On regular quasi-cyclic LDPC codes from binomials

In the past, several authors have considered quasi-cyclic LDPC codes whose circulant matrices in the parity-check matrix are cyclically shifted identity matrices. By composing a parity-check matrix not only with such matrices but also with sums of two cyclically shifted identity matrices and with zero matrices, one can increase the minimum distance while keeping the same regularity. Specifically, whereas for (3,4)-regular codes in the first class the best minimum distance is 24, the best minimum distance in the second class is 32. We give examples of codes that achieve these bounds.