Abstract
In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma =7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction
Highlights
The authors in [13] have defined G-codes which are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group
We have extended the idea of G-codes to composite G-codes
We have shown that as the G-codes, the composite G-codes are ideals in the group ring RG
Summary
The authors in [13] have defined G-codes which are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group This idea is based on applying the matrix σ (v), where v is a group ring element, which was first introduced in [21]. The authors define a number of composite constructions which they apply to search for extremal self-dual codes, but no general theory is given. We apply the more general and rigorous definition of the composite matrices to define composite G-codes This is an extension of G-codes mentioned earlier. We combine the ideas of composite matrices, the well known extension method and the neighbour construction and its generalization (see [18] for details), to search for extremal binary selfdual codes of length 68. We finish with concluding remarks and directions for possible future research
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