  Independent Component Analysis (ICA) is a statistical tool that decomposes an
observed random vector into components that are as statistically independent as
possible. ICA over finite fields is a special case of ICA, in which both the
observations and the decomposed components take values over a finite alphabet.
This problem is also known as minimal redundancy representation or factorial
coding. In this work we focus on linear methods for ICA over finite fields. We
introduce a basic lower bound which provides a fundamental limit to the ability
of any linear solution to solve this problem. Based on this bound, we present a
greedy algorithm that outperforms all currently known methods. Importantly, we
show that the overhead of our suggested algorithm (compared with the lower
bound) typically decreases, as the scale of the problem grows. In addition, we
provide a sub-optimal variant of our suggested method that significantly
reduces the computational complexity at a relatively small cost in performance.
Finally, we discuss the universal abilities of linear transformations in
decomposing random vectors, compared with existing non-linear solutions.
