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In this work, we define a modification of a bordered construction for self-dual codes which utilises $ \lambda $-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist. [ABSTRACT FROM AUTHOR]

In this work, we give a new technique for constructing self-dual codes over commutative Frobenius rings using λ -circulant matrices. The new construction was derived as a modification of the well-known four circulant construction of self-dual codes. Applying this technique together with the building-up construction, we construct singly-even binary self-dual codes of lengths 56, 58, 64, 80 and 92 that were not known in the literature before. Singly-even self-dual codes of length 80 with β ∈ { 2 , 4 , 5 , 6 , 8 } in their weight enumerators are constructed for the first time in the literature. [ABSTRACT FROM AUTHOR]

Abstract We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + u F 2 , using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper. Highlights • A novel bordered construction for self-dual codes can be obtained using groups rings. • Extremal binary self-dual codes of different lengths can be constructed using the constructions. • 41 new additions to the set of known extremal binary self-dual codes of length 68 have been made. • There is a strong connection between groups rings and constructions for self-dual codes. [ABSTRACT FROM AUTHOR]