• Double Bordered Constructions of Self-Dual Codes from Group Rings over Frobenius Rings

      Gildea, Joe; Kaya, Abidin; Taylor, Rhian; Tylyshchak, Alexander; University of Chester; Sampoerna University; Uzhgorod State University
      In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F2 + uF2 and F4 + uF4. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables
    • G-Codes, self-dual G-Codes and reversible G-Codes over the Ring Bj,k

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; Sahinkaya, Serap; Tarsus University; University of Chester (Springer, 2021-05-03)
      In this work, we study a new family of rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj,k to a code over Bl,m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.
    • 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.