Ensemble Analysis on Minimum Span of Stopping Sets

In this paper, the minimum span of stopping sets of regular LDPC codes are defined and analyzed. The minimum span of an LDPC code is closely related to immunity to burst erasures when an LDPC code is decoded with belief propagation (BP) for erasure channels. The minimum span of stopping sets is the smallest span of a non-empty stopping set. If a single erasure burst of length shorter than the minimum span, the erasure burst can be corrected with BP. In order to study the nature of the minimum span, we use ensemble analysis for the bipartite graph ensemble. I. I NTRODUCTION Recent progress on the studies of LDPC codes and decoding algorithms shows that LDPC codes are not only theoretically interesting but also practically promising error correcting codes. An LDPC code is now considered to be a candidate of an error correction scheme for high speed wireless channel, wired channel (such as 10G-baseT) and magnetic recording systems. In such an application, we have to take care of burst errors occurred on the channel as well as random (Gaussian) noise. For example, fading with time-correlation in wireless communication causes burst errors, which severely limits the throughput of the system. In magnetic recording system such as hard disk system, burst errors due to thermal asperity, media defect and off-track should be corrected using an error correcting code to attain high reliability of recording. Some works on LDPC codes for burst channels have been made. There are two approaches: One approach is to devise a decoding algorithm for burst channel. Another is to construct LDPC codes which have high burst error correcting capability. The works by Garcia-Frias [1][2][3], Wadayama [4], Ratzer [5] belong to the first category. They presented iterative decoding algorithms suitable for burst channels. The works by Hosoya et al.[6] and Yang and Ryan[7] lie in the second category. In order to construct a code with burst error immunity, Hosoya et al. proposed a method to obtain a new parity check matrix from an original parity check matrix by permuting its columns. Yang and Ryan proposed Lmax algorithm which efficiently evaluates maximum correctable erasure burst length of a given LDPC code. In some situations, burst errors can be detected before decoding. Namely, in such a case, a burst detector can inform the location of burst errors to the decoder. For example, in magnetic recording system, burst errors caused by thermal asperity can be detected by observing the magnitude of the received signals. When we know the channel state information (CSI), i.e., the location of burst errors, we can regard the burst errors as burst erasures. Another example is packet loss channel which is a channel model for a packet based network. In such a network, burst erasures occur at a router when the network traffic is high. The introduction of stopping sets by Di et al.[9] opened a way to finite length performance analysis on LDPC codes over erasure channels. A stopping set is a subset of variable nodes which dominates the error performance of belief propagation(BP)-based decoder over an erasure channel. In this paper, the minimum span of stopping sets of regular LDPC codes are defined and analyzed. The minimum span of an LDPC code is closely related to immunity to burst erasures when an LDPC code is decoded with BP for erasure channels. The minimum span of stopping sets is the smallest pan of a non-empty stopping set. If a single erasure burst of length shorter than the minimum span, the erasure burst can be corrected with BP. In order to study the nature of the minimum span, we use ensemble analysis for the bipartite graph ensemble. II. PRELIMINARIES In this section, some notations required in this paper will be introduced and several known results on ensemble average of stopping sets will be reviewed. A. Bipartite graph ensemble In this paper, we consider an ensemble of regular bipartite graphs withn-variable nodes andm-check nodes [6]. The variable node and check node degrees are denoted by dv and dc, respectively. A graphG belonging toE is associated with qual probabilityP (G) = 1/|E|. We call such an ensemble a bipartite graph ensembleand it is denoted byE . The set of variable nodes and check nodes are denoted by V andC, respectively. In this paper, a node is represented by a integer such that V = {1, 2, 3, . . . , n}, C = {1, 2, 3, . . . , m}. Let E = {(v, c) : v ∈ V, c ∈ C} be a set of edges which is called an edge set. A graphG = (V ∪ C, E) ∈ E naturally corresponds to an m×n binary parity check matrix which is the adjacent matrix of G. The matrix is denoted by H(G). B. Probability on stopping sets Stopping sets of a bipartite graph are basis of finite length performance analysis of LDPC codes under erasure channels. The definition of stopping sets is as follows. Definition 1: Let S be a subset of V . The subsetS is a stopping set if and only if every check nodes connected to S has at least two edges connected to S. The most important property of stopping sets is that they dominates the correctability of BP-based decoding algorithm under erasure channels. If an erasure pattern covers a stopping set, a BP-based decoding procedure fails. Di et al. [9], Orlitsky et al. [10] presented ensemble analysis on stopping set distribution. The following lemma is due to Orlitsky et al.[10]. Lemma 1:Assume a bipartite graph ensemble E . Let U ⊂ V . The probability thatU is a stopping set is given by ∑ G∈E P (G)I[U is a stopping set ] = coef(((1 + x)c − dcx), xdv|U |) ( dvn dv|U | ) , where coef(f(x), x) means the coefficient of (x) corresponding to the termx. The functionI[predicate] is 1 if the predicate is true; otherwise I[predicate] takes 0. Since this probability depends only on the size of U , we define Q(w)(w ∈ [0, n]), for convenience, by Q(w) = coef(((1 + x)c − dcx), xv) ( dvn dvw ) , (1) which mean the probability of a subset of V with sizew is a stopping set. The notation [a, b] denotes the set of consecutive integers froma to b. III. M INIMUM SPAN OF STOPPINGSETS