LDPC absorbing sets, the null space of the cycle consistency matrix, and Tanner's constructions

Dolecek et al. introduced the cycle consistency condition, which is a necessary condition for cycles — and thus the absorbing sets that contain them — to be present in separable circulant-based (SCB) LDPC codes. This paper introduces a cycle consistency matrix (CCM) for each possible absorbing set in an SCB LDPC code. The CCM efficiently enforces the cycle consistency condition for all cycles in a specified absorbing set by spanning its associated binary cycle space. Under certain conditions, a CCM not having full column rank is a necessary and sufficient condition for the LDPC code to contain the absorbing set associated with that CCM. This paper uses the CCM approach to carefully analyze LDPC codes based on the Tanner construction for r = 4 rows of sub-matrices (i.e., Tanner-construction LDPC codes with column weight 4).