Three-edge type LDPC code ensembles with exponentially few codewords with linear small weight

Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. Nakasendo et al. derived the condition that code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. In this paper, we derive the condition for three-edge type LDPC code ensembles whose edge-types does not share any variable node and does share all of the check nodes with exponentially decreasing small linear weight codewords. The condition is necessary for the existence of the average relative minimum distance of ensembles. Our method is expected to derive the condition for multi-edge type LDPC codes.