New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singly-even binary self-dual codes of lengths 80, 84 and 96 as well as doubly-even binary self-dual codes of length 96 that were not known in the literature before.

Double bordered constructions of self-dual codes from group rings over Frobenius rings

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 $\mathbb {F}_{2}+u\mathbb {F}_{2}$ F 2 + u F 2 and $\mathbb {F}_{4}+u\mathbb {F}_{4}$ F 4 + u F 4 . 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.

Notes on quasi-Frobenius rings

We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R, the following statements are equivalent: (1) R is a quasi-Frobenius ring, (2) {M_{2}(R)} is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that {R/S_{l}} is left Goldie, (5) R is a right YJ-injective right minannihilator ring with ACC on right annihilators.

On C-co-epi-retractable Modules

In this paper, we introduce the notion of c-co-epi-retractable modules. An R-module M is called c-co-epi-retractable if it contains a copy of its factor module by a complement submodule. The ring R is called c-co-pri if RR is c-co-epi-retractable. Conditions are found under which, a c-coepi-retractable module is extending, retractable, semi-simple, quasi-injective, injective and simple. Also, we investigate when c-co-epi-retractable modules have finite uniform dimension. Finally, right SI-rings, semi-simple artinian rings and quasi-Frobenius rings are characterized in termes of c-co-epi-retractable modules.