• New Self-dual Codes from 2 x 2 block circulant matrices, Group Rings and Neighbours of Neighbours

      Gildea, Joe; Kaya, Abidin; Roberts, Adam; Taylor, Rhian; Tylyshchak, Alexander; University of Chester; Harmony Public Schools; Uzhgorod National University (American Institute of Mathematical Sciences, 2021-09-01)
      In this paper, we construct new self-dual 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 self-dual codes. The theory is supported by the construction of self-dual 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 self-dual codes of length $68$. We construct 48 new best known singly-even self-dual codes of length 96.
    • New Self-Dual Codes of Length 68 from a 2 × 2 Block Matrix Construction and Group Rings

      Bortos, Maria; Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; Uzhgorod National University; University of Chester; Harmony School of Technology
      Many generator matrices for constructing extremal binary self-dual 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 self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to con- struct many new binary self-dual [68,34,12]-codes with the rare parameters $\gamma = 7$; $8$ and $9$ in $W_{68,2}$: In particular, we find 92 new binary self-dual [68,34,12]-codes.
    • On hereditary reducibility of 2-monomial matrices over commutative rings

      Bondarenko, 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 2-monomial matrix over a commutative ring $R$ is by definition any matrix of the form $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0&tI_{n-k}\end{smallmatrix}\right)$, $0<k<n$, where $t$ is a non-invertible element of $R$, $\Phi$ the compa\-nion matrix to $\lambda^n-1$ 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 Self-Dual Codes from Group Rings

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; University of Scranton; University of Chester; Sampoerna University (Springer Verlag, 2019-07-05)
      In this paper, we introduce a new bordered construction for self-dual 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 self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68.
    • Self-Dual Codes using Bisymmetric Matrices and Group Rings

      Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; University of Chester ; Sampoerna University ; University of Chester: Uzhgorod National University (Elsevier, 2020-08-14)
      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 self-dual 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 group

      Gildea, Joe; University of Chester (Springer, 2015-12-28)
      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 Groups

      Gildea, Joe; University of Chester (Springer, 2016-06-25)
      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 Groups

      Gildea, Joe; University of Chester (2015-12-31)
      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 type

      Gildea, 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, 2015-08-31)
      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, 2018-07-05)
      In this paper, we establish the structure of the unit group of the group algebra ${\FF}_{3^t}(C_n\times D_6)$ for $n \geq 1$.