Transmission channels underlying modern memory systems, e.g., Flash memories, possess a signicant amount of asymmetry. While existing LDPC codes optimized for

symmetric, AWGN-like channels are being actively considered for Flash applications, we

demonstrate that, due to channel asymmetry, such approaches are fairly inadequate. We

propose a new, general, combinatorial framework for the analysis and design of non-binary LDPC (NB-LDPC) codes for asymmetric channels.We introduce a refined definition of absorbing sets, which we call general absorbing sets (GASs), and an important subclass of GASs, which we refer to as general absorbing sets of type two (GASTs). Additionally, we study the combinatorial properties of GASTs. We then present the weight consistency matrix (WCM), which succinctly captures key properties in a GAST. Based on these new concepts, we then develop a general code optimization framework, and demonstrate its effectiveness on the realistic Flash channels. Our optimized designs enjoy over one order of magnitude performance gain in the uncorrectable BER (UBER) relative to the unoptimized codes.