Hermite normal form and basis for the nullspace

Question

Define the Hermite Normal form $H$ for a matrix $A\in Z^{n\times n}$ as following:

$$H=UA$$ which in Mathematica is given as {u, h} = HermiteDecomposition[a]

Then for a system of homogeneous equations $Ax = [0]_{n}$ we normally compute a basis for the Nullspace of $A$, so any vector $x$ in $Ker(A)$ satisfies the system.

It results that $$Ax=[0]_{n} \rightarrow U^{-1}Hx =[0]_{n} \rightarrow Hx = [0]_{n}$$

Then computing such a basis for $A$ can be done in terms of $H$, where $H$ is an upper triangular matrix, so at first glance NullSpace runs faster for $H$ instead of $A$.

But I'm not very familiarized with the time-complexity of Mathematica's implementation of HermiteDecomposition and Nullspace (this last one over a prime modulus). I don't think this would be a good idea for big dimensions, this is, big $n$.

The thing is that I need a $\textit{fast}$ and efficient method for computing a basis for the nullspace of $A$ over $F_2$ when $n$ is considerably big. What can I do?

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P.D = As a recommendation from MarcoB the time lapses are given with an example matrix of size $100\times 100$:

NullSpace only call: $0.010054$s

HermiteDecomposition + NullSpace: $0.071549$s

LatticeReduce + NullSpace: $0.258836$s

where the NullSpace call is always taken over $F_2$.

Answer 1

There are at least a few ways to improve matters. First is to use

NullSpace[mat,Modulus->2]

This is still memory hungry though. There is a splitting method, useful when the row count is large, that is shown here.

Another method, which can be used in tandem with splitting, is to use bit strings to represent rows. The null space can then be extracted with ResourceFunction["BitStringNullSpace"], documentation for which is found here.