Centrala begrepp
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.
Sammanfattning
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.