Quadruple Bordered Constructions of Self-Dual Codes from Group Rings

This is a post-peer-review, pre-copyedited version of an article published in Cryptography and Communications. The final authenticated version is available online at: https://doi.org/10.1007/s12095-019-00380-8In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring F4 + uF4. We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68

2^n Bordered Constructions of Self-Dual codes from Group Rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes length 64 and 68, constructing 30 new extremal self-dual codes of length 68

Group rings: Units and their applications in self-dual codes

The initial research presented in this thesis is the structure of the unit group of the group ring Cn x D6 over a field of characteristic 3 in terms of cyclic groups, specifically U(F3t(Cn x D6)). There are numerous applications of group rings, such as topology, geometry and algebraic K-theory, but more recently in coding theory. Following the initial work on establishing the unit group of a group ring, we take a closer look at the use of group rings in algebraic coding theory in order to construct self-dual and extremal self-dual codes. Using a well established isomorphism between a group ring and a ring of matrices, we construct certain self-dual and formally self-dual codes over a finite commutative Frobenius ring. There is an interesting relationships between the Automorphism group of the code produced and the underlying group in the group ring. Building on the theory, we describe all possible group algebras that can be used to construct the well-known binary extended Golay code. The double circulant construction is a well-known technique for constructing self-dual codes; combining this with the established isomorphism previously mentioned, we demonstrate a new technique for constructing self-dual codes. New theory states that under certain conditions, these self-dual codes correspond to unitary units in group rings. Currently, using methods discussed, we construct 10 new extremal self-dual codes of length 68. In the search for new extremal self-dual codes, we establish a new technique which considers a double bordered construction. There are certain conditions where this new technique will produce self-dual codes, which are given in the theoretical results. Applying this new construction, we construct numerous new codes to verify the theoretical results; 1 new extremal self-dual code of length 64, 18 new codes of length 68 and 12 new extremal self-dual codes of length 80. Using the well established isomorphism and the common four block construction, we consider a new technique in order to construct self-dual codes of length 68. There are certain conditions, stated in the theoretical results, which allow this construction to yield self-dual codes, and some interesting links between the group ring elements and the construction. From this technique, we construct 32 new extremal self-dual codes of length 68. Lastly, we consider a unique construction as a combination of block circulant matrices and quadratic circulant matrices. Here, we provide theory surrounding this construction and conditions for full effectiveness of the method. Finally, we present the 52 new self-dual codes that result from this method; 1 new self-dual code of length 66 and 51 new self-dual codes of length 68. Note that different weight enumerators are dependant on different values of β. In addition, for codes of length 68, the weight enumerator is also defined in terms of γ, and for codes of length 80, the weight enumerator is also de ned in terms of α

An Altered Four Circulant Construction for Self-Dual Codes from Group Rings and New Extremal Binary Self-dual Codes I

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F2 + uF2 and F4 + uF4; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with \gamma=5 in W68,2, which is the first instance of such a value in the literature

Modified Quadratic Residue Constructions and New Exermal Binary Self-Dual Codes of Lengths 64, 66 and 68

In this work we consider modified versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary field and the rings F2 +uF2 and F4 +uF4 for different prime values of p. Using these constructions with extensions and neighbors we are able to construct a number of extremal binary self-dual codes of different lengths with new parameters in their weight enumerators. In particular we construct 2 new codes of length 64, 4 new codes of length 66 and 14 new codes of length 68. The binary generator matrices of the new codes are available online at [8]

Type II Codes over F4

AbstractThe natural analogues of Lee weight and the Gray map over F4 are introduced. Self-dual codes for the Euclidean scalar product with Lee weights multiple of 4 are called Type II. They produce Type II binary codes by the Gray map. All extended Q-codes of length a multiple of 4 are Type II. This includes quadratic residue codes attached to a prime p≡3 (mod8), certain double circulant codes, and some affine invariant codes. A general mass formula is derived, a new upper bound for Euclidean self-dual codes over F4 is given, and the first extremal self-dual [92, 46, 16] binary code is built

Constructing Self-Dual Codes from Group Rings and Reverse Circulant Matrices

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Advances in Mathematics of Communications following peer review. The definitive publisher-authenticated version Gildea, J., Abidin, K., Adrian, K. & Bahattin, Y. (2020). Constructing Self-Dual Codes from Group Rings and Reverse Circulant Matrices. Advances in Mathematics of Communications. is available online at: https://www.aimsciences.org/article/doi/10.3934/amc.2020077?viewType=htmlIn this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92

Composite Constructions of Self-Dual Codes from Group Rings and New Extremal Self-Dual Binary Codes of Length 68

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Advances in Mathematics of Communications following peer review. The definitive publisher-authenticated version Dougherty, S, T., Gildea, J., Korban, A. & Kaya, A. (2019 - forthcoming). Composite Constructions of Self-Dual Codes from Group Rings and New Extremal Self-Dual Binary Codes of Length 68. Advances in Mathematics of Communications, is available online at: 10.3934/amc.2020037.We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over F4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These are lifted to codes over F4 + uF4, to obtain codes with Gray images extremal self-dual binary codes of length 64. Finally, we use a building-up method over F2 + uF2 to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors