Consider bipartite graph. Consider its incidence matrix. It will have a form

0 A^t

A 0

Take matrix $A$. Consider the null-space $L$ of $A$ over $F_2^N$.

Question Can we say something about the $L$ from graph theoretic perspective ? For example to determine what is minimum Hamming weight for vectors in $L$ ?

In error correction codes community the following words are used: Original graph is called Tanner graph for $A$. Matrix $A$ is called parity-check matrix. Let dim(L)=k, any linear map $F_2^k \to L\subset F_2^N$ is called "encoder".

Consider bipartite graph. Consider its adjacency matrix. It will have a form

0 A^t

A 0

Take matrix $A$. Consider the null-space $L$ of $A$ over $F_2^N$.

Question Can we say something about the $L$ from graph theoretic perspective ? For example to determine what is minimum Hamming weight for vectors in $L$ ?

In error correction codes community the following words are used: Original graph is called Tanner graph for $A$. Matrix $A$ is called parity-check matrix. Let dim(L)=k, any linear map $F_2^k \to L\subset F_2^N$ is called "encoder".