LDPC Codes From Generalized Polygons - IEEE Xplore

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 11, NOVEMBER 2005

LDPC Codes From Generalized Polygons Zhenyu Liu, Student Member, IEEE, and Dimitris A. Pados, Member, IEEE

Abstract—We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation. Index Terms—Belief propagation, cyclic codes, generalized polygons, low-density parity-check (LDPC) codes, sum–product algorithm.

I. INTRODUCTION

L

OW-density parity-check (LDPC) codes [1]–[3] are currently capturing the interest of coding theory researchers and communication systems practitioners alike. LDPC codes can be very long, and yet can be decoded effectively and practically via the iterative belief-propagation (BP) algorithm (also known as sum–product) [2]. “Good” LDPC codes are usually generated randomly by computer. Decoder-aware (BP processing) generator tuning [4] has produced codes performing extremely close to the Shannon limit. These codes, however, also exhibit several disadvantages. Frequently, they have small minimum distance inducing an early error floor. Most are irregular codes with no easy way to describe. Their generator matrices are not sparse at all, hence, the encoding complexity is by all accounts high. Lately, more and more LDPC codes with explicit algebraic/ combinatorial construction have been reported [5]–[9]. Compared with random LDPC codes, algebraic LDPC codes have Manuscript received December 29, 2004; revised May 16, 2005. This work was supported in part by the U.S. Air Force Research Laboratory under Agreement FA8750-04-2-0179 and the Air Force Office of Scientific Research under Grant FA9550-04-1-0256. The material in this paper was presented in part at the First IEEE Upstate New York Workshop on Communications and Networking, Rochester, NY, November 2004. Z. Liu was with the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA. He is now with Marvell Semiconductor, Inc., Sunnyvale, CA 94089 USA (e-mail:[email protected]). D. A. Pados is with the Department of Electrical Engineering, State University of New York at Buffalo, Buffalo, NY 14260 USA (e-mail:pados@eng. buffalo.edu). Communicated by Ø. Yitrehus, Associate Editor for Coding Techniques. Digital Object Identifier 10.1109/TIT.2005.856936

a more symmetric structure, better minimum distance, and are easier to describe and encode. A commonly used approach when searching for good LDPC code designs is the maximization of the girth of the associated Tanner graph [6], [10], [11]. It is well known that high girth decreases the dependence between passing messages in the BP algorithm and improves the minimum distance of the code. An opposing, less used, objective may be the minimization of the diameter of the Tanner graph. Low diameter enables efficient BP message passing and reduces the required number of decoding iterations. For example, the codes based on two-dimensional finite geometries [5] have diameter six or eight and are known to converge very fast compared with similar rate and length random codes. Obviously, maximum girth and minimum diameter are conflicting requirements since in any graph the girth is always upper-bounded by two times the diameter length. The graphs achieving exactly this bound are called generalized polygons (GPs) and were, to the best of our knowledge, first researched by Tits [12]. According to the celebrated theorem of Feit and Higman [13], nontrivial finite GPs exist only with or . In this paper, we call these objects finite diameter generalized triangles, quadrangles, hexagons, and octagons, respectively, and use them to derive corresponding LDPC codes. Interestingly, among them the finite generalized triangle codes are precisely the two-dimensional projective geometry LDPC (PG-LDPC) codes in [5]. These codes were seen to perform extremely well especially at low-to-medium lengths. However, having the diameter at its smallest value (three, that is) makes the connectivity of the Tanner graph very strong and the row/column weight of the parity-check matrix grows quickly with the code length (roughly proportional to the square root). This makes the parity matrix not as “sparse” as in random codes and greatly increases the decoding complexity. By using higher diameter/girth GPs, the connections of the Tanner graph become weaker with reduced associated decoding cost. Some codes based specifically on generalized quadrangles were discussed earlier in [14]. In this paper, we derive and study LDPC codes based on all classical generalized polygons [15]. In particular: i) We show or ) polygons define codebooks of that even-diameter ( even-weight codewords. ii) We observe that even-diameter polygons with equal number of points and lines have a regular/nonregular dual form that gives rise to two LDPC codes of the same length and rate but of different minimum distance. Theoretically, we see that the nonregular polygon code has minimum distance that is better at least by two over its regular dual. Numerical upper-bound studies indicate that the difference in minimum distance may be increasing significantly in favor of the nonregular polygon code as the code length increases. iii) For LDPC codes from even-diameter nonregular polygons with

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LIU AND PADOS: LDPC CODES FROM GENERALIZED POLYGONS

equal number of points and lines we produce a tighter minimum distance bound than the one in [16]. iv) Using the intersection of the Singer group and the automorphism group [17] of the polygon, we prove that all codes based on classical generalized quadrangles are quasi-cyclic and give the explicit size of circulant blocks in the parity-check matrix. Our simulation studies demonstrate that the LDPC codes that we derived from generalized polygons, especially from split Cayley hexagons, have superior performance, fast convergence, and very low decoding complexity (due to the low weight of the parity-check matrix). We also find that many GP codes can be decoded very effectively by the low-complexity normalized BP-based algorithm of [18] and, somewhat surprisingly, the induced bit-error rate (BER) can be even lower than full BP decoding. The rest of the paper is organized as follows. In Section II, we present a brief overview of the theory of finite generalized polygons and introduce our notation. In Section III, we examine basic properties of LDPC codes derived from GPs. Simulation findings on the performance of some selected codes are presented in Section IV. A few concluding remarks are given in Section V.

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is regular and has order . Generalized polygons with are called finite. It can be proved finite point and line sets that every finite generalized polygon has an order. Therefore, a finite weak generalized polygon is a finite generalized polygon and . The incidence matrix of if and only if is the matrix with rows labeled by the lines of , columns labeled by the points of , and entries or depending on the corresponding point-to-line incidence. , by exchanging For each incidence structure the roles of and we obtain a dual structure , where if and only if . It is obvious that the dual of a (weak) generalized polygon is again a (weak) generalized polygon. The incidence matrices of and are the transpose of each other. are Finally, we say that two generalized polygons and isomorphic if there exists a one-to-one mapping such that and if and only if is called an isomorphism. The automorphism is an isomorphism from to itself. All automorphisms of put together form the automorphism group . C. Finite Generalized Polygons

II. GENERALIZED POLYGONS: DEFINITIONS AND NOTATION A. Finite Fields as Projective Spaces GF and an -dimenConsider a finite field sional vector space over . Define the -dimensional prodenoted by PG or PG with jective space over for points the equivalence classes all (all one-dimensional subspaces with are the two-dithe zero vector excluded). The lines of PG , mensional subspaces . As in [5] and [17], for example, we may set denoted by equal to the extension field GF which can also with dimension . In be viewed as a vector space over this form, if is a primitive element of then is a primitive element of . The equivalence classes for points become

and the line

is simply

B. Incidence Structures is incident to a line if We say that a point and we call and only if an incidence structure. Two points incident to a common line are called collinear. The point-line incidence graph or simply incidence graph of is an undirected bipartite graph with vertex partitions the sets and and edges the elements of . We say that is a weak generalized polygon if and only if the diameter of is half of its girth. is a generalized polygon if and only if it is a weak generalized polygon and each vertex of has degree at least three. If all point vertices have the same degree and all line vertices have the same degree , then

are trivial Finite generalized polygons with diameter (each point is incident to every line). By Feit and Higman [13], and order finite generalized polygons with diameter exist only in the following cases: , 1) , 2) is a square, 3) is a square. 4) If , then is a projective plane. This case encompasses the two-dimensional projective geometry developments (and derived codes) in [5], as explained in the Introduction. We will . focus in the sequel on the cases D. Classical Finite Generalized Quadrangles (GQs) The so-called classical finite generalized quadrangles (GQs) are constructed based on either the totally isotropic subspaces of a nondegenerate (unitary or symplectic) polarity1 of index two or a nondegenerate quadric2 of index two on some finite projective space PG . The point set is the set of all totally isotropic points (also called absolute points) of or the set of points of , respectively. The line set is the set of all totally isotropic lines of or the set of all lines that are contained in , respectively. The incidence set is the set of all pairs such that in PG point is on line . In the case of a unitary polarity, symplectic polarity, or a quadric, we have unitary, symplectic, or orthogonal classical GQs, respectively. , a nondegenerate quadric of index For every finite field exists (uniquely) only if or and two on PG 1A nondegenerate polarity on an n-dimensional projective space is an involutary permutation of its subspaces that sends a rank r subspace to a rank n +1 r subspace and inverts the inclusion relation. A totally isotropic subspace with respect to a nondegenerate polarity is completely contained in its image under the polarity. The index of the polarity is the rank of the maximal totally isotropic subspaces with respect to the polarity. 2A nondegenerate quadric is the set of all points x that satisfy a nondegenerate quadratic form q(x ) = 0. The index of is the rank of the maximal subspaces in .

0

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TABLE I GP-LDPC CODES: RATE AND MINIMUM DISTANCE

the quadric is of hyperbolic, parabolic, or elliptic type, respectively. For two points in the space represented by vectors and , the canonical quadratic forms are if if if

(hyperbolic) (parabolic) elliptic) (1)

and the corresponding symmetric bilinear forms are (2) . In this where is an irreducible binary quadratic form in and two points setup, a point is in if and only if and are collinear if and only if . The GQs genand erated from such quadrics are usually denoted by [15]. Note that when , the order behave order ; therefore, are weak. comes A nondegenerate unitary polarity of PG exists if and . The polarity has only if is a square. Assume then that or and is again unique for each index if and only if up to isomorphism. The canonical Hermitian forms are (3) is the unique involutary field automorphism of where . A point is absolute if and only if and two . The genpoints and are collinear if and only if erated GQs are denoted by and have order [15]. It can be shown that is isomorphic to the dual of . with index A nondegenerate symplectic polarity of PG exists (uniquely) only when . The canonical alternating form is (4)

The absolute and collinear conditions are the same as in unitary for any ; types. However, it can be shown that are absolute in this case. The therefore, all points in PG generated GQs are denoted by and have order [15]. is isomorphic to the dual of is self-dual (isomorphic to its dual) if and only if is even. E. Classical Finite Generalized Hexagons and Octagons known as split The classical generalized hexagons known as twisted triality Cayley hexagons and and , respectively [15]. A hexagons have order split Cayley hexagon is self-dual if and only if is a multiple of have three. The classical Ree–Tits generalized octagons and exist only when . order Explicit constructions for these hexagons and octagons can be found in [15] and will not be needed here.

III. LDPC CODES FROM GENERALIZED POLYGONS In the remainder of this paper, is always a fiand even diameter , nite generalized polygon with order GF is the base field of the corresponding underand lying projective space. A. Code Construction It is straightforward to construct an LDPC code from a given generalized polygon . We simply use the incidence graph of as the Tanner graph of the code. Without loss of generality, we view the point-set vertices as bit nodes and the line-set vertices as check nodes. It is easy to verify that the parity-check matrix of the constructed code is exactly the incidence matrix of . In the sequel, when there is no fear of ambiguity, we will call our codes by their polygon-origin name, for example etc.


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Fig. 1. Tree structure example of a generalized quadrangle. Active (inactive) bits, checks, and edges are denoted by solid (open) circles, squares, and lines, respectively. (a) Tree structure. (b) Tree structure after removing C and all inactive elements.

B. Rate and Minimum Distance It is known [14] that the length of polygon codes is

and the number of checks is

Since all finite generalized polygons have order, all GP-LDPC codes are regular LDPC codes with fixed row weight and column weight . Code rates or rate bounds can be found/extracted from the early works by Bagchi et al. [16], [19], [20] on “codes associated with generalized polygons” before the rediscovery and wide-spread recognition of the importance of BP-decoded LDPC codes. The minimum distance bound of [21] is in fact tight (holds with equality) and for the regular polygons [16]. We observed and note that the recent “bit-oriented bound” in [22] is tighter for some very low rate codes and ). These findings are ( summarized in Table I. A new result on polygon LDPC codes is given below in the form of a proposition. Proposition 1: If a generalized polygon has even diameter , then all codewords of the corresponding LDPC code have even Hamming weight.

Proof: Using an arbitrary check node as the root we arrange the incidence graph of the polygon in the form of a tree excluding the deepest check-node level, as depicted in Fig. 1(a) where check nodes are represented by squares and bit nodes by circles. Following the terminology of [22], we say that a bit node is active for a codeword if ; an edge is active if it connects to an active bit node; a check node is active if it is connected to at least one active bit node. In Fig. 1, active bits, edges, and checks are indicated by solid circles, lines, and squares, respectively. The set of nodes at each depth level from the bottom up (checks, bits, checks, bits, ) is denoted . If an active node is connected to its by parent by an inactive edge or has no parent, then we call that node an active root. Since all edges connected to an active bit are active, an active root is necessarily an active check. We reand all inactive elements, as in move now the bottom level Fig. 1(b). The graph structure becomes a forest and each active root node becomes the root of a tree. As a result, every active bit node is connected to a unique active root. We denote the number of active bit nodes that are descendents . For any , of a node , including itself if active, by we claim that if is in level and is not an active root, then (5) , the result is The claim can be proved by induction. For . If clear. Assume (5) holds true for some arbitrary level is odd and is in level , then is a check node (but not an

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active root) and has an odd number, say , of bit node children. So

transformation by the primitive element ) and ically, if the resulting subgroup is generated by has order

. Specif, then it

(6) , then is a bit node with If is even and is in level check node children (none of them is an active root). So

(7) Expressions (6) and (7) establish that (5) is valid for any when is not an active root. If is an active root, then it has an even number, say , of bit node children and (8)

where The codeword weight is active roots. By (8), the weight is even.

is the set of all

Bagchi and Sastry [16] showed that the minimum-distance bound for generalized polygon codes is tight only if subpolygon. the corresponding polygon contains a proper As a particularly interesting case, we recognize that with odd are nonregular (in fact, anti-regular) and do not subpolygon. The same holds true contain any proper . Therefore, we can state the for following two corollaries. Corollary 1: The LDPC codes derived from the orthogonal classical finite generalized quadrangles odd, have . Corollary 2: The LDPC codes derived from the split Cayley classical finite generalized hexagons have . The results for odd, and not a power of , are also included in Table I. Experimentally, numerical upper bound (small-weight codeword) search via the procedure may in fact be growing much larger in [23] suggests that and , respectively, as the code length than increases.

and each orbit of the set has the same size . If, in addition, we require that each circulant block has the same size , then every orbit of the set should have the same size as well under this subgroup. We continue with the following lemma. and , if acts Lemma 1: For any , then each line of on fixed-point-freely and has the same orbit size under the action of . In particular, acts fixed-point-freely on as well. is such that for any . Proof: Suppose , the order of , say , must divide . Since Since and is fixed by acts fixed-point-freely on the under points of . Therefore, every point of has orbit size and divides . We conclude that divides and is the identity element. According to Lemma 1, if we arrange the parity-check matrix so that all bits in the same orbit under such a group are next to each other and all checks in the same orbit are next to each other, then the parity-check matrix is divided into circulant blocks, each of size . As explained earlier, the classical GQ codes are of the following six types: and . and are the dual of Among them, the others. If a parity-check matrix is block-wise cyclic, then its transpose, i.e., the parity-check matrix of the code from the dual polygon, is also block-wise cyclic. Hence, we only need and to consider three quadrangle types, say . First we deal with . Consider in the projective where the points of PG are identified by space PG GF . Define the form the element classes of where and, for convenience, denote the field GF by . We claim that is a nondegen. Certainly, is -bilinear erate symplectic form on is a basis of over , for any and alternating. Since we have (note that )

C. Encoding of Finite GQ LDPC Codes In [24], it was shown that it is possible to have a quasi-cyclic structure for certain finite GQ codes. In this subsection, we prove that a quasi-cyclic structure exists for all classical finite GQ codes and we give the exact size of the circulant blocks. In our derivations, we will be following the common approach of arranging the parity-check matrix of the code in a block-wise cyclic form and applying the “transform theory” acting [25]. In doing so, we use a cyclic subgroup of fixed-point-freely on both and . A choice of such an automorphism group that is fixed-point-free on is the intersection of the Singer group (that is, the cyclic group generated from the

Thus, is a symplectic form on nonzero suppose

. In addition, for any . Then

Therefore, is nondegenerate. Define where is the trace function from to . Then, is -linear, alternating, and nondegenerate since is nondegenerate [17]. Therefore, is a nondegenerate symplectic form on PG and generates the unique GQ . We can now verify that is an element of since it


preserves is

. The order of

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in

, so

even odd

(9)

and for both even and odd . According to in Lemma 1, we can arrange the parity-check matrix of a block-wise cyclic form where each circulant block is of size or , if is even or odd, respectively. in the projective space PG Next, we consider whose points are identified by the element classes of GF . Define the form where is the unique involutary automorphism of the field and is the trace function from to . For any ; hence, is -Hermitian. Similar to the case, is nondegenerate as well. Therefore, is a nondegenerate unitary form on PG and generates the GQ . is an element of since it preserves . The order of in is , so (10) and . By Lemma 1, the size of each circulant . block is in the projective space PG Finally, we consider whose points are identified by the element classes of GF . The form is a nondegenerate quadratic since is in fact the norm function from form on to GF . Then, is the nondegenerate quadratic form on PG where is the trace function from to . According to [17], and are both elliptic; thus, defines the unique GQ . is an element of since it preserves . The order of is and even odd. However, this time so we try a smaller subgroup. If , we take . Then, the circulant block size is (11) as take

. If . Then

, we

(12) . and We have established that all classical finite generalized quadrangles create quasi-cyclic LDPC codes and we have found the size of the circulant blocks. We summarize these findings in the form a proposition as follows. Proposition 2: All classical finite generalized quadrangle LDPC codes are quasi-cyclic. Their circulant block size is as follows.

and even and odd, respectively; and , for respectively; and

or

, for or ,

and .

The following section is devoted to simulation studies and demonstrations. IV. SIMULATION RESULTS In this section, we examine the error rate performance of a few selected generalized polygon LDPC codes. In Fig. 2 we code with rate . Decoding consider the is performed by either BP or the low complexity “normalized BP-based” procedure of [18] (the normalization factor is set ). For code comparison purposes, we generate a second to LDPC code using the progressive edge-growth (PEG) algorithm [10]. The PEG code generator has been known to produce some of the best LDPC codes. The bit node degree sequence used in the construction is density evolution optiand was seen by experimental mized with maximum degree trials to offer the best performing PEG LDPC codes for this setup. The exact degree sequence used, as returned by [26], is . We can see that at , the BP decoded generalized polygon code is BER only 2.7 dB away from the Shannon limit and 0.25 dB away from the PEG code. Moreover—and quite interestingly—the normalized BP-based algorithm works exceptionally well on and outperforms BP decoding in the high signal-to-noise ratio (SNR) area with significantly lower computational cost. For illustration purposes, Fig. 2 also includes the regular , which is of the same length polygon dual of is and rate (decoded by BP). The minimum distance of . Using the algorithm provided in [23] to search for the small-weight codeword, we see that the minimum may be as high as (returned upper bound). distance of and in Fig. 2 is not While the BER difference of that pronounced, we do observe an error floor forming in the . Fig. 3 shows that the block-error-rate high-SNR area for is clearly inferior to . In fact, in decoding of we find a small amount of undetected errors (incorrect codeall errors are detected word decisions) while in decoding errors (decoding failures). Since GP-LDPC codes have the strongest connectivity in their Tanner graph for a given girth, convergence of the BP decoder is very fast. In Fig. 4, we plot the BER curve of under maximum BP iteration number 10, 15, and 50. At BER , the performance gap between 10 and 50 iterations is about 0.2 dB; the gap between 15 and 50 iterations narrows down to about 0.1 dB. Rapid BP convergence is a GP-LDPC property shared with some of the other algebraically generated codes such as EG-LDPC codes [5]. GP-LDPC codes, in addition, have small row and column weight (for example, five code) which coupled with normalized BP-based for this decoding makes the decoding cost exceptionally low.

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Fig. 2.

BER performance of the (1365; 378) H (4) and H (4) hexagon codes and a best available (1365; 378) PEG generated code.

Fig. 3.

Block-error-rate performance comparison of the H (4) and H (4) (1365; 378) codes.

As a final example, we plot in Fig. 5 the BER performance of code with rate . Its dual polygon the , is also included in the study as well as a best availcode, able PEG code of the same length/rate generated with bit node degree sequence from [27]. Normalized BP-based decoded outperforms the PEG code. The minimum distance of searched with [23] is upper-bounded by , which—if

(nearly) tight—can be a big improvement over the minimum of for which an obvious error distance floor can be seen. It is arguably surprising to report that low-complexity normalized BP-based decoding [18] performs ; at BER that much better than full BP decoding on the gain is about 0.25 dB. The identified quasi-cyclic nature of quadrangle codes makes the encoding cost of very manageable as well.


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Fig. 4.

Performance progression of belief-propagation decoding of H (4).

Fig. 5.

BER performance of the (400; 175) Q(4; 7) and W (7) quadrangle codes and a best available (400; 175) PEG generated code.

V. CONCLUSION We derived algebraic LDPC codes from finite classical generalized polygons and established new codebook properties for even-diameter (four, six, or eight) polygon codes. In particular, we showed that all codewords have even Hamming weight. We observed that the dual regular/nonregular quadrangles and and the dual regular/nonregular (split Cayley) and give rise to corresponding classes hexagons

of LDPC codes with the same length and rate but different minimum distance that favors the nonregular polygon code constructions. For such LDPC codes (from the nonregular and polygons) a new minimum-distance bound was presented. Finally, we were able to prove that all quadrangle LDPC codes are quasi-cyclic and we found their exact circulant block size. Simulation studies demonstrated that very low error rates can be achieved by nonregular polygon LDPC codes even for

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small code lengths. For all polygon LDPC codes, BP decoding exhibits rapid convergence with low cost per iteration due to the inherently small row and column weights. Interestingly, low-cost normalized BP-based decoding was seen to outperform in error rate full-cost BP decoding on nonregular polygon codes. If we also take advantage of the reported quasi-cyclic structure of quadrangle-based codes for encoding purposes, then the overall performance versus encoding/decoding complexity tradeoff point appears quite attractive. REFERENCES [1] R. G. Gallager, “Low-density parity-check codes,” IRE Trans. Inf. Theory, vol. IT-8, no. 1, pp. 21–28, Jan. 1962. , Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, [2] 1963. [3] D. J. C. MacKay and R. M. Neal, “Near Shannon limit performance of low-density parity-check codes,” Electron. Lett., vol. 32, pp. 1645–1646, Aug. 1996. [4] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 619–637, Feb. 2001. [5] Y. Kou, S. Lin, and M. P. C. Fossorier, “Low-density parity-check codes based on finite geometries: A rediscovery and new results,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2711–2736, Nov. 2001. [6] G. A. Margulis, “Explicit constructions of graphs without short cycles and low-density codes,” Combinatorica, vol. 2, pp. 71–78, 1982. [7] S. J. Johnson and S. R. Weller, “Resolvable 2-designs for regular lowdensity parity-check codes,” IEEE Trans. Commun., vol. 51, no. 9, pp. 1413–1419, Sep. 2003. [8] B. Vasic and O. Milenkovic, “Combinatorial constructions of low-density parity-check codes for iterative decoding,” IEEE Trans. Inf. Theory, vol. 50, no. 5, pp. 1156–1176, Jun. 2004. [9] J.-L. Kim, U. N. Pelde, I. Perepelitsa, V. Pless, and S. Friedland, “Explicit construction of families of LDPC codes with no 4-cycles,” IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2378–2388, Oct. 2004. [10] X.-Y. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregular progressive edge-growth Tanner graphs,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 386–398, Jan. 2005.

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