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For LDPC codes operating over additive white Gaussian noise channels anddecoded using message-passing decoders with limited precision, absorbing setshave been shown to be a key factor in error floor behavior. Focusing on thisscenario, this paper introduces the cycle consistency matrix (CCM) as apowerful analytical tool for characterizing and avoiding absorbing sets inseparable circulant-based (SCB) LDPC codes. SCB codes include a wide variety ofregular LDPC codes such as array-based LDPC codes as well as many commonquasi-cyclic codes. As a consequence of its cycle structure, each potentialabsorbing set in an SCB LDPC code has a CCM, and an absorbing set can bepresent in an SCB LDPC code only if the associated CCM has a nontrivial nullspace.  CCM-based analysis can determine the multiplicity of an absorbing set in anSCB code and CCM-based constructions avoid certain small absorbing setscompletely. While these techniques can be applied to an SCB code of any rate,lower-rate SCB codes can usually avoid small absorbing sets because of theirhigher variable node degree. This paper focuses attention on the high-ratescenario in which the CCM constructions provide the most benefit. Simulationresults demonstrate that under limited-precision decoding the new codes havesteeper error-floor slopes and can provide one order of magnitude ofimprovement in the low FER region.

The Cycle Consistency Matrix Approach to Absorbing Sets in Separable Circulant-Based LDPC Codes