Binary linear codes with a fixed point free permutation automorphism of order three

We investigate the structural properties of binary linear codes whose permutation automorphism group has a fixed point free automorphism of order $3$. We prove that up to dimension or codimension $4$, there is no binary linear code whose permutation automorphism group is generated by a fixed point free permutation of order $3$. We also prove that there is no binary $5$-dimensional linear code whose length is at least $30$ and whose permutation automorphism group is generated by a fixed point free permutation of order $3$.Comment: 10 page

Sobre automorfismos de códigos extremales de tipo II

En el presente artículo se muestran algunas técnicas para obtener tipos de automorfismos de los códigos binarios auto-duales, doblemente pares y extremales, también denominados extremales de tipo II, con parámetros [24, 12, 8], [48, 24, 12] y [120, 60, 24]. El objetivo central es obtener información sobre el correspondiente grupo de automorfismos a partir de la exclusión de algunos números primos de su orde

Sobre automorfismos de códigos extremales de tipo II

En el presente artículo se muestran algunas técnicas para obtener tipos de automorfismos de los códigos binarios auto-duales, doblemente pares y extremales, también denominados extremales de tipo II, con parámetros [24, 12, 8], [48, 24, 12] y [120, 60, 24]. El objetivo central es obtener información sobre el correspondiente grupo de automorfismos a partir de la exclusión de algunos números primos de su ordenIn this article we present some techniques to determine the types of automorphisms of extremal doubly even binary self-dual codes, also called extremal type II codes, with parameters [24, 12, 8], [48, 24, 12] and [120, 60, 24]. We aim to obtain information about the automorphism group considering the exclusion of some prime numbers from its order

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