The cost of solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the system size.

I am wondering about the lower bound for solving a linear system. An obvious lower bound is $\mathcal{\Omega}(N^2)$ (since the information content is $\mathcal{O}(N^2)$). Are there better lower bounds other than $\mathcal{\Omega}(N^2)$ for solving the linear system? Is there a way to prove that the lower bound of $\mathcal{\Omega}(N^2)$ can never be hit for a matrix with no special structure? (assume that we are solving a system with only one/few right hand side).

Also are there other algorithm which solve these system "exactly" whose cost is less than $\mathcal{O}(N^3)$? I am aware of Strassen algorithm which perform matrix multiplications in $\mathcal{O}(N^{\log_27})$. I assume this can be used to solve a linear system in $\mathcal{O}(N^{\log_27})$. (?)

(The system has no special structure. I am not worried about the stability and other numerical intricacies of the method as of now. I would appreciate if someone could point to some work done in this regard.)

Thanks