Controlling LDPC Absorbing Sets via the Null Space of the Cycle Consistency Matrix

Regular LDPC codes tend to have better error-floor behavior than irregular LDPC codes. However, for moderate block lengths and high rates, the error floor remains a concern even for regular LDPC codes. This is especially the case for applications such as memory media that require very low frame error rates (FERs). This paper focuses on a class of regular LDPC codes: separable, circulant-based (SCB) codes. For a specified circulant matrix, SCB codes all share a common mother matrix and include array-based LDPC codes as well as many common quasi-cyclic codes. SCB codes retain standard properties of quasi-cyclic LDPC codes such as girth, code structure, and compatibility with existing high-throughput hardware implementations. This paper introduces a cycle consistency matrix (CCM) for each possible absorbing set in an SCB LDPC code. For an absorbing set to be present in an SCB LDPC code, the associated CCM must not be full column-rank. Using this novel observation, a new code construction approach selects rows and columns from the SCB mother matrix to systematically eliminate dominant absorbing sets by forcing the associated CCMs to be full column-rank. Simulation results demonstrate that the new codes have steeper error-floor slopes and provide at least one order of magnitude of improvement in the low FER region. Identifying absorbing-set-spectrum equivalence classes within the family of SCB codes with a specified circulant matrix significantly reduces the search space of possible code matrices.