where Q is the quantization function defined as set is the minimal number of variable nodes that have to be initially in error for the decoder to end up in the trapping set. Note that these definitions are for the case of BSC. Also in analysis of decoders in this paper, it is implicitly assumed that the all-zero codeword is transmitted. This is a valid assumption since we consider only symmetric decoders, as explained in [3]. Readers can refer to the work of Chilappagari et al. [8], [12] for more details on these notions. where i = 1 , 2 , . . . , M and T$_{M}$$_{+1}$ = ∞ . Remark: Note that although one might consider these de- coders as special instances of a quantized min-sum decoder, the messages are still not beliefs. Also particular message sets and threshold sets for Φ$_{v}$ are defined using knowledge of trapping sets which will be discussed later. Non-linear-threshold (NLT) decoders: For these decoders, the function Φ$_{v}$ determines its output by taking the sum of its incoming messages and a weighted channel value, and then comparing with a set of thresholds. The weight ω$_{c}$ assigned to the channel value y$_{i}$ is computed using a symmetric non- linear function Ω : M d$_{v}$ − 1 → { 0 , 1 } . Multilevel decoders are a new class of quantized message- passing decoders for LDPC codes. For this class, a decoder F is defined as a 4-tuple given by F = ( M , Y , Φ$_{v}$ , Φ$_{c}$) . The message set M consists of all the levels under which the messages are confined to and is defined as M = { 0 , ± L$_{i}$ : 1 ≤ i ≤ M } , where L$_{i}$ ∈ R + and L$_{i}$ > L$_{j}$ for any i > j . The set Y denotes the set of possible values called channel values , that are computed by the decoder based on the vector r received from the channel. For the case of BSC, Y is defined as Y = {± C } , and for each variable node v$_{i}$ in G , the channel value y$_{i}$ ∈ Y is determined by y$_{i}$ = ( − 1) $^{r$_{i}$}$C . Φ$_{v}$ : Y × M d$_{v}$ − 1 → M denotes the update rule used at a variable node with degree d$_{v}$ , which is a simple map derived using knowledge of trapping sets. Φ$_{c}$ : M d$_{c}$ − 1 → M denote the check node map for a check node with degree d$_{c}$ . We shall restrict ourselves to d$_{v}$ -left-regular codes where the same map is used at every variable node. Also the update rules we consider are time-invariant rules, i.e., they do not change from iteration to iteration. Remark: Due to the nonlinearity, Φ$_{v}$ can output different outgoing messages for any two distinct sets of incoming messages even though the sum of them is the same. Hence these decoders are different from quantized min-sum decoders or any other existing quantized message-passing decoders. It is evident from the expressions for Φ$_{v}$ and Φ$_{c}$ , that the rules are symmetric for both LT and NLT decoders. We now introduce a key concept called isolation assumption that enables us to analyze decoding on isolated subgraphs induced by trapping sets of a code. Analyzing decoders on isolated subgraphs is a crucial strategy required for deriving update rules. It is important to note that this is different from the independence assumption considered by Gallager [2]. Before we delve into this, we first provide some motivation for the need of defining such a concept. Remark: In this paper, by the term “trapping sets”, we mean trapping sets that are known for message-passing decoders such as Gallager-A/B and/or BP decoders. In this paper, we consider the check node update function Φ$_{c}$ to be where sgn denotes the standard signum function. Remark: Φ$_{c}$ defined in this manner corresponds exactly to the check node update rule of the min-sum algorithm. Based on the definition of Φ$_{v}$ , we propose two subclasses of multilevel decoders: linear-threshold decoders and non-linear- threshold decoders. Linear-threshold (LT) decoders: For these decoders, the function Φ$_{v}$ determines its output by taking a sum of its inputs and comparing with a set of thresholds. A threshold set T = { T$_{1}$, T$_{2}$, · · · , T$_{M}$ } where T$_{i}$ ∈ R $^{+}$, is defined in a way such that for any T$_{p}$, T$_{q}$ ∈ T , T$_{p}$ > T$_{q}$ if p > q . The function Φ$_{v}$ is defined as The design of multilevel decoders is based on the knowledge of trapping sets for existing decoders. Given a list of trapping sets, the aim is to design a variable node update rule that guarantees correction of all of them. Consider the subgraph of a trapping set contained in the Tanner graph with some nodes initially in error. In order to verify whether a given rule succeeds on a trapping set, it is necessary not only to know the subgraph induced by the trapping set, but also the neigh- borhood of the induced subgraph and messages coming from this neighborhood. Since this neighborhood can be different for different Tanner graphs, the design of multilevel decoders becomes very complex. Therefore, to facilitate the design process, we work under the assumption that the messages to the nodes of the trapping set from the nodes outside the induced subgraph of the trapping set are known. We call this assumption the isolation assumption to signify the fact that the trapping set can be considered in isolation from the rest of the graph and can be analyzed as an independent entity. Φ$_{c}$( m$_{1}$, . . . , m$_{d}$$_{c}$$_{−}$$_{1}$ ) =   d$_{c}$ − 1 ∏ j =1 sgn ( m$_{j}$ )$^{}$  min j ∈{ 1 ,...,d$_{c}$ − 1 $_{}}$( | m$_{j}$ | ) Φ$_{v}$( m$_{1}$, m$_{2}$, · · · , m$_{d}$$_{v}$$_{−}$$_{1}$, y$_{i}$ ) = Q$^{}$  d$_{v}$ − 1 ∑ j =1 m$_{j}$ + y$_{i}$$^{}$  Φ$_{v}$( m$_{1}$, m$_{2}$, · · · , m$_{d}$$_{v}$$_{−}$$_{1}$, y$_{i}$ ) = Q$^{}$  d$_{v}$ − 1 ∑ j =1 m$_{j}$ + ω$_{c}$ · y$_{i}$$^{}$ $_{}$. B. Isolation assumption A. Update rules III. MULTILEVEL DECODERS: GENERAL FRAMEWORK Q ( x ) = { L$_{i}$ , T$_{i}$ ≤ x < T$_{i}$$_{+1}$ − L$_{i}$ , − T$_{i}$$_{+1}$ < x ≤ − T$_{i}$ 0, otherwise