Scaling Exponent of Sparse Random Linear Codes over Binary Erasure Channels

The problem of analyzing the finite-length scaling behavior of sparse random linear codes is considered. Random linear codes with random generator matrices whose entries are picked according to i.i.d. Bernoulli distribution with parameter q = o(1) are called sparse. The parameter q is referred to as the sparsity of the random linear code. We develop a methodology to show the optimality of the scaling exponent of uniform random linear codes, i.e., q = 1/2, with high probability. The results are then extended to sparse random linear codes with sparsity q = Theta(n(-1/2)), where n is the code block length. The encoding complexity of such sparse random linear codes is reduced from O(n(2)), in uniform random linear codes, to O(n(3/2)). It is also conjectured that q = log n/n is the lowest sparsity of random linear codes with optimal scaling exponent. The connection of these results to an open problem regarding finding binary polar codes with optimal scaling exponent are also discussed. In particular, we point out that as the size of the polarization kernel increases it can be used as the generator matrix for a code with optimal scaling exponent, without the need to do further polarization.