Bhattacharyya parameter of monomials codes for the Binary Erasure Channel: from pointwise to average reliability

Monomial codes were recently equipped with partial order relations, fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of Binary Erasure Channel. The new order relation takes us closer to the ultimate order relation induced by the pointwise evaluation of the Bhattacharyya parameter of the synthetic channels. The best we can hope for is still a partial order relation. To overcome this issue we appeal to related technique from network theory. Reliability network theory was recently used in the context of polar coding and more generally in connection with decreasing monomial codes. In this article, we investigate how the concept of average reliability is applied for polar codes designed for the binary erasure channel. Instead of minimizing the error probability of the synthetic channels, for a particular value of the erasure parameter p, our codes minimize the average error probability of the synthetic channels. By means of basic network theory results we determine a closed formula for the average reliability of a particular synthetic channel, that recently gain the attention of researchers.