Recent Developments in Low-Density Parity-Check Codes

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2 Recent Developments in Low-Density Parity-Check Codes Wen-Ching Winnie Li 1, Min Lu 2 and Chenying Wang 3 1 Department of Mathematics, The Pennsylvania State University, University Park PA 16802, USA 2 Department of Mathematics, The Pennsylvania State University, University Park PA 16802, USA 3 Department of Mathematics, The Pennsylvania State University, University Park PA 16802, USA Abstract. In this paper we prove two results related to low-density parity-check (LDPC) codes. The first is to show that the generating function attached to the pseudo-codewords of an LDPC code is a rational function, answering a question raised in [6]. The combinatorial information of its numerator and denominator is also discussed. The second concerns an infinite family of q-regular bipartite graphs with large girth constructed in [8]. The LDPC codes based on these graphs have attracted much attention. We show that the first few of these graphs are Ramanujan graphs. 1 Introduction Introduced by Gallager [3], a low-density parity-check (LDPC) code C is a binary linear code equipped with a sparse parity-check matrix H. The check equations included in H are graphically described by a bipartite Tanner graph T (H) whose two vertex sets consist of the bit nodes and check nodes, corresponding to the columns and rows of H, respectively. A bit node v j is adjacent to a check node f i if and only if the ijth entry of H is 1, in other words, the jth variable occurs in the ith check equation. Thus a codeword is an assignment of 0 and 1 of the bit nodes such that the neighbors of each check node sum to zero. The vertices in T (H) have low degree because the matrix H is sparse. To capitalize on this feature, an efficient decoding scheme is designed, first by Gallager [3], to send information back and forth between bit nodes and check nodes across the Tanner graph as follows [9]. The bit nodes are initialized by the received word, which is first screened to see if it is already a codeword. If it is, then stop. If not, the information is passed to the neighboring check nodes, which compute new estimates based on the received information and error probability, and then send the new information to their neighboring bit nodes. After one round of message passing, the algorithm computes the error probability and outputs an assignment of 0 and 1 at each bit node which is again checked to see if it is a codeword. If yes, then stop; if not, repeat the above procedure. To date there are several messagepassing iterative decoding (MPID) algorithms, such as min-sum and sum-product algorithms [16, 17]. Some questions arise naturally: how does MPID compare with the traditional maximum likelihood (ML) decoding? how to avoid mis-correction? and what will happen if MPID is repeatedly applied to a codeword? These questions are addressed in 2. Since the MPID algorithms operate locally, they converge very fast. On the other hand, an algorithm acting locally can not distinguish the Tanner graph T (H) from its finite unramified covers, which are the same locally. Let T m be an m-fold unramified cover of T (H) and C m the code whose Tanner graph is T m. Since each vertex of T (H) is covered by m vertices in T m, the length of C m is m times the length of C, which is the number, n, of the bit nodes in T (H). Given a codeword c of C m, The research of the first author is supported in part by the NSF grants DMS and DMS , and by the DARPA grant HR

3 the unscaled pseudo-codeword p(c) = (c 1,..., c n ) associated to c is a vector in Z n such that for each 1 j n, c j counts the number of 1 s in c occurring at the bit nodes of T m above the bit node v j of T (H). Notice that p(c) comes from a codeword in C if and only if its normalization 1 mp(c) lies in C. If the Tanner graph T (H) is a tree, then all normalized pseudo-codewords are codewords. Otherwise, the relation between pseudo-codewords and MPID algorithms is seen through the computation tree, as explained in [16, 17]. Suffice it to say that the pseudo-codewords record all decoding errors, hence it is of utmost importance to understand them. The smallest cone containing all pseudo-codewords, called the fundamental cone K(H) attached to the LDPC code C, is determined by Koetter and Vontobel [7] in terms of inequalities derived from the check equations contained in H. Two characterizations for pseudo-codewords are given by Koetter-Li-Vontobel-Walker in [5, 6]. The first one says that they are the integral points in the cone which, modulo 2, reduce to codewords. The second is in terms of the monomials occurring in the edge zeta function of a graph. When the code C is a cycle code on a graph X, the pseudo-codewords are described by the monomials in the edge zeta function of X. But for a general C, they correspond to monomials in the edge zeta function of the Tanner graph of C satisfying a special property. In view of this, a question was raised in [6] to find a zeta function attached to the code C whose monomials describe the pseudo-codewords of C, and the zeta function itself is a rational function providing interesting combinatorial information. The first topic of this paper is to present an answer to this question. A full discussion is contained in Lu s thesis [10]. Given a vector of n variables u = (u 1,..., u n ) and a vector w = (w 1,..., w n ) in IR n, write u w for the monomial u w1 1 uwn n. The zeta function attached to an LDPC code C of length n is the zeta function attached to its fundamental cone K(H), defined as the generating function of its pseudo-codewords: Z C (u 1,..., u n ) = u p. p pseudo codeword of C Using the aforementioned first criterion of pseudo-codewords, we shall prove Theorem 1. Let C be an LDPC code of length n with parity check matrix H and fundamental cone K(H). Then its zeta function is a rational function of the form Z C (u 1,..., u n ) = σ(u) (1 u w1 ) (1 u w d ), where w 1,..., w d are pseudo-codewords with even components which generate the cone, namely K(H) = {λ 1 w λ d w d : λ i 0 for i = 1,..., d}, and σ(u) is a polynomial. To describe the numerator, more information about the cone K(H) is needed. First assume that the cone K(H) is d-dimensional generated by d linearly independent pseudo-codewords w 1,..., w d, called a simplicial d-cone. In this case σ(u) = σ Π (u) := p pseudo codeword in Π up, where Π is the half-open fundamental parallelopiped Π = {λ 1 w λ d w d : 0 λ i < 1, i = 1,..., d}. In general, the cone is triangulated into a finite union of simplicial cones, each generated by a subset of w 1,..., w d. The generating function of the pseudo-codewords contained in each simplicial cone

4 is a rational function, and Z C is a finite sum and difference of these rational functions. The least common denominator of these rational functions is the denominator of Z C as described Theorem 1. Thus the first step to explicitly determine the zeta function of the code is to find generators of the fundamental cone. To avoid the subsequent time-consuming steps of triangulation and computing the generating function of each simplicial cone, we propose a much simpler alternative Z C(u 1,..., u n ) = σ Π (u) (1 u w1 ) (1 u w d ) using the same notation as above. Its power series expansion is the sum over the pseudo-codewords p of C of monomials u p with positive integral coefficients. See 5 for terminologies and details. Now we turn to the second topic of this paper. As explained in 2, in MPID half of the girth of the Tanner graph plays the role of minimum distance in the traditional theory of error correcting codes. Thus it is desirable to constructed codes from graphs with large girth. Ramanujan graphs introduced by Lubotzky-Phillips-Sarnak [11] are known to have this property. These are connected k-regular graphs whose nontrivial eigenvalues, namely eigenvalues other than ±k, have absolute value at most 2 k 1. The extremal spectral property makes them optimal expanders, leading to wide applications. In addition, they are sparse and have large girth. Hence they appear to be good candidates for Tanner graphs. Rosenthal and Vontobel [12] were the first to construct LDPC codes based on Ramanujan graphs. For a fixed prime power q, Lazebnik and Ustimenko [8] constructed an infinite q-regular bipartite graph D(q) with vertex sets X = {x = [x, x 1, x 2,...] : x, x i IF q for i 1} and Y = {y = [y, y 1, y 2,...] : y, y i IF q for i 1}; a vertex x in X and a vertex y in Y are adjacent if their coordinates satisfy the following relations: and for i 1, y 1 = xy + x 1, y 2 = xy 1 + x 2, y 3 = yx 1 + x 3, y 4 = yx 2 + x 4, (1) y 4i+1 = xy 4i 1 + x 4i+1, y 4i+2 = xy 4i + x 4i+2, y 4i+3 = yx 4i+1 + x 4i+3, y 4i+4 = yx 4i+2 + x 4i+4. By deleting all except the first m coordinates of the vertices, one obtains a truncation graph D(m, q), which is q-regular bipartite of size 2q m. Each D(m + 1, q) is a q-fold unramified cover of D(m, q). So they form an infinite tower of covering graphs. It was shown in [8] that for m 3, the girth of D(m, q) is large, at least m + 5. Recently, using D(m, q) as the Tanner graph, Kim et al [4] constructed and studied the associated LDPC code LU(m, q). Sin and Xiang [13] determined the dimension of the code LU(3, q) for odd q, and Arslan [1] very recently settled the case of even q. In view of the nice properties that D(m, q) possess, we wonder if they happen to be Ramanujan graphs. It turns out that the first few are. Theorem 2. The nontrivial eigenvalues of D(2, q) have absolute value at most q and those of D(3, q) at most 2q. Consequently, they are Ramanujan graphs.

5 The proof given in 5 is a new version of the joint work by two of the authors (Li and Wang). More information on eigenvalues and eigenvectors can be found in the thesis of Wang [15]. This paper is organized as follows. In 2 we summarize the behavior of MPID. The conclusion is that, if not too many errors occurred, then after a definite number of iterations, the outputs remain the same and the errors are corrected. The two criteria on pseudo-codewords obtained in [6] are reviewed in 3. The rationality of the zeta function of the code (Theorem 1) is proved in 4, where examples are also exhibited. Finally in 5 we prove Theorem 2. 2 Stability of MPID and Comparison with ML Decoding Wiberg proves in [16] that if the Tanner graph is a tree, then MPID algorithm stabilizes, that is, gives the same output after finitely many iterations, and it agrees with the ML decoding. In this case, to guarantee the agreement with ML decoding, in an unpublished paper, M-H. Kang shows that the number of iterations should be at least half of the diameter of the tree. Furthermore, he constructs an example to show that, while a codeword is obtained after fewer number of iterations, it is not the one sent. In other words, mis-correction occurs. In real life applications, it is important to avoid mis-corrections. The case where the Tanner graph is not a tree is considered by Wang in her thesis [15]. Denote by g half of the girth of the Tanner graph, and set d = g 1 2. Wang proves that a similar result holds provided that not too many errors occurred to begin with. Theorem 3 (Wang). (1) Suppose each bit node has degree 2. If the number of errors in a received word is at most d, then the min-sum decoding will stabilize and can correct the errors after 2d2 +3d 2 iterations. (2) Assume the minimal degree of the bit nodes is c 3. If the number of errors in a received word is at most (c 1)d+1 1 2c 4, then the min-sum decoding will stabilize and can correct the errors after d + 1 iterations. 3 Pseudo-codewords and Graph Edge Zeta Functions 3.1 Edge Zeta Function of a Graph Let X be an undirected graph with edges e 1,..., e n. Assign the variable u i to the edge e i. To each edge path E = (e i1,..., e ik ), associate the monomial The edge zeta function of X is defined as g(e) = u i1... u ik. Z X (u 1,..., u n ) = [E](1 g(e)) 1, where [E] runs through all equivalence classes of backtrackless, tailless and primitive cycles in X. Endow two directions on each edge of X. Define the out-neighbors of a directed edge x y to be the directed edges y z with z x. The edge adjacency matrix A e associated to X is a 2n 2n matrix whose rows and columns are indexed by the directed edges of X such that ee entry of A e records the number of out-edges from e to e. Let U denote the corresponding column variable matrix such that the same variable u i is attached to both directions of e i. In their paper [14], Stark and Terras gave an explicit expression of the edge zeta function as a rational function:

6 Theorem 4 (Stark-Terras). Z X (u 1,..., u n ) = 1 det(i A e U). 3.2 Pseudo-codewords of a Cycle Code and Graph Edge Zeta Function Let C be an LDPC code with parity-check matrix H and Tanner graph T (H). If all bit nodes in T (H) has degree two, then we may regard each bit node of T (H) as representing an edge connecting the two check nodes adjacent to this bit node. Thus we obtain a graph X whose vertices are the check nodes of T (H) and whose edges are marked by the bit nodes of T (H). The codewords in C are then in one-to-one correspondence to the simple edge cycles in X, and C is called a cycle code. The relation between pseudo-codewords of a cycle code and the zeta function of the underlying graph was studied in [5]. Theorem 5 (Koetter-Li-Vontobel-Walker). Let C be a cycle code of length n on graph X with edge zeta function Z X (u 1,..., u n ). Then p = (p 1,..., p n ) is an unscaled pseudo-codeword of C if and only if there is a disjoint union of backtrackless tailless cycles in X which use the edge e i exactly p i times for i = 1,..., n. In other words, p = (p 1,..., p n ) is an unscaled pseudo-codeword of C if and only if the monomial u p = u p1 1 upn n occurs in the power series expansion of Z X (u 1,..., u n ) with nonzero coefficient. Recall that the Newton polygon of a polynomial f(x, Y ) is the convex hull of the exponents of the monomials occurring in f. Define the Newton polytope of a power series in several variables in a similar way. In particular, there is a Newton polytope associated to the edge zeta function of a graph. As mentioned in 1, the fundamental cone of an LDPC code is the smallest cone containing all pseudo-codewords of the code. Combined with the above characterization of the pseudo-codewords of a cycle code, we get Theorem 6. The fundamental cone of the cycle code C on graph X is the Newton polytope of the edge zeta function Z X associated to X. 3.3 Pseudo-codewords of a General LDPC Code and Edge Zeta Function of Its Tanner Graph Cycle codes are the simplest LDPC codes. For a general LDPC code C with parity-check matrix H, a similar result relating pseudo-codewords of C and the edge zeta function of the Tanner graph T (H) was obtained in [6]. In this case, only those monomials in the edge zeta function satisfying a special property are relevant. Theorem 7 (Koetter-Li-Vontobel-Walker). Let C be an LDPC code of length n, parity-check matrix H and Tanner graph T (H). Suppose that every bit node has even degree. Then (1) The codewords of C are in one-to-one correspondence with the simple edge cycles in T (H) such that at each bit node, either all or none of the edges from that node occur. (2) p = (p 1,..., p n ) is an unscaled pseudo-codeword of C if and only if in the edge zeta function Z T (H) of T (H) there occurs a monomial in which the degree of the variables representing the edges incident to the ith bit node is p i for 1 i n.

7 4 Zeta Function of an LDPC Code In this section we fix an LDPC code C of length n with parity-check matrix H = (h ji ) and fundamental cone K(H). The edge zeta function of its Tanner graph defined in 3 contains monomials that do not correspond to pseudo-codewords. In contrast, we shall prove in this section that Z C (u 1,..., u n ) = u p p=(p 1,...,p n) pseudo codeword of C is a rational function as described in Theorem 1. Moreover, we shall propose a much simpler rational function which also enumerates the pseudo-codewords of C, but allows a pseudo-codeword to appear multiple times. 4.1 Preliminary Results We will use the following characterization of pseudo-codewords given in [6]. Theorem 8 (Koetter-Li-Vontobel-Walker). Let p = (p 1,..., p n ) be a vector with integral components. Then the following two statements are equivalent: (1) p is an unscaled pseudo-codeword of C; (2) p K(H) and Hp t 0 (mod 2). To facilitate our proof, we review some basic results concerning the structure of cones which will be used later. We follow the terminologies in [2]. A pointed cone K IR d is a set of the form K = {v + λ 1 w 1 + λ 2 w λ m w m : λ i 0, i = 1,..., m}, where v, w 1, w 2,..., w m IR d are such that there exists a hyperplane P for which P K = {v}, in other words, K lies strictly on one side of P. The vector v is called the apex of K, and the w k s are the generators of K. The cone is called rational if v, w 1, w 2,..., w m Q d, in which case we may choose w 1,..., w m Z d by clearing the denominators. The dimension of K is the dimension of the affine space spanned by K; if K is of dimension d, we call it a d-cone. The d-cone K is simplicial if K has precisely d linearly independent generators. A collection T of simplicial d-cones is a triangulation of the d-cone K if it satisfies (a) K = S T S; and (b) For any S 1, S 2 T, S 1 S 2 is a face common to both S 1 and S 2. We say that K can be triangulated using no new generators if there exists a triangulation T such that the generators of any S T are generators of K. An important conclusion on pointed cone is Theorem 9 ([2], p. 60). Any pointed cone can be triangulated into simplicial cones using no new generators. The following result says that cones defined by generators are the same as those defined by linear inequalities. Theorem 10 ([18], p. 30). A cone K IR d is a finitely generated combination of vectors in IR d if and only if it is a finite intersection of closed linear half-spaces.

8 Our fundamental cone K(H) is defined by inequalities derived from the check equations as follows. The ith check equation j h ijx j = 0 gives rise to n inequalities h ij x j h il x l, for 1 l n, j l and the cone consists of points (x 1,..., x n ) IR n satisfying all x j 0 and all the inequalities above. By Theorem 10 it is a pointed cone with apex at the origin. Its properties are summarized in Theorem 11. The fundamental cone K(H) is a pointed rational cone with apex at the origin, it is generated by finitely many pseudo-codewords, and it can be triangulated into simplicial cones using no new generators. Proof. The generators of the fundamental cone lie on its 1-dimensional boundaries, which are the intersections of closed linear half-spaces defined by linear equations with integral coefficients, hence the cone is rational. Moreover, each 1-dimensional boundary contains pseudo-codewords, for instance, the lattice points with all even components; choose as generators a nonzero pseudo-codeword from each 1-dimensional boundary. The third assertion follows from Theorem Generating Functions of Rational Cones The generating function of a subset S IR d is the sum of monomials recording the lattice points in S: σ S (z) = σ S (z 1, z 2,..., z d ) := z m. m S Z d The following theorem describes how to enumerate the lattice points in a simplicial cone. Theorem 12 ([2], p. 62). Suppose K := {λ 1 w 1 + λ 2 w λ d w d : λ i 0, i = 1,..., d} is a simplicial rational d-cone, where w 1, w 2,..., w d Z d. Then for v IR d, the generating function σ v+k of the shifted cone v + K is the rational function σ v+k (z) = σ v+π (z) (1 z w1 )(1 z w2 ) (1 z w d ), where Π is the (half-open) fundamental parallelepiped of K: Π := {λ 1 w 1 + λ 2 w λ d w d : 0 λ 1, λ 2,..., λ d < 1}. By Theorem 9 and the fact that the intersection of simplicial cones in a triangulation is again a simplicial cone, the following consequence is evident. Corollary 13 ([2], p. 63). The generating function of a finitely generated pointed rational cone is a rational function. Here is a simple example from [2], p. 60, which illustrates how to enumerate lattice points in a two-dimensional cone.

9 Example 1. Consider the two-dimensional cone K := {λ 1 (1, 1) + λ 2 ( 2, 3) : λ 1, λ 2 0} IR 2. Its fundamental parallelogram Π is Π = {λ 1 (1, 1) + λ 2 ( 2, 3) : 0 λ 1, λ 2 < 1} IR 2. The cone K can be exactly covered by the translations of Π by nonnegative linear combinations of (1, 1) and ( 2, 3). The lattice points in Π are (0, 0), (0, 1), (0, 2), ( 1, 2) and ( 1, 3). Hence σ K (x, y) = 1 + y + y2 + x 1 y 2 + x 1 y 3 (1 xy)(1 x 2 y 3. ) 4.3 Generating Functions of Pseudo-codewords As recalled in 4.1, pseudo-codewords of C are the lattice points in K(H) congruent to codewords modulo 2. We prove the first main result of this paper, Theorem 1, by adapting the proof of Theorem 12. Theorem 14. The zeta function is rational. Z C (u 1,..., u n ) = p=(p 1,...,p n) pseudo codeword of C Proof. By Theorem 11 the fundamental cone is generated by finitely many pseudo-codewords. We only need to show that the generating function of pseudo-codewords in a simplicial rational cone is rational. Assume that a simplicial cone K is generated by the pseudo-codewords w 1, w 2,..., w d and that all generators have even components. (Multiply them by 2 if necessary). We claim that every pseudocodeword p in K can be uniquely written as p = p 0 + λ 1 w 1 + λ 2 w λ d w d (2) for some pseudo-codeword p 0 in the fundamental parallelepiped Π of K and some non-negative integers λ 1, λ 2,..., λ d. Indeed, it follows from the definition of a simplicial cone given in 4.1 that we may write p = α 1 w 1 + α 2 w α d w d uniquely for some nonnegative real numbers α 1, α 2,..., α d. Let x and {x} denote the integral part and fractional part of a non-negative real number x, respectively. Then p = p 0 + α 1 w 1 + α 2 w α d w d, where p 0 = {α 1 }w 1 + {α 2 }w {α d }w d is a lattice point since it is the difference of p and a lattice point. Clearly p 0 lies in the fundamental parallelepiped Π since 0 {α i } < 1. Again, this representation is unique. Because all the generators have even components, p is congruent to p 0 modulo 2. Therefore, by Theorem 8, p 0 is a pseudo-codeword because p is. This proves the claim with λ i = α i for i = 1, 2,..., d. u p

10 The above unique representation of each pseudo-codeword in K allows us to express the generating function σ K (z) of the pseudo-codewords in a simplicial (rational) cone K as σ K (u) := u p p = ( pseudo codeword in K u p0 ) ( u λ1w1 ) ( p 0 pseudo codeword in Π λ 1 0 λ d 0 σ Π (u) = (1 u w1 )(1 u w2 ) (1 u w, d ) u λ dw d ) where σ Π (u) = p 0 u p0 is a finite polynomial summing over the pseudo-codewords in the fundamental parallelepiped Π. Therefore σ K (u) is a rational function. By Theorem 11, the fundamental cone can be triangulated into simplicial cones. Since the intersections of simplicial cones in a triangulation are again simplicial, the generating function of the pseudo-codewords in K(H), which is Z C, can be obtained using inclusion-exclusion. It is rational because it is the sum and difference of rational functions. Remark. From the proof we can see that if we do not require that each pseudo-codeword appear exactly once in the generating function, that is, the coefficient of each monomial equal to 1, we do not have to decompose the fundamental cone into simplicial cones, which really saves a lot of time. 4.4 An Example Consider the parity-check matrix It defines the code H = ( ) C(H) = {(0, 0, 0, 0), (1, 0, 1, 0), (1, 1, 0, 1), (0, 1, 1, 1)}, and the fundamental cone K(H) is generated by the following 5 vectors with even components: w 1 = (2, 2, 0, 2), w 2 = (2, 4, 2, 0), w 3 = (2, 0, 2, 0), w 4 = (2, 0, 2, 4), w 5 = (0, 2, 2, 2). (These generators are found by the software package POLYMAKE.) A triangulation of K(H) is T (K(H)) = {S 1, S 2 }, where and S 1 = cone generated by{w 1, w 2, w 3, w 5 } S 2 = cone generated by{w 1, w 3, w 4, w 5 }.

11 Let σ S (u) be the generating function of pseudo-codewords in the region S and Π C be the fundamental parallelepiped of the cone C. then σ K(H) (u) = where σ ΠS1 (u) (1 u w1 )(1 u w2 )(1 u w3 )(1 u w5 ) + σ (u) ΠS2 (1 u w1 )(1 u w3 )(1 u w4 )(1 u w5 ) σ ΠS1 S 2 (u) (1 u w1 )(1 u w3 )(1 u w5 ), σ ΠS1 (u) = 1 + u 1 u 3 + u 2 u 3 u 4 + u 1 u 2 u 4 + u 1 u 2 2u 3 + u 1 u 2 u 2 3u 4 + u 2 1u 2 u 3 u 4 +u 1 u 2 2u 3 u u 2 1u 2 2u u 1 u 3 2u 2 3u 4 + u 2 1u 3 2u 3 u 4 + u 2 1u 2 2u 2 3u 2 4 +u 2 1u 3 2u 3 3u 4 + u 3 1u 3 2u 2 3u 4 + u 2 1u 4 2u 2 3u u 3 1u 4 2u 3 3u 2 4, and σ ΠS2 (u) = 1 + u 1 u 3 + u 1 u 2 u 4 + u 2 u 3 u 4 + u 1 u 3 u 2 4 +u 1 u 2 u 2 3u 4 + u 2 1u 2 u 3 u 4 + u 1 u 2 2u 3 u u 2 1u 2 3u 2 4 +u 2 1u 1 u 3 u u 1 u 2 u 2 3u u 2 1u 2 2u 2 3u 2 4 +u 2 1u 2 u 3 3u u 3 1u 2 u 2 3u u 2 1u 2 2u 2 3u u 3 1u 2 2u 3 3u 4 4, σ ΠS1 S 2 (u) = 1 + u 1 u 3 + u 1 u 2 u 4 + u 2 u 3 u 4 +u 1 u 2 u 2 3u 4 + u 2 1u 2 u 3 u 4 + u 1 u 2 2u 3 u u 2 1u 2 2u 2 3u 2 4. The initial terms in the Taylor series of Z C(H) (u) are Z C(H) (u) = 1 + u 1 u 3 + u 1 u 2 u 4 + u 2 u 3 u 4 + u 1 u 2 2u 3 + u 1 u 3 u u 2 1u 2 3 +u 2 1u 2 u 3 u 4 + u 1 u 2 u 2 3u 4 + u 1 u 2 2u 3 u u 2 1u 2 2u u 2 1u 2 2u u 2 1u 2 3u u 2 2u 2 3u u 3 1u 3 3 +u 2 1u 3 2u 3 u 4 + u 3 1u 2 u 2 3u 4 + u 1 u 3 2u 2 3u 4 + u 2 1u 2 u 3 3u 4 + u 2 1u 2 u 3 u u 1 u 2 u 2 3u 3 4 +u 2 1u 4 2u u 2 1u 2 3u u 3 1u 2 2u u 3 1u 3 3u u 3 1u 2 2u 3 u u 1 u 2 2u 3 3u u 2 1u 2 2u 2 3u u 4 1u Another Zeta Function Enumerating Pseudo-codewords For a general LDPC code, it is usually complicated to triangulate the fundamental cone, especially when the number of generators of the cone is much greater than the dimension of the cone. So it is desirable to find another rational function that not only records the pseudo-codewords but also has a simpler form. The proof of Theorem 14 shows that if we only want to enumerate pseudocodewords and have no restrictions on the coefficients of the monomials, then the rational function in the following theorem will serve our purpose.

12 Theorem 15. Suppose that the fundamental cone K(H) of C is generated by the pseudo-codewords w 1,..., w m with even components. Then the rational function Z C(u 1,..., u n ) = σ Π (u 1,..., u n ) (1 u w1 ) (1 u wm ) enumerates all the pseudo-codewords of C, where σ Π (u 1,..., u n ) enumerates the pseudo-codewords in Π = {λ 1 w λ m w m : 0 λ i < 1 for 1 i m}. This is because (2) still holds, although the expression may not be unique. In fact, the number of different representations of p is the coefficient of u p in Z C. Example 2. Consider again the code studied in 4.4. We compute Z C(H). There are 52 pseudocodewords in Π = {λ 1 w λ 5 w 5 : 0 λ i < 1 for 1 i 5}, and σ Π (u) = 1 + u 1 u 3 + u 2 u 3 u 4 + u 1 u 2 u 4 + u 1 u 3 u u 5 1u 6 2u 5 3u u 5 1u 6 2u 5 3u u 6 1u 5 2u 5 3u u 6 1u 6 2u 6 3u 6 4. Thus Z C(H) (u) = σ Π (u) (1 u w1 ) (1 u w5 ) = 1 + u 1 u 3 + u 1 u 2 u 4 + u 2 u 3 u 4 + u 1 u 2 2u 3 + u 1 u 3 u u 2 1u u 2 1u 3 2u 3 3u u 4 1u 3 2u 3 3u u 3 1u 3 2u 4 3u u 4 1u 5 2u 3 3u u 5 1u 5 2u 4 3u u 4 1u 5 2u 5 3u The monomials occurring in Z C(H) agree with those in Z C(H). Shown above are the lowest degree monomials whose coefficients are more than 1. 5 Eigenvalues of D(2, q) and D(3, q) A Proof of Theorem 2 In this section we prove Theorem 2. Fix a prime power q. As the eigenvalues of a q-regular graph lie between q and q, and 2 q 1 = q when q = 2, all 2-regular graphs are automatically Ramanujan. We shall assume q 3 from now on. 5.1 The Strategy By definition, D(m, q), m 1, is a q-regular bipartite graph with vertex sets the truncated X and Y, namely X(m) = {x = ([x, x 1,..., x m 1 ]} ) and Y (m) = {y = [y, y 1,..., y m 1 ]}. The adjacency matrix 0 H(m) of D(m, q) is A(m) = H(m) t, where the rows and columns of A(m) are first indexed by 0 vertices in X(m) and then those in Y (m), and H(m) t denotes the transpose of H(m). Since D(m, q) is bipartite, the eigenvalues of A(m) are symmetric, that is, ±λ occur as eigenvalues simultaneously ( and with the same ) multiplicity. Hence it suffices to check the eigenvalues of H(m)H(m) A(m) 2 t 0 = 0 H(m) t. Further, since H(m) H(m) t H(m) and H(m)H(m) t have the same characteristic polynomial, it suffices to prove that the nontrivial eigenvalues of H(m)H(m) t, which are nonnegative, are no greater than (2 q 1) 2 = 4(q 1).

13 When m = 1, D(1, q) is a complete q-regular bipartite graph so that all entries in H(1) are 1 and hence all entries in H(1)H(1) t are q. This shows that the eigenvalues of H(1)H(1) t are q 2 (the trivial eigenvalue) of multiplicity one, and 0 of multiplicity q 1. Hence D(1, q) is Ramanujan. The natural projection π m : D(m+1, q) D(m, q) omitting the last coordinate is a covering map. Thus the lifting of a function f on D(m, q) to a function F on D(m+1, q) given by F (v) = f(π m (v)) for all vertices v of D(m + 1, q) sends an eigenfunction f of A(m) to an eigenfunction F of A(m + 1) with the same eigenvalue. Notice that F takes the same value on each fibre of the projection. The space V (m + 1) of the lifted functions F has the same dimension as the space of functions f on D(m, q), which is equal to 2q m, the size of D(m, q). Since the space of functions on D(m + 1, q) constant on each fibre has dimension equal to 2q m, the number of fibres, and it contains V (m+1), we conclude that V (m + 1) is generated by functions which are constant on each fibre of the projection π m. The space W (m + 1) of the orthogonal complement of V (m + 1) in the space of functions on D(m + 1, q) has dimension 2q m+1 2q m, and it is generated by functions which sum to zero on each fibre. Moreover, W (m + 1) is invariant under A(m + 1) since A(m + 1) is diagonalizable by orthogonal functions on D(m + 1, q). Having shown that D(1, q) is Ramanujan, we shall prove that D(2, q) and D(3, q) are Ramanujan inductively. Since the space of functions on D(m + 1, q) decomposes into the direct sum V (m + 1) W (m + 1) and the eigenvalues of A(m + 1) on V (m + 1) are the same as those of A(m), it suffices to check the eigenvalues of A(m + 1) on W (m + 1). As analyzed above, this amounts to showing that the eigenvalues of H(m + 1)H(m + 1) t on the subspace W X (m + 1) of functions on X(m + 1) which sum to zero on the fibre above each vertex in X(m) are no greater than 4(q 1) for m = 1, 2. Observe that, for m 2, H(m)H(m) t = qi(m) + A 2 (m), where I(m) is the identity q m q m matrix and A 2 (m) is a q m q m matrix with rows and columns indexed by the vertices x in X(m) such that the xx -entry is 1 if the two vertices have distance 2 in D(m, q), and 0 otherwise. We proceed to compute the eigenvalues of A 2 (m) for m = 2 and The Case D(2, q) Two distinct vertices x = [x, x 1 ] and x = [x, x 1] in X(2) have distance 2 in D(2, q) if and only if they share a common neighbor y = [y, y 1 ] in Y (2). In view of the condition (1), their coordinates satisfy the relations y 1 = xy + x 1 and y 1 = x y + x 1. As x and x are two distinct neighbors of y, we have x x and x 1 = y(x x ) + x 1. To find all distance 2 neighbors of x, we let y run through all neighbors of x, which are uniquely determined by the leading coordinate y of y running through IF q ; and for each y, we collect its neighbors x with leading coordinates x running through all elements in IF q but x. Hence x has q(q 1) distance 2 neighbors, which are [x, x 1] with x IF q \ {x} and x 1 IF q. So as an operator on W X (2), A 2 (2) maps a function f W X (2) to another function A 2 (2)f defined by A 2 (2)f([x, x 1 ]) = f([x, x 1]) = 0 x 1 IFq x IF q\{x} since the points [x, x 1] with x 1 running through IF q are the points on the fibre above x X(1) over which f sums to zero. Therefore A 2 (2) is the zero operator and the eigenvalues of H(2)H(2) t on W X (2) are all equal to q. This proves Theorem 16. The eigenvalues of D(2, q) are q of multiplicity one, 0 of multiplicity 2q 1, and ± q of multiplicity q 2 q. Consequently, it is a Ramanujan graph.

14 5.3 The Case D(3, q) The procedure is similar to the previous case, but the situation is more complicated. Two distinct vertices x = [x, x 1, x 2 ] and x = [x, x 1, x 2] in X(3) have distance 2 in D(3, q) if and only if they share a common neighbor y = [y, y 1, y 2 ] in Y (3). By (1), their coordinates satisfy the relations y 1 = xy + x 1, y 2 = xy 1 + x 2, y 1 = x y + x 1, and y 2 = x y 1 + x 2. This implies that x 1 = y(x x ) + x 1 and x 2 = y 1 (x x ) + x 2 = xx 1 x x 1 + x 2. Hence [x, x 1, x 2 ] has q(q 1) distance 2 neighbors, which are [x, x 1, xx 1 x x 1 + x 2 ] with x IF q \ {x} and x 1 IF q. The operator A 2 := A 2 (3) on W X (3) sends f to A 2 f defined by A 2 f([x, x 1, x 2 ]) = x IF q\{x} x 1 IFq f([x, x 1, xx 1 x x 1 + x 2 ]). To proceed, we need another operator C on W X (3) which sends f to Cf defined by Cf([x, x 1, x 2 ]) = x 1 IFq f([x, x 1, x(x 1 x 1 ) + x 2 ]). The invariance of W X (3) under the action of C is easily checked by summing over x 2 IF q in the above formula. The properties of A 2 and C are summarized below. Lemma 17. On the space W X (3) there hold (1) (A 2 + C) 2 = q 2 I; (2) A 2 C = CA 2 = 0; and (3) C 2 = qc. Proof. (1) By definition, we have (A 2 + C) 2 f([x, x 1, x 2 ]) = x, x 1 IFq(A 2 + C)f([x, x 1, xx 1 x x 1 + x 2 ]) = x, x 1 IFq z, z 1 IF q f([z, z 1, x z 1 zx 1 + xx 1 x x 1 + x 2 ]) = z, z 1 IF q x, x 1 IFq f([z, z 1, x 1(x z) + x (z 1 x 1 ) + x 2 ]). With z and z 1 fixed, if either z x or z 1 x 1, the sum over x 1 or x is over the points on the fibre above [z, z 1 ] and hence is equal to zero. When z = x and z 1 = x 1, the inner sum is independent of x and x 1 and the summand is f([x, x 1, x 2 ]), hence we obtain (A 2 + C) 2 f([x, x 1, x 2 ]) = q 2 f([x, x 1, x 2 ]), as desired. (2) We compute A 2 Cf([x, x 1, x 2 ]) = x IF q\{x} x Cf([x, x 1, xx 1 IFq 1 x x 1 + x 2 ]) x 1 IFq = x IF q\{x} z IF q f([x, z, x (z x 1) + xx 1 x x 1 + x 2 ]) = x IF q\{x} z IF q x 1 IFq f([x, z, x 1(x x ) + x (z x 1 ) + x 2 ]) = 0. This is because x x 0, therefore as x 1 runs through IF q, the point [x, z, x 1(x x )+x (z x 1 )+x 2 ] runs through all points on the fibre above [x, z] X(2).

15 Similarly, CA 2 f([x, x 1, x 2 ]) = z IF q A 2 f([x, z, x(z x 1 ) + x 2 ) = z IF q x IF q\{x} x 1 IFq f([x, x 1, xx 1 x z + x(z x 1 ) + x 2 ]) = x IF q\{x} x 1 IFq z IF q f([x, x 1, z(x x ) + x(x 1 x 1 ) + x 2 ]) = 0. (3) This identity follows from the definition of C. Indeed, C 2 f([x, x 1, x 2 ]) = x 1 IFq Cf([x, x 1, x(x 1 x 1 ) + x 2 ]) = x 1 IFq z IF q f([x, z, x(z x 1) + x(x 1 x 1 ) + x 2 ]) = x 1 IFq z IF q f([x, z, xz xx 1 + x 2 ]) = qf([x, x 1, x 2 ]). It follows from the above lemma that (A 2 + C) 2 = A qc = q 2 I. Moreover, the minimal polynomial of C on W X (3) divides x(x q), which implies that C on W X (3) is diagonalizable and the eigenvalues are among q and 0. It is easy to verify that both occur as eigenvalues. Since A 2 and C commute with each other and are both diagonalizable, they can be simultaneously diagonalized. This shows that the eigenvalues of A 2 2 on W X (3) are 0 and q 2, which in turn implies that those of A 2 are 0 and ±q, and consequently, those of H(3)H(3) t are 2q, q and 0. We have shown Theorem 18. The eigenvalues of D(3, q) are q (of multiplicity one), ± 2q, ± q, and 0. Consequently D(3, q) is a Ramanujan graph. This completes the proof of Theorem 2. References 1. Arslan, O.: The dimension of LU(3,q) codes. preprint (2008), available online under 2. Beck, M., Robins, S.: Computing the Continuous Discretely : integer-point enumeration in polyhedra. Springer, New York, Gallager, R. G.: Low-density parity-check codes. IRE Trans. Inform. Theory 8 (1962), Kim, J-L., Peled, U. N., Perepelitsa, I., Pless, V., Friedl, S.: Explicit construction of families of LDPC codes with no 4-cycles. IEEE Trans. Inform. Theory 50(2004), Koetter, R., Li, W-C. W., Vontobel, P. O., Walker J. L.: Pseudo-codewords of cycle codes via zeta functions. Proc. IEEE Inform. Theory Workshop, San Antonio, TX, USA (2004), Koetter, R., Li, W-C. W., Vontobel, P. O., Walker J. L.: Characterizations of pseudo-codewords of (lowdensity) parity check codes. Adv. in Math. 213 (2007), Koetter, R., Vontobel, P. O.: Graph covers and iterative decoding of finite-length codes. Proc. 3rd Intern. Conf. on Turbo Codes and Related Topics, Brest, France (2003), Lazebnik, F., Ustimenko, V. A.: Explicit construction of graphs with arbitrary large girth and of largh size. Discrete Applied Math. 60 (1997), Loeliger, H.-A.: An introduction to factor graphs. IEEE Sig. Proc. Mag. 21 (2004), Lu, M.: Low-density parity-check codes: asymtotic behavior and zeta functions. Thesis, Penn State University, U.S.A. (2009). 11. Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8 (1988), Rosenthal, J., Vontobel, P. O.: Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis. Proceedings of the 38-th Allerton Conference on Communication, Control, and Computing (2000),

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