The neighbor graph of binary self-dual codes

Abstract

We study the neighbor graph $\Gamma _n$ of binary self-dual codes, where two codes are connected by an edge if and only if they share a subcode of co-dimension 1. We show it is a connected, regular graph with $\prod _{i=1}^{\frac{n}{2} -1} (2^i+1)$ vertices, $\left( \prod _{i=1}^{\frac{n}{2}-1} (2^i+1) \right) (2^{\frac{n}{2} -1} -1)$ edges, and degree $2^{\frac{n}{2}} -2.$ We prove that the number of distance k-neighbors in the graph $\Gamma _n$ is $\frac{\prod _{j=0}^{k-1} (2^{n-1-j} - 2^{\frac{n}{2}} )}{\prod _{h=0}^{k-1} (2^{\frac{n}{2}} -2^{\frac{n}{2}-1-h} ) }$. For $n \equiv 0 \pmod {8}$, we define the neighbor graph $\Delta _n$ of Type II codes, which is a subgraph of $\Gamma _n$. We show it is a connected, regular graph with $2 \prod _{i=1}^{\frac{n}{2} -2} (2^i+1)$ vertices, $\left( \prod _{i=1}^{\frac{n}{2} -2} (2^i+1)\right) (2^{\frac{n}{2} -1} -1)$ edges, and degree $(2^{\frac{n}{2} -1} -1)$. We prove that the number of distance k-neighbors in the graph $\Delta _n$ is $\frac{\prod _{j=0}^{k-1} (2^{n-2-j} - 2^{\frac{n-2}{2}} )}{\prod _{h=0}^{k-1} (2^{\frac{n}{2}} -2^{\frac{n}{2}-1-h} ) }$.