Browsing Mathematics by Authors
Now showing items 2131 of 31

New Selfdual Codes from 2 x 2 block circulant matrices, Group Rings and Neighbours of NeighboursGildea, Joe; Kaya, Abidin; Roberts, Adam; Taylor, Rhian; Tylyshchak, Alexander; University of Chester; Harmony Public Schools; Uzhgorod National University (American Institute of Mathematical Sciences, 20210901)In this paper, we construct new selfdual codes from a construction that involves a unique combination; $2 \times 2$ block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields selfdual codes. The theory is supported by the construction of selfdual codes over the rings $\FF_2$, $\FF_2+u\FF_2$ and $\FF_4+u\FF_4$. Using extensions and neighbours of codes, we construct $32$ new selfdual codes of length $68$. We construct 48 new best known singlyeven selfdual codes of length 96.

New SelfDual Codes of Length 68 from a 2 × 2 Block Matrix Construction and Group RingsBortos, Maria; Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; Uzhgorod National University; University of Chester; Harmony School of TechnologyMany generator matrices for constructing extremal binary selfdual codes of different lengths have the form G = (In  A); where In is the n x n identity matrix and A is the n x n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F2 +uF2 and by employing the extension method for codes, we were able to construct new extremal binary selfdual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to con struct many new binary selfdual [68,34,12]codes with the rare parameters $\gamma = 7$; $8$ and $9$ in $W_{68,2}$: In particular, we find 92 new binary selfdual [68,34,12]codes.

On hereditary reducibility of 2monomial matrices over commutative ringsBondarenko, Vitaliy M.; Gildea, Joe; Tylyshchak, Alexander; Yurchenko, Natalia; Institute of Mathematic, Kyiv; University of Chester; Uzhgorod National University (Taras Shevchenko National University of Luhansk, 2019)A 2monomial matrix over a commutative ring $R$ is by definition any matrix of the form $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0&tI_{nk}\end{smallmatrix}\right)$, $0<k<n$, where $t$ is a noninvertible element of $R$, $\Phi$ the compa\nion matrix to $\lambda^n1$ and $I_k$ the identity $k\times k$matrix. In this paper we introduce the notion of hereditary reducibility (for these matrices) and indicate one general condition of the introduced reducibility.

Quadruple Bordered Constructions of SelfDual Codes from Group RingsDougherty, Steven; Gildea, Joe; Kaya, Abidin; University of Scranton; University of Chester; Sampoerna University (Springer Verlag, 20190705)In this paper, we introduce a new bordered construction for selfdual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring F4 + uF4. We use groups of order 4, 12 and 20. We construct some extremal selfdual codes and nonextremal selfdual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal selfdual codes of length 68.

SelfDual Codes using Bisymmetric Matrices and Group RingsGildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; University of Chester ; Sampoerna University ; University of Chester: Uzhgorod National University (Elsevier, 20200814)In this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4 + uF4 together with the well known extension and neighbour methods, we construct new selfdual codes of length 68: In particular, we find 41 new codes of length 68 that were not known in the literature before.

Torsion Units for a Ree group, Tits group and a Steinberg triality groupGildea, Joe; University of Chester (Springer, 20151228)We investigate the Zassenhaus conjecture for the Steinberg triality group ${}^3D_4(2^3)$, Tits group ${}^2F_4(2)'$ and the Ree group ${}^2F_4(2)$. Consequently, we prove that the Prime Graph question is true for all three groups.

Torsion Units for Some Almost Simple GroupsGildea, Joe; University of Chester (Springer, 20160625)We prove that the Zassenhaus conjecture is true for $Aut(PSL(2,11))$. Additionally we prove that the Prime graph question is true for $Aut(PSL(2,13))$.

Torsion Units for some Projected Special Linear GroupsGildea, Joe; University of Chester (20151231)In this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently, we prove that the Prime graph question is true for both groups.

Torsion units for some untwisted exceptional groups of lie typeGildea, Joe; O'Brien, Killian; University of Chester ; Manchester Metropolitan University (Bolyai Institute, University of Szeged, 2016)In this paper, we investigate the Zassenhaus conjecture for exceptional groups of lie type $G_2(q)$ for $q=\{3,4\}$. Consequently, we prove that the Prime graph question is true for these groups.

Torsion units in the integral group ring of PSL(3,4)Gildea, Joe; Tylyshchak, Alexander; University of Chester ; Uzhgorod State University (World Scientific Publishing, 20150831)We investigate the Zassenhaus Conjecture for the integral group ring of the simple group PSL(3,4).

Units of the group algebra of the group $C_n\times D_6$ over any finite field of characteristic $3$Gildea, Joe; Taylor, Rhian; University of Chester (International Electronic Journal of Algebra, 20180705)In this paper, we establish the structure of the unit group of the group algebra ${\FF}_{3^t}(C_n\times D_6)$ for $n \geq 1$.