# Matrix multiplication when one matrix is fixed

Let $$A$$ be a fixed positive entried integer matrix of size $$a\times n$$ with $$\ell$$ bits per entry

One is allowed to pre-process this matrix as appropriate.

Given another positive integer entried $$B$$ of size $$n\times b$$ with $$\ell'$$ bits per entry, what is the complexity of multiplication $$AB$$?

Note we already have $$(\max(a,n,b))^{2+\theta}polylog(\ell\ell')$$ time and space algorithms where $$2+\theta=\omega$$ and $$\omega$$ is exponent of matrix multiplication.

1. The query here is whether we can do take $$\theta=0$$ or at least have flexibility to take any $$\theta>0$$ by anything cleverer?

2. Can we attain $$\theta=0$$ or at least have flexibility to take any $$\theta>0$$ with number of operations independent of $$\ell$$ and $$\ell'$$ in the $$RAM$$ model?