  We consider irregular product codes.In this class of codes, each codeword is
represented by a matrix. The entries in each row (column) of the matrix should
come from a component row (column) code. As opposed to (standard) product
codes, we do not require that all component row codes nor all component column
codes be the same. As we will see, relaxing this requirement can provide some
additional attractive features including 1) allowing some regions of the
codeword be more error-resilient 2) allowing a more refined spectrum of rates
for finite-lengths and improved performance in some of these rates 3) more
interaction between row and column codes during decoding. We study these codes
over erasure channels. We find that for any $0 < \epsilon < 1$, for many rate
distributions on component row codes, there is a matching rate distribution on
component column codes such that an irregular product code based on MDS codes
with those rate distributions on the component codes has asymptotic rate $1 -
\epsilon$ and can decode on erasure channels (of alphabet size equal the
alphabet size of the component MDS codes) with erasure probability $<
\epsilon$.
