Incidence Structures, Graphs, and Codes
- IncidenceStructure(G): Grph -> Inc
Construct the incidence structure \(D\) corresponding to the graph \(G\), where the blocks of \(D\) correspond to the edges of \(G\).
- PointGraph(D): Inc -> Grph
The point graph \(G\) of the incidence structure \(D\). The graph \(G\) has the same point set as \(D\) and two vertices \(u\) and \(v\) of \(G\) are adjacent whenever there is a block of \(D\) containing both \(u\) and \(v\).
- BlockGraph(D): Inc -> Grph
The block graph of the incidence structure \(D\), i.e., the point graph of the dual of \(D\).
- IncidenceGraph(D): Inc -> Grph
The incidence graph of the incidence structure \(D\). This bipartite graph has as vertex set the union of the point set \(P\) and block set \(B\) of \(D\). A vertex \(p \in P\) is adjacent to a vertex \(b \in B\) whenever \(p \in b\).
- LinearCode(D, K): Inc, FldFin -> Code
Given an incidence structure \(D\) with \(v\) points and a finite field \(K\), this function returns the linear code \(C\) of length \(v\) generated by the characteristic functions of the blocks of \(D\) considered as vectors of the \(K\)-space \(K^{(v)}\).