Abidin Kaya

TL;DR: It is proved that a group ring element corresponds to a self-dual code if and only if it is a unitary unit, and that the double-circulant and four- Circulant constructions come from cyclic and dihedral groups.

TL;DR: 41 new binary extremal self-dual codes of length 68 are obtained from groups of orders 15 and 33 using neighboring and extensions using a bordered construction over group rings for self- dual codes.

TL;DR: New extremal doubly even binary codes of lengths 80 and 88 are found by the F 2 + u F 2 -lifts of binary four circulant codes and thus a lower bound on the number of non-isomorphic 3-(80, 16, 665) designs is modified.

TL;DR: This work proposes a modified four circulant construction for self-dual codes and a bordered version of the construction using the properties of λ -circulant and λ-reverse circulent matrices and finds new binary codes of lengths 64 and 68 that were not known to exist in the literature.

TL;DR: In this paper, a duality and distance preserving Gray map from F2 + uF2+u^2F2 to (F_2)^3 is defined by using quadratic double circulant constructions and their extensions, and a substantial number of extremal binary self-dual binary codes are found.