| Title: | The distance function on Coxeter-like graphs and self-dual codes |
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| Authors: | ID Orel, Marko (Author)
ID Višnjić, Draženka (Author) |
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| Files: |  PDF - Presentation file, download (1,96 MB)
MD5: 601559481BB70482626E3250B98F2611
 URL - Source URL, visit https://doi.org/10.1016/j.ffa.2025.102580
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Let $SGL_n(\mathbb{F}_2)$ be the set of all invertible $n\times n$ symmetric matrices over the binary field $\mathbb{F}_2$. Let $\Gamma_n$ be the graph with the vertex set $SGL_n(\mathbb{F}_2)$ where a pair of matrices $\{A,B\}$ form an edge if and only if $\textrm{rank}(A-B)=1$. In particular, $\Gamma_3$ is the well-known Coxeter graph. The distance function $d(A,B)$ in $\Gamma_n$ is described for all matrices $A,B\in SGL_n(\mathbb{F}_2)$. The diameter of $\Gamma_n$ is computed. For odd $n\geq 3$, it is shown that each matrix $A\in SGL_n(\mathbb{F}_2)$ such that $d(A,I)=\frac{n+5}{2}$ and $\textrm{rank}(A-I)=\frac{n+1}{2}$ where $I$ is the identity matrix induces a self-dual code in $\mathbb{F}_2^{n+1}$. Conversely, each self-dual code $C$ induces a family ${\cal F}_C$ of such matrices $A$. The families given by distinct self-dual codes are disjoint. The identification $C\leftrightarrow {\cal F}_C$ provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogonal group ${\cal O}_n(\mathbb{F}_2)$ acts transitively on the set of all self-dual codes in $\mathbb{F}_2^{n+1}$. |
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| Keywords: | Coxeter graph, invertible symmetric matrices, binary field, rank, distance in graphs, alternate matrices, self-dual codes |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.03.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 1-51 |
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| Numbering: | Vol. 103, article 102580 |
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| PID: | 20.500.12556/DiRROS-21403  |
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| UDC: | 519.17 |
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| ISSN on article: | 1071-5797 |
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| DOI: | 10.1016/j.ffa.2025.102580 |
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| COBISS.SI-ID: | 223842819 |
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| Publication date in DiRROS: | 31.01.2025 |
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| Views: | 508 |
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| Downloads: | 323 |
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