Characterizing the Distance Function on Coxeter-like Graphs and Its Connection to Self-Dual Codes

Khái niệm cốt lõi

The paper describes the distance function on the graph Γn, where the vertices are the invertible symmetric binary matrices, and two matrices are connected by an edge if their rank difference is 1. It also shows that for odd n, certain matrices in this graph correspond to self-dual codes in Fn+1 2, and vice versa.

Tóm tắt

The paper focuses on the graph Γn, where the vertices are the set of all invertible n×n symmetric matrices over the binary field F2, and two matrices A and B are connected by an edge if rankpA-B) = 1.
The main results are:

The paper provides a complete characterization of the distance function dpA,B) between any two vertices A and B in Γn, depending on the structure of the matrix B-A.

It is shown that for odd n, each matrix A in SGLn(F2) such that dpA,I) = (n+5)/2 and rankpA-I) = (n+1)/2, where I is the identity matrix, induces a self-dual code in Fn+1 2. Conversely, each self-dual code C in Fn+1 2 induces a family FC of such matrices A.

It is proved that the orthogonal group OnpF2) acts transitively on the set of all self-dual codes in Fn+1 2, improving a previous result of Janusz.

The paper also includes several auxiliary results from linear algebra that are used in the proofs of the main theorems.

Thống kê

None.

Trích dẫn

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Thông tin chi tiết chính được chắt lọc từ

The distance function on Coxeter-like graphs and self-dual codes

by Mark... lúc arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17067.pdf

Yêu cầu sâu hơn

How can the techniques developed in this paper be extended to study the distance function and other properties of graphs induced by invertible hermitian or alternate matrices over finite fields

The techniques developed in the paper can be extended to study the distance function and other properties of graphs induced by invertible hermitian or alternate matrices over finite fields by adapting the methodology used for symmetric matrices. For hermitian matrices, the distance function can be defined based on the rank of the difference between two matrices, similar to the approach taken for symmetric matrices. The properties of the induced graph, such as diameter, automorphisms, and homomorphisms, can be investigated using similar graph theoretical techniques. Additionally, the connection between invertible alternate matrices and the corresponding graphs can be explored to understand the distance function and structural properties in that context as well.

Can the connection between the graph Γn and self-dual codes be further exploited to obtain new insights or results in coding theory

The connection between the graph Γn and self-dual codes can be further exploited to obtain new insights or results in coding theory. By studying the relationship between matrices in SGLnpF2q that induce self-dual codes in Fn+1 2, additional properties of self-dual codes can be uncovered. This could involve investigating the structure of the matrices that correspond to self-dual codes, exploring the properties of the codes induced by different families of matrices, and potentially identifying new families of self-dual codes based on specific characteristics of the matrices in SGLnpF2q. Furthermore, the graph theoretical description of self-dual codes provided by the identification C Ø FC can be leveraged to analyze the properties of self-dual codes in a more systematic and structured manner.

Are there any applications or implications of the transitive action of the orthogonal group OnpF2) on the set of self-dual codes in Fn+1 2 beyond the enumeration problem

The transitive action of the orthogonal group OnpF2q on the set of self-dual codes in Fn+1 2 has implications beyond the enumeration problem in coding theory. This transitive action implies that the orthogonal group can transform any self-dual code into another self-dual code, highlighting the symmetry and structure present in self-dual codes. This property can be utilized in code construction and manipulation, potentially leading to the development of new coding techniques or algorithms. Additionally, the transitive action can aid in understanding the relationships between different self-dual codes and their equivalence classes under orthogonal transformations, providing insights into the diversity and complexity of self-dual codes in Fn+1 2.