We consider the problem of analyzing the scaling exponent of random linear codes. In particular, we consider transmission over binary erasure channels and show that random linear codes have optimal scaling exponent, similar to that of random block codes, over binary erasure channels, with high probability. We further consider random linear codes with random generator matrices whose entries are picked according to i.i.d. Bernoulli distribution with parameter $q = Theta(n^{-1/2})$, where $n$ is the code block length. In other words, such generator matrices are sparse with high probability and reduce the $O(n^2)$ encoding complexity of uniform random linear codes to $O(n^{3/2})$ encoding complexity. It is shown that random linear codes with such sparse generator matrices also have optimal scaling exponent, with high probability.