Speaker
Description
The matrix equation $rank(A-B)=1$ is well studied in linear algebra and combinatorics within preserver problems and the theory of distance-regular graphs/association schemes. In the talk I will present how this equality is related to coding theory, namely to binary self-dual codes.
Let $\widehat{\Gamma}_{n}$ be the graph with the vertex set formed by all $n\times n$ symmetric matrices with coefficients in the binary field $\mathbb{F}_2=\{0,1\}$ where two matrices $A$ and $B$ form an edge if and only if $rank(A-B)=1$. Graph $\widehat{\Gamma}_n$ was studied in numerous papers and the distance function $d_{\widehat{\Gamma}_{n}}(A,B)$ between arbitrary vertices is well known and easy to compute. On the other hand, its subgraph $\Gamma_n$, which is induced by invertible matrices, has not attained much attention till recently. In fact, graph $\Gamma_n$ was introduced in 2015 [1]. It generalizes the well-known Coxeter graph (obtained if $n=3$).
Recently, we computed the distance function $d_{\Gamma_n}(A,B)$ [2], which turns out to be related to coding theory. Namely, for odd $n\geq 3$, each linear self-dual code $C$ in $\mathbb{F}_2^{n+1}$ can be identified with a certain subset $\mathcal{F}_C$ of the set
$\mathcal{SD}_n=\left\{A\in V(\Gamma_n): d_{\Gamma_n}(A,I)=\frac{n+5}{2}, rank(A-I)=\frac{n+1}{2} \right\}$ where $I$ is the identity matrix. In particular, the matrices $A\in V(\Gamma_n)$, which represent self-dual codes, are fully determined by the values of two graph parameters: $d_{\Gamma_n}(A,I)$ and $d_{\widehat{\Gamma}_n}(A,I)$. In the talk, I will describe the identification $C\leftrightarrow \mathcal{F}_C$ in more details.
References
[1] M. Orel, On generalizations of the Petersen and the Coxeter graph. Electron. J. Combin. 22(4) (2015), Paper #P.4.27.
[2] M. Orel, D. Višnjić, The distance function on Coxeter-like graphs and self-dual codes. Finite Fields Appl. 103 (2025), paper 102580, 51 pp., https://doi.org/10.1016/j.ffa.2025.102580