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Subfield codes and trace codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
It should now be clear when two codes are equivalent under permutations of coordinates. A highly interesting situation occurs if we find a permutation σ such that σ() = (in words: the image of our code under σ is the code itself). In this case we call σ an automorphism of . The automorphisms of form a subgroup, the stabilizer or automorphism group of . We saw a large example in Chapter 7. The generator matrix for the binary Golay code [23, 12, 7]2 given there has the special feature that all rows are cyclic shifts of the first row. In other words, the cyclic permutation σ : 1 ↦ 2, 2 ↦ 3,... 22 ↦ 23, 23 ↦ 1
Group Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
William Cocke, Meng-Che ‘Turbo’ Ho
When dealing with an object, both in the physical world and within cyber space, understanding the symmetries inherent in the definition of that object can increase our understanding of the object itself. The symmetries of an object are captured in a corresponding group known as the automorphism group of the object. We give the following example to demonstrate how automorphisms are often implicitly used to assist in problem solving.
On the equivariance properties of self-adjoint matrices
Published in Dynamical Systems, 2020
Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus
An isomorphism from a graph to itself is called an automorphism. Let A be the adjacency matrix of an undirected graph , then the automorphism group (or symmetry group) of is defined as A graph is called asymmetric if is trivial, i.e. . Since A is self-adjoint, we can use Corollary 4.5 to identify the orthogonal commutator of A. Permutation matrices are orthogonal, hence .
Selecting energy efficient inputs using graph structure
Published in International Journal of Control, 2023
Isaac Klickstein, Francesco Sorrentino
Let be a graph and let be a bijection on the set of nodes of a graph. After applying a permutation to the nodes of a graph, define the new set of edges as where if then . A permutation π is a symmetry if . The set of all such symmetries along with function composition form the automorphism group of a graph, .
Super extra edge-connectivity in regular networks with edge faults
Published in International Journal of Parallel, Emergent and Distributed Systems, 2021
For a graph G, use to denote the automorphism group of G. A graph G is edge transitive if for any two edges , there is an automorphism such that . Note that , and are examples of edge transitive graphs [24].