Menon Difference Sets, K. T. Arasu, James A. Davis, Jonathan Jedwab, Surinder K. Sehgal
New Constructions Of Menon Difference Sets, K. T. Arasu, James A. Davis, Jonathan Jedwab, Surinder K. Sehgal
Department of Math & Statistics Faculty Publications
Menon difference sets have parameters (4N2, 2N2 − N, N2 − N). These have been constructed for N = 2a3b, 0 ⩽ a,b, but the only known constructions in abelian groups require that the Sylow 3-subgroup be elementary abelian (there are some nonabelian examples). This paper provides a construction of difference sets in higher exponent groups, and this provides new examples of perfect binary arrays.
Symmetric Designs, Difference Sets, And A New Way To Look At Macfarland Difference Sets, John Bowen Polhill Jr.
Symmetric Designs, Difference Sets, And A New Way To Look At Macfarland Difference Sets, John Bowen Polhill Jr.
Honors Theses
In this paper, the topics of symmetric designs and difference sets are discussed both separately and in relation to each other. Then an approach to MacFarland Difference Sets using the theory behind homomorphisms from groups into the complex numbers is introduced. This method is contrasted with the method of finding this type of difference set used by E.S. Launder in his book Symmetric Designs: An Algebraic Approach.
A Generalization Of Kraemer's Result On Difference Sets, James A. Davis
A Generalization Of Kraemer's Result On Difference Sets, James A. Davis
Department of Math & Statistics Faculty Publications
Kraemer has shown that every abelian group of order 22d+ 2 with exponent less than 22d+ 3 has a difference set. Generalizing this result, we show that any nonabelian group with a central subgroup of size 2d+ 1 together with an exponent-like condition will have a difference set.
Difference Sets In Abelian 2-Groups, James A. Davis
Difference Sets In Abelian 2-Groups, James A. Davis
Department of Math & Statistics Faculty Publications
Examples of difference sets are given for large classes of abelian groups of order 22d + 2. This fills in the gap of knowledge between Turyn's exponent condition and Dillon's rank condition. Specifically, it is shown thatℤ/(2d)×ℤ/(2d+2) andℤ/(2d+1)×Z/(2d+1) both admit difference sets, and these have many implications.
A Note On Intersection Numbers Of Difference Sets, K. T. Arasu, James A. Davis, Dieter Jungnickel, Alexander Pott
A Note On Intersection Numbers Of Difference Sets, K. T. Arasu, James A. Davis, Dieter Jungnickel, Alexander Pott
Department of Math & Statistics Faculty Publications
We present a condition on the intersection numbers of difference sets which follows from a result of Jungnickel and Pott [3]. We apply this condition to rule out several putative (non-abelian) difference sets and to correct erroneous proofs of Lander [4] for the nonexistence of (352, 27, 2)- and (122, 37, 12)-difference sets.