Recently, the quantum information theory attracts much attention. In quantum information theory, the existence of superadditivity in capacity of a quantum channel was foreseen conventionally. So far, some examples of codes which show the superadditivity in capacity have been clarified. However in present stage, characteristics of superadditivity are not still clear up enough. The reason is as follows. All examples were shown by calculating the mutual information by quantum combined measurement, so that one had to solve the eigenvalue and the eigenvector problems. In this paper, we construct a simplification algorithm to calculate the mutual information by using square-root measurement as decoding process of quantum combined measurement. The eigenvalue and the eigenvector problems are avoided in the algorithm by using group covariancy of binary linear codes. Moreover, we derive the analytical solution of the mutual information for parity check codes with any length as an example of applying the simplification algorithm.

We have researched high dimensional parity-check (HDPC) codes that give good performance over a channel that has a very high error rate. HDPC code has a little coding overhead because of its simple structure. It has hard-in, maximum detected bit flipping (MDBF) decoding that has reasonable decoding performance and computational cost. In this paper, we propose a modified algorithm for MDBF decoding and compare the proposed MDBF decoding with conventional hard-in decoding.