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In this work, we present a number of generator matrices of the form [ I 2 n | τ 2 (v) ] , where I 2 n is the 2 n × 2 n identity matrix, v is an element in the group matrix ring M 2 (R) G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [ 72 , 36 , 12 ] self-dual codes directly over the finite field F 2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings. [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]

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the n × n matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element v , one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element v , one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters. [ABSTRACT FROM AUTHOR]

In this work, we use the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k -neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with γ = 8 in their W 68 , 2 and 40 with γ = 9 in their W 68 , 2. These examples are the first in the literature for these γ values. This completes the theoretical list of possible values for γ in W 68 , 2. [ABSTRACT FROM AUTHOR]