This work studies the design of MET-LDPC code ensembles at low code rates with good threshold performances and exponentially few small weight codewords. There was a notable study on the condition, so called the ` t-value condition', for exponentially few small weight codewords. Meanwhile, it was shown that by introducing degree-one variable nodes, the threshold performances of MET-LDPC code ensembles of low rates can be significantly improved. However, the degree-one variable nodes may result in small weight codewords and thus must be carefully introduced to MET-LDPC code ensembles. Despite the importance, the existing t-value condition was developed only for MET-LDPC code ensembles without degree-one variable nodes. This work extends the t-value condition to MET-LDPC code ensembles with degree-one variable nodes, which includes the existing work as a special case. The extended t-value condition provides useful insights into the contributions of degree-one variable nodes to the distribution of small weight codewords. Thus, the results of this work allow us to design MET-LDPC code ensembles of low rates with both good threshold performances and exponentially few small weight codewords. In addition, it will be demonstrated that MET-LDPC codes at finite lengths based on the designed code ensembles have good error-rate performances both in the waterfall and low-error-rate regions.

Since multi-edge type low-density parity-check (MET-LDPC) codes were first proposed, the design of MET-LDPC codes has been extensively studied for various applications. However, the existing design rules assume that check node degrees are in the so-called concentrated form which enables one to conveniently find pairs of edge and node distributions satisfying the socket count equalities (SCEs). However, it in return makes the code design conducted in a limited search space, which may lead to a sub-optimal design. This paper proposes a novel design rule for MET-LDPC codes together with a scheme to efficiently sort out invalid distributions, i.e., ones not satisfying the SCEs, in the design process. The proposed design rule does not require check node degrees to be in the concentrated form and can find out optimized MET-LDPC codes without any restriction on the search space. Based on the proposed design rule, MET-LDPC codes are designed across a wide range of code rates on binary-erasure channel and binary-input additive-white Gaussian channel. The designed codes are compared with ones reported in the open literature, which confirms that MET-LDPC codes with the proposed design rule have better threshold performances regardless of channel and code rate even with sets of limited code parameters.