Lower Bounds on Stopping Distance of Linear Codes and Their Applications

One of the most important open problems in the communication society is to determine the performance of iterative message passing algorithms over loopy graphs. Some recent work addresses this problem for loopy Tanner graphs by introducing the concepts of stopping distance and stopping redundancy. By analyzing the eigenvalues and eigenvectors of the normalized incidence matrix representing a Tanner graph, we derive lower bounds on its stopping distance. Using these lower bounds, an upper bound on stopping redundancy of the difference-set codes is derived as well. I. I NTRODUCTION Considering an [ n, k, dmin] binary linear codeC specified by ap×n incidence matrixHp with columns representing bit variables, rows representingparity-checksandp ≥ n−k = p0, the corresponding Tanner graph [1] G is: G = (B ∪ Y,E) = ({b0, b1, ., bn−1} ∪ {y0, y1, ., yp−1}, E) (1) where B is the set of variables, Y is the set of single parity-check constraints and E = {(b, y) : b ∈ B, y ∈ Y } is the set of edges. It can be shown that the correspondence between Hp and the traditional parity-check matrixrepresentingC is one-to-one. Furthermore, as Y may contain more than necessary parity-checks, we also refer Hp as parity-check matrix whenp = n− k = p0, and redundant parity-check matrix otherwise. ConsideringS ⊆ B, define bit variables inS as active bits and parity-checks in the neighborhood ofS asactive parity-checks[2], respectively. We say that S is a stopping set if all the neighbors ofS, i.e., all active parity-checks, are connected to S at least twice. It is known that, for binary erasure channels (BEC), the performance of iterative decoding over G is completely determined by its stopping sets [3]. The size of the smallest stopping sets was defined as topping distance[4], which is usually denoted as s(Hp) to emphasize that it is a function of the specific (redundant) parity-check matrix representing the code. Previous investigations [3], [5] have considered the properties of random ensembles of linear codes. In contrast, we focus on the parameters of an arbitrary linear code and will analyze eigenvalues and eigenvectors of the “normalized” incidence matrixrepresenting the code. Using this technique, we will derive two lower bounds on stopping distance. Since the stopping distance is always no larger than dmin, these lower bounds are also lower bounds on minimum distance. In particular, if the graph is regular, they are Tanner’s bit-oriented bound and parity-oriented bound [2], respectively, i.e., we demonstrate that Tanner’s bounds are actually lower bounds on s(Hp) for regular Tanner graphs. Also, these lower bounds can be used to derive upper bounds on stopping redundancy , denoted asρ(C), which is defined as the minimum number of rows in a (redundant) paritycheck matrixHp for C such thats(Hp) = dmin [4]. Previously, Schwartz and Vardy [4] proved This work was supported in part by the Army Research Office DAAD19-01-1-0477 that stopping redundancy is well defined and provided bounds on ρ(C) for the family of binary Reed-Muller codes, extended Golay Codes and maximum distance separable (MDS) codes. In this work, we will provide an upper bound on ρ(C) for the family ofsimple difference-set codes, i.e., ρ(C) ≤ n, wheren is the length of the code. After introducing the elements of graphical representation of linear codes and the associated matrices, we will prove a lemma by analyzing eigenvalues of the normalized incidence matrix. Next, this lemma will be used to provide lower bounds on stopping distance for linear codes, which will also lead to Tanner’s bit-oriented bound and parity-oriented bound on min for regular Low-Density Parity-Check (LDPC) codes [6]. We continue in Section IV to show connections between our work and the work of Schwartz and Vardy by providing an upper bound on stopping redundancy of the difference-set codes. Conclusions and future work are discussed in Section V. II. GRAPH REPRESENTATIONS ANDEIGENVALUE ANALYSIS Let dv denote the degree of vertex v ∈ B ∪ Y , andS ⊆ B ∪ Y ri = weight of row i of Hp = dyi 0 ≤ i ≤ p− 1 (2) cj = weight of columnj of Hp = dbj 0 ≤ j ≤ n− 1 (3) N(v) = the set of neighbors of v = {u : (v, u) ∈ E or (u, v) ∈ E} (4) N(S) = the set of neighbors of S (5) vol(S) = the volume ofS = ∑