Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 1 Shortened Arry Codes of Lrge Girth Olgic Milenkovic, Member, IEEE, Nvin Kshyp, Member, IEEE, nd Dvid Leyb rxiv:cs/0504016v2 [cs.dm] 26 Apr 2006 Abstrct One pproch to designing structured low-density prity-check (LDPC) codes with lrge girth is to shorten codes with smll girth in such mnner tht the deleted columns of the prity-check mtrix contin ll the vribles involved in short cycles. This pproch is especilly effective if the prity-check mtrix of code is mtrix composed of blocks of circulnt permuttion mtrices, s is the cse for the clss of codes known s rry codes. We show how to shorten rry codes by deleting certin columns of their prity-check mtrices so s to increse their girth. The shortening pproch is bsed on the observtion tht for rry codes, nd in fct for slightly more generl clss of LDPC codes, the cycles in the corresponding Tnner grph re governed by certin homogeneous liner equtions with integer coefficients. Consequently, we cn selectively eliminte cycles from n rry code by only retining those columns from the prity-check mtrix of the originl code tht re indexed by integer sequences tht do not contin solutions to the equtions governing those cycles. We provide Rmsey-theoretic estimtes for the mximum number of columns tht cn be retined from the originl prity-check mtrix with the property tht the sequence of their indices void solutions to vrious types of cycle-governing equtions. This trnsltes to estimtes of the rte penlty incurred in shortening code to eliminte cycles. Simultion results show tht for the codes considered, shortening them to increse the girth cn led to significnt gins in signlto-noise rtio in the cse of communiction over n dditive white Gussin noise chnnel. Index Terms Arry codes, LDPC codes, shortening, cyclegoverning equtions I. INTRODUCTION Despite their excellent error-correcting properties, lowdensity prity-check (LDPC) codes with rndom-like structure [8], [17, pp. 556 572] hve severl shortcomings. The most importnt of these is the lck of mthemticl structure in the prity-check mtrices of such codes, which leds to incresed encoding complexity nd prohibitively lrge storge requirements. These issues cn usully be resolved by using structured LDPC codes, but t the cost of some performnce loss. This performnce loss my be ttributed to the fct tht lgebric code design techniques introduce vrious constrints on the set of code prmeters influencing the performnce of belief propgtion decoding, so tht it is hrd to optimize the overll structure of the code. One prmeter tht is usully trgeted for optimiztion in the process of designing structured LDPC codes is the girth Mnuscript submitted Mrch 30, 2005, revised December 11, 2005. This work NSF Grnt CCF-0514857 wrded to O. Milenkovic, Lincoln Lbortory Fellowship wrded to D. Leyb, nd n NSERC Discovery Grnt wrded to N. Kshyp. Some of the results in this work were presented t the 42nd Allerton Conference on Communiction, Control nd Computing, Monticello, IL, USA, Sept. 2004. Olgic Milenkovic nd Dvid Leyb re with the Deprtment of Electricl nd Computer Engineering, University of Colordo, Boulder, CO 80309, USA. (emil: olgic.milenkovic@colordo.edu, dvid.leyb@colordo.edu) Nvin Kshyp is with the Deprtment of Mthemtics nd Sttistics t Queen s University, Kingston, ON K7L 3N6, Cnd. (emil: nkshyp@ mst.queensu.c). of the underlying Tnner grph. Severl clsses of structured LDPC codes with moderte nd lrge vlues of girth nd good performnce under itertive decoding re known, exmples of which cn be found in [10], [12] [14], [19], [22], [26], [29]. In this pper, we focus our ttention on clss of LDPC codes termed rry codes [5] (or equivlently, lttice codes [29]). These codes re qusi-cyclic, nd hve prity-check mtrices tht re composed of circulnt permuttion mtrices. Generl forms of such prity-check mtrices were investigted in [6] nd [27], nd codes of girth eight, ten nd twelve were obtined primrily through extensive computer serch. Fossorier [6] considered fmily of qusi-cyclic LDPC codes closely relted to rry codes, nd derived simple necessry nd sufficient conditions for such codes to hve girth lrger thn six or eight. Subsequently, codes with lrge girth were constructed with the id of computer serch strtegies which rely on rndomly generting integers until the conditions of the theorem re met. We generlize nd extend the rry code design methods in slightly different direction, nd provide less computtionintensive pproch to constructing codes with lrge girth (including vlues exceeding eight). Our pproch is bsed on the observtion tht the existence of cycles in the Tnner grph of n rry code is governed by certin homogeneous liner equtions. We show tht it is possible to exhustively list ll the equtions governing cycles of length six, eight nd ten in n rry code hving prity-check mtrix with smll number of ones in ech column. Thus, by shortening n rry code in such wy s to only retin those columns of its prity-check mtrix whose indices form sequence tht voids solutions to some of these cycle-governing equtions, one cn obtin rry codes with pre-specified distribution of cycles of vrious lengths. This provides mens of studying the effects of different clsses of cycles on rry code performnce. In prticulr, this technique cn be used to entirely eliminte cycles of short lengths, resulting in codes of girth up to twelve. One specil form of n rry code of girth eight nd column-weight three ws first described in [29] nd [30], where good choice for the set of columns to be retined from the originl prity-check mtrix ws determined using geometricl rguments. Using techniques from grph theory nd Rmsey theory, we provide nlyticl estimtes of the designed code rtes chievble by shortening n rry code to improve girth, nd present some useful lgorithms for identifying lrge sets of columnindices tht void solutions to cycle-governing equtions. Simultion results show tht eliminting short cycles using this technique leds to significnt signl-to-noise rtio (SNR) gins, over the dditive white Gussin nose (AWGN) chnnel. These codes lso compre fvorbly with other clsses of structured LDPC codes in the literture, nd in fct show mrked improvement in performnce in some cses.
2 Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 The reminder of the pper is orgnized s follows. Section 2 describes generliztion of the rry code construction nd provides some definitions needed for the subsequent exposition. In Section 3, we explicitly show how cycles in the Tnner grphs of these codes re governed by certin homogeneous liner equtions with integer coefficients. We then go on to list the equtions governing cycles of length six, eight nd ten in rry codes with prity-check mtrices of smll column-weight. Section 4 contins bounds on the size of the mximl sequence of column indices tht contins no solutions to certin homogeneous liner equtions. A greedy lgorithm for constructing such sequences, s well s some simple extensions thereof, re discussed in Section 5. Simultion results re given in Section 6, with some concluding remrks presented in Section 7. The proofs of some of the results of Section 4 re provided in the Appendix. II. ARRAY CODES Arry codes [5] re structured LDPC codes with good performnce under itertive messge-pssing decoding. Their prity-check mtrix hs the form H rr = I I I I P P q 1 I P r 1 P (r 1)(q 1), (1) where q is n odd prime, r is n integer 1 in [1, q], I is the q q identity mtrix, nd P is q q circulnt permuttion mtrix distinct from I. Recll tht permuttion mtrix is squre mtrix composed of 0 s nd 1 s, with single 1 in ech row nd column. A circulnt permuttion mtrix is permuttion mtrix tht is lso circulnt, i.e., the ith row of the mtrix cn be obtined by cycliclly shifting the (i 1)th row by one position to the right. Typiclly, the mtrix P in (1) is chosen to be the mtrix 0 1 0 0... 0 0 0 1 0... 0 P =............ 0 0... 0 0 1 1 0 0... 0 0. An LDPC code described by such prity-check mtrix is regulr, with length q 2 nd co-dimension r q. The row nd column weights of such code re q nd r, respectively. Consequently, the rte R of such codes is t lest 1 r/q. We will consider the following more generl form for prity-check mtrix: H = P 0 0 P 0 1 P 0 (q 1) P 1 0 P 1 1 P 1 (q 1) P r 1 0 P r 1 1 P r 1 (q 1) (2) where 0, 1,..., r 1 is some sequence of r distinct integers from [0, q 1]. Ech such prity-check mtrix defines code. If the sequence 0, 1,..., r 1 forms n rithmetic progression 1 In this pper, we will use the nottion [, b] to denote the set {x Z : x b}. (A.P.), i.e., if there exists n integer 0 such tht i+1 i = for i = 0, 1, 2,..., r 2, then we cll the corresponding code proper rry code (PAC). Note tht if 0 = 0, then the PAC is simply n rry code with prity-check mtrix H rr s in (1), since the prity-check mtrix in (2) hs the sme form s H rr, s cn be seen by replcing P in H rr by P. If the sequence 0, 1,..., r 1 does not form n A.P., then the corresponding code will be referred to s n improper rry code (IAC). The term rry code without further qulifiction will henceforth be used to men n IAC or PAC. Throughout the reminder of the pper, we will use the following definitions/terminology: The odd prime q used in defining the prity-check mtrix of n rry code will be referred to s the modulus of the code. A block-column (block-row) of prity-check mtrix, H, of n rry code is the submtrix formed by column (row) of permuttion mtrices from H. The q blockcolumns of H re indexed by the integers from 0 to q 1, nd the r block-rows re indexed by the integers from 0 to r 1. For exmple, the jth block-column of H is the mtrix [P 0 j P 1 j P 2 j... P r 1 j ] T. The term block-row lbels will be used to denote the integers in the sequence 0, 1,..., r 1 tht define the mtrix H in (2). A block-column-shortened rry code, or simply shortened rry code, is code whose prity-check mtrix is obtined by deleting prescribed set of block-columns from the prity-check mtrix of n rry code. The lbels of the block-columns retined in the pritycheck mtrix of the shortened code re simply their indices in the prent code. For the prent code itself, the terms lbel nd index for block-column cn be used interchngebly. A closed pth of length 2k in ny prity-check mtrix of the form in (2) is sequence of block-row nd blockcolumn index pirs (i 1, j 1 ), (i 1, j 2 ), (i 2, j 2 ), (i 2, j 3 ),..., (i k, j k ), (i k, j 1 ), with i l i l+1, j l j l+1, for l = 1, 2,..., k 1, nd i k i 1, j k j 1. The significnce of closed pths rises from the following simple but importnt result from [5] (see lso [6, Theorem 2.1]): Theorem 1. A cycle of length 2k exists in the Tnner grph of n rry code with prity-check mtrix H nd block-row lbels 0, 1,..., r 1 if nd only if there exists closed pth (i 1, j 1 ), (i 1, j 2 ), (i 2, j 2 ), (i 2, j 3 ),..., (i k, j k ), (i k, j 1 ) in H such tht P i 1 j1 (P i 1 j2 ) 1 P i 2 j2 (P i 2 j3 ) 1 P i k j k (P i k j1 ) 1 evlutes to the identity mtrix I. In fct, since P is q q circulnt permuttion mtrix, P I, nd q is prime, we cn hve P n = I if nd only if n 0 (mod q). So, the condition in the theorem is equivlent to i1 (j 1 j 2 )+ i2 (j 2 j 3 )+ + ik (j k j 1 ) 0 (mod q), (3)
MILENKOVIC, KASHYAP AND LEYBA: SHORTENED ARRAY CODES 3 which cn lso be written s j 1 ( i1 ik )+j 2 ( i2 i1 )+ +j k ( ik ik 1 ) 0 (mod q). (4) Bsed on Theorem 1, it is esily seen [5] tht rry codes re free of cycles of length four. This is becuse cycle of length four exists if nd only if there exist indices i 1, i 2, j 1, j 2, i 1 i 2, j 1 j 2 such tht ( i1 i2 )(j 1 j 2 ) 0 (mod q). which is clerly impossible since i 1 i 2 nd j 1 j 2. On the other hnd, n rry code with prity-check mtrix of the form in (1), with q 5, r 3, hs cycles of length six. An exmple is the closed pth described by the coordintes (1, 1), (1, 2), (2, 2), (2, q+3 q+3 2 ), (0, 2 ), (0, 1), which stisfies (3), since i = i in this cse, nd 1 (1 2)+2 (2 q + 3 + 3 )+0 (q 1) = q 0 (mod q). 2 2 In generl, closed pth of length six in the prity-check mtrix of n rry code must pss through three different block-rows, indexed by r 1, r 2, r 3, nd three different blockcolumns, indexed by i, j, k. In the cse of PAC, the block-row lbels 0, 1,..., r 1 form n A.P. with common difference, 0 < < q, nd hence (4) reduces to [i(r 1 r 3 ) + j(r 2 r 1 ) + k(r 3 r 2 )] 0 (mod q). Thus, PAC hs cycle of length six if nd only if there exist distinct block-row indices r 1, r 2, r 3 nd distinct block-column indices i, j, k such tht i(r 1 r 3 ) + j(r 2 r 1 ) + k(r 3 r 2 ) 0 (mod q). (5) Therefore, by shortening the PAC so s to only retin blockcolumns with lbels such tht (5) is never stisfied, we eliminte ll cycles of length six, obtining code of girth t lest eight. It is nturlly of interest to extend this kind of nlysis to cover the cse of cycles of length lrger thn six, nd utilize it to ppropritely shorten n rry code to increse its girth. The next section dels with the subject of identifying sequences of block-column lbels leding to codes with lrge girth. III. ARRAY CODES OF GIRTH EIGHT, TEN, AND TWELVE For clrity of exposition, in ll subsequent derivtions we will focus only on the two specil cses of rry codes with column weight three nd four. The results presented cn be extended in strightforwrd, lbeit tedious, mnner to codes with lrger column weights. Theorem 2. Let C be PAC with modulus q whose pritycheck mtrix, H, hs column weight r. If r = 4, then C contins cycle of length six if nd only if there exist three distinct block columns in H whose lbels i, j, k stisfy t lest one of the following two congruences: 2i + j + k 0 3i + j + 2k 0 (mod q), (mod q). If r = 3, then C contins cycle of length six if nd only if there exist three distinct block columns whose lbels i, j, k stisfy the first of the two equlities. (6) Proof. The clim for r = 4 follows immeditely from (5) once we note tht ny three block-row indices r 1, r 2, r 3 {0, 1, 2, 3}, r 1 < r 2 < r 3, must stisfy one of the following: (i) r 1 r 3 = 2, r 3 r 2 = 1, (ii) r 1 r 3 = 3, r 3 r 2 = 1, or (iii) r 1 r 3 = 3, r 3 r 2 = 2. The proof for the r = 3 cse similrly follows from the fct tht the only possible choice for the set of three distinct block-row lbels in this cse is {0, 1, 2}. A useful consequence of the bove result is Corollry 3 below, to stte which it is convenient to introduce the following definition. Here, nd in the rest of the pper, the set of positive integers is denoted by Z +, nd given n N Z +, the ring of integers modulo N is denoted by Z N. Definition 1. A sequence of distinct non-negtive integers n 1, n 2, n 3,... is defined to be non-verging sequence if it contins no term tht is the verge of two others, i.e., n i +n j = 2n k only if i = j = k. Similrly, given n N Z +, sequence of distinct integers n 1, n 2, n 3,... in [0, N 1] is non-verging over Z N if n i + n j 2n k (mod N) implies tht i = j = k. It is cler from the definition tht sequence is nonverging if nd only if it contins no non-constnt threeterm A.P. The following result is simple consequence of Theorem 2 nd Definition 1. Corollry 3. Let H be the prity-check mtrix of PAC with modulus q, consisting of three block-rows, nd let A be the 3q mq mtrix obtined by deleting some q m block-columns from H. The shortened rry code with prity-check mtrix A hs girth t lest eight if nd only if the sequence of lbels of the block-columns in A forms non-verging sequence over Z q. To extend the bove result to PAC s with four block-rows, we require the following generliztion of Definition 1. Definition 2. Let c be fixed positive integer. A sequence of distinct non-negtive integers n 1, n 2, n 3,... is defined to be c-non-verging sequence if n i +cn j = (c+1)n k implies tht i = j = k. We extend this definition s before to sequences over Z N, for n rbitrry N Z +. Note tht sequence is c-non-verging if nd only if it does not contin three elements of the form n, n+t, n+(c+1)t, for some integers n, t, with t > 0. We cn now stte the following corollry to Theorem 2. Corollry 4. Let H be the prity-check mtrix of PAC with modulus q, consisting of four block-rows, nd let A be the 4q mq mtrix obtined by deleting some q m block-columns from H. The shortened rry code with prity-check mtrix A hs girth t lest eight if nd only if the sequence of blockcolumn lbels in A is non-verging nd 2-non-verging over Z q. We next consider the cse of cycles of length eight. By the resoning used to derive (5), it follows from Theorem 1 tht PAC contins cycle of length eight if nd only if its prity-check mtrix contins closed pth of the form
4 Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 (r 1, i), (r 1, j), (r 2, j), (r 2, k), (r 3, k), (r 3, l), (r 4, l), (r 4, i) such tht i(r 1 r 4 )+j(r 2 r 1 )+k(r 3 r 2 )+l(r 4 r 3 ) 0 (mod q) (7) Note tht closed pths of length eight my pss through two, three or four different block-columns of the prity-check mtrix of the PAC. Let us first consider the sitution where closed pth psses through exctly two different block-columns. Let i nd j be the lbels of these block-columns. This closed pth forms cycle of length eight if nd only if (7) is stisfied with k = i nd l = j. A re-grouping of terms results in the eqution (i j)(r 1 + r 3 r 2 r 4 ) 0 (mod q) which, for i j, is stisfied if nd only if r 1 + r 3 r 2 r 4 0 (mod q). (8) Now, observe tht for PAC with column-weight r 3, the bove eqution is lwys stisfied by tking r 1 = 0, r 2 = 1, r 3 = 2 nd r 4 = 1. This shows tht in PAC with columnweight r 3, ny pir of block-columns is involved in cycle of length eight. Hence, shortening will never be ble to eliminte cycles of length eight from such PAC (except obviously in the trivil cse where we delete ll but one blockcolumn), implying tht shortened PAC s cn hve girth t most eight. We record this fct in the lemm below. Lemm 5. A shortened PAC of column-weight t lest three hs girth t most eight. The following theorem provides the constrining equtions tht govern cycles of length eight involving three of four different block-columns in PAC with row-weight q nd column-weight three or four. The proof is long the lines of tht of Theorem 2, nd is omitted. Theorem 6. In PAC with modulus q nd column-weight r = 3, the constrining equtions, over the ring Z q, for the block-column lbels i, j, k, l specifying cycles of length eight involving three or four different block-columns re i j k + l = 0, 2i j 2k + l = 0 2i + j 3k = 0, 2i j k = 0 For PAC s with modulus q nd column-weight r = 4, the set of constrining equtions, over Z q, for the lbels i, j, k, l tht describe cycles of length eight involving three or four different block-columns is 3i j k l = 0, 3i 2j 2k + l = 0, 3i 3j + k l = 0, 3i 3j + 2k 2l = 0, 2i 2j + k l = 0, i + j k l = 0, 2i j k = 0, 4i 3j k = 0, 3i 2j k = 0 (9) (10) Figure 1 shows the structures of some cycles of lengths six nd eight, nd provides the modulo-q eqution governing ech such cycle. The generic vribles, b, c nd i, j, k, l represent the block-row nd block-column lbels, respectively. The equtions governing ll such cycles re lso summrized in Tbles II nd III. It should be bundntly cler by now tht we cn eliminte lrge number of cycles of length eight from PAC by b c b c b c b c i j k l i j k l P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 b c P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 j i k l j i l k P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 i j k l i j k l j i P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 ( b)i + (2b c)j + (c b)k = 0 k l b c b c P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 ( c)i + (b )j + (c b)k = 0 ( c)i + (b )j + ( b)k + (c ) l = 0 c b P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 i j k l P 1 P 2 P 3 P 4 5 6 7 8 9 10 11 12 i j+k l = 0 Fig. 1. Some cycles of lengths six nd eight, nd their governing equtions. selectively deleting some of its block-columns, retining only those block-columns the set of whose lbels does not contin solutions to some or ll of the equtions listed in Theorem 6. Note lso tht the equtions listed in (6), upon relbeling the vribles if necessry, form subset of the equtions listed in (9), s well s of those in (10). Hence, if we shorten PAC in such wy s to retin only those block-columns whose lbels form non-verging nd 2-non-verging sequence, not only does the resultnt shortened code hve no cycles of length six, but it lso hs fewer cycles of length eight thn the originl code. As observed erlier, shortened PAC s cnnot hve girth lrger thn eight. This is direct consequence of the fct tht the block-row lbels of shortened PAC with column-weight t lest three lwys contin solution to (8), nd hence ny such code lwys contins cycles of length eight tht pss through pirs of distinct block-columns. On the other hnd, IAC s cn be constructed in such wy s to void cycles of length eight tht involve only two different block-columns. Anlogous to (8), the eqution governing such cycles in n IAC is r1 + r3 r2 r4 0 (mod q). (11) Thus, if the block-row lbels of the IAC re chosen so tht they do not contin solutions to (11), then such eight-cycles cnnot rise. Exmples of such sets of block-row lbels re {0, 1, 3} for n IAC with three block-rows, nd {0, 1, 3, 7} for n IAC with four block-rows. Such IAC s cn be shortened to yield codes with girth ten or twelve, provided tht the blockcolumn lbels retined in the shortened code void set of constrining equtions nlogous to (6), (9) nd (10). The
MILENKOVIC, KASHYAP AND LEYBA: SHORTENED ARRAY CODES 5 equtions governing cycles of lengths six, eight nd ten for IAC s with three block-rows (r = 3) nd lbel set {0, 1, 3} re listed in Tble IV. Similrly, Tble V lists the twentyeight equtions governing cycles of lengths six nd eight in IAC s with four block-rows (r = 4) nd lbel set {0, 1, 3, 7}. There re more thn fifty equtions governing cycles of length ten in IAC s with r = 4. These equtions were obtined vi n exhustive computer-ided nlysis of ll the possible structures tht cycles cn hve in these codes. It is worth pointing out tht Tbles II V need not only be used to construct codes with prescribed girth, but cn lso be used to design codes with pre-specified set of cycles. This cn help in studying the effects of vrious cycle clsses on the performnce of code. The structure of the prity-check mtrix in n rry code llows us to use existing results in the literture to obtin upper nd lower bounds on the minimum distnce, d, of such codes. A lower bound on d for regulr LDPC codes ws derived in [28]: { 2 (r 1)(g 2)/4 1 r 2 + 2 r (r 1)(g 2)/4, g/2 odd d 2 (r 1)g/4 1 r 2, g/2 even (12) where g is the girth of the code nd r is the column-weight of the prity-check mtrix (i.e., the degree of ny vrible node). This bound cn be improved slightly in some cses by noting tht the minimum distnce of n rry code must be even, since the code cn hve even-weight codewords only. This is consequence of the fct tht within ny block-row, [P i 0 P i 1 P i 2... P i (q 1) ], of the prity check mtrix of n rry code, the rows sum to [1 1 1... 1], nd hence the dul of n rry code lwys contins the ll-ones codeword. For bounding d from bove, we mke use of prticulrly elegnt result due to McKy nd Dvey [18], which shows tht prity-check mtrices contining n r (r + 1) grid of permuttion mtrices P i,j tht commute (i.e., for which P i,j P k,l = P k,l P i,j ) must hve minimum distnce t most (r+1)!. Tble I lists the lower nd upper bounds on minimum distnce for rry codes with column-weight r {3, 4} nd girth g {8, 10, 12}. A. The Code Msk Arry codes, s well s the generl clss of qusi-cyclic LDPC codes with prity-check mtrices consisting of blocks of circulnt permuttion mtrices, cnnot hve girth exceeding twelve [6]. This is most esily seen by exmining the exmple in Figure 2. There, sub-mtrix of prity check mtrix consisting of circulnt permuttion blocks P i, i = 1, 2,..., 6, is shown, long with directed closed pth lbeled bcdefghijkl tht trverses the blocks. Setting P i = P bi for some circulnt permuttion mtrix P nd exponents b i, we see tht the condition in Theorem 1 is stisfied, since b 1 b 4 +b 5 b 2 +b 3 b 6 +b 4 b 1 +b 2 b 5 +b 6 b 3 = 0. (13) Thus, length-12 cycles re gurnteed to exist in ny qusicyclic LDPC code with prity-check mtrix consisting of blocks of circulnt permuttion mtrices. l P P P 1 h 2 d 3 g i c e k b j 4 5 6 f P P P Fig. 2. Cycle of length twelve in n rry code. Nevertheless, using msking pproch, rry codes cn be modified so tht their girth exceeds twelve. Msks were introduced in [20] for the purpose of incresing the girth of codes s well s for constructing irregulr LDPC codes. As n illustrtive exmple, consider the mtrix M in (14) below. It consists of q q zero mtrices 0 nd q q circulnt permuttion blocks P i = P bi, for some integers b i. One cn view M s rising from prity-check mtrix of n rry code, or more generlly, qusi-cyclic code with circulnt permuttions blocks, from which some blocks re zeroed out ccording to given msk. The mtrix M does not contin submtrix of the form depicted in Figure 2. Consequently, there exist no length-12 cycles tht trverse exctly six permuttion mtrix blocks. Of course, this is chieved t the expense of incresed code length (for the given exmple, the length hs to be doubled). Other kinds of length-12 cycles my still exist, but these re governed by non-trivil homogeneous liner equtions similr in form to those governing shorter cycles, nd cn be eliminted by judicious choice of the exponents b i. M = P 1 P 2 P 3 0 0 0 P 4 0 0 0 0 P 5 P 6 P 7 0 P 8 P 9 P 10 0 P 11 P 12 0 0 0 0 0 P 13 0 0 P 14 P 15 P 16 0 P 17 P 18 P 19 0 P 20 0 0 P 21 0 0 0 P 22 0 P 23 P 24 IV. AVOIDING SOLUTIONS TO CYCLE-GOVERNING EQUATIONS (14) Cycle-governing equtions, such s those listed in Tbles II V, re lwys of the following type: m c i u i 0 (mod q), (15) i=1 the integer m being the number of distinct block-columns through which the cycle psses, the u i s being vribles 2 tht denote the lbels of those m block-columns, nd the c i s being fixed nonzero integers (independent of q) such tht m i=1 c i = 0. This is becuse ll such equtions rise s specil cses of n eqution of the form (4), nd clerly, 2 To void the sloppiness of using u i to denote both vrible nd vlue it cn tke, we will mke typogrphicl distinctions between the two whenever necessry.
6 Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 TABLE I BOUNDS ON THE MINIMUM DISTANCE, d, OF ARRAY CODES FOR VARIOUS VALUES OF COLUMN-WEIGHT, r, AND GIRTH, g. r = 3 r = 3 r = 4 r = 4 Girth g Lower bound on d Upper bound on d Lower bound on d Upper bound on d 8 6 24 8 120 10 10 24 14 120 12 14 24 26 120 ( i1 ik ) + ( i2 i1 ) + + ( ik ik 1 ) = 0. Any solution u = (u 1, u 2,...,u m ) to (15), with u i [0, q 1], nd such tht u i u j when i j, represents cycle pssing through the m block-columns whose lbels form the solution vector u. To void potentil mbiguity, we estblish some terminology tht we will use consistently in the rest of the pper. Given homogeneous liner eqution of the form m i=1 c iu i = 0, we refer to vector (u 1, u 2,...,u m ) [0, q 1] m s solution over Z q to the eqution if m i=1 c iu i 0 (mod q). If u = (u 1, u 2,...,u m ) Z m is such tht m i=1 c iu i = 0, then u is referred to s n integer solution to the eqution. In both cses, solution u = (u 1, u 2,...,u m ) to (15), with ll the u i s distinct, will be referred to s proper solution. The design of shortened rry code typiclly involves determining the smllest prime q for which there exists sequence of integers S [0, q 1] of some desired crdinlity s, such tht there is no proper solution with entries in S to ny eqution within certin set of cycle-governing equtions. This choice of q would gurntee the smllest possible code length, equl to q s, for PAC or n IAC with prescribed girth, column-weight r nd designed code rte R = 1 r/s. For exmple, if we seek n IAC with r = 3, designed rte R = 1/2 nd girth ten, then we need the smllest q tht gurntees the existence of set S of crdinlity t lest six tht does not contin proper solution to ny of the equtions listed in Tble IV. It is therefore useful to estimte, s function of q, the size of the lrgest subset of [0, q 1] tht voids proper solutions to certin liner equtions of the form given in (15). In this section, we provide number of results tht bound the size of such lrgest subset. Equtions of the form m i=1 c iu i = 0, with m i=1 c i = 0, hve been extensively studied in Rmsey theory [9, Chpter 3], [15, Chpter 9]. It is known [7, Fct 3] tht ny such eqution tht is not of the form u 1 u 2 = 0 (or n integer multiple of it) hs proper solution. In fct [7, Theorem 2], for ny ǫ > 0 nd sufficiently lrge N, if L [1, N] is such tht L ǫn, then L contins proper solution to such n eqution. This implies the following result: Theorem 7. Let m 3, nd let c i, i = 1, 2,...,m, be nonzero integers such tht m i=1 c i = 0. For n rbitrry q 1, let s(q) be the size of the lrgest subset of [0, q 1] tht does not contin proper solution to m i=1 c iu i 0 (mod q). Then, s(q) lim = 0. q q Proof. Let S(q) [0, q 1] be set of size s(q) tht does not contin ny proper solution to m i=1 c iu i 0 (mod q). Clerly, S(q) does not contin proper solution to m i=1 c iu i = 0 (without the modulo-q reduction) s well. Note tht since (1, 1,...,1) is solution to m i=1 c iu i = 0, (u i ) is solution iff (u i + 1) is solution. Thus, L(q) = S(q) + 1 = {j + 1 : j S(q)} is set of crdinlity s(q) in [1, q] tht does not contin proper solution to m i=1 c iu i = 0. Hence, for ny ǫ > 0, we must hve s(q) < ǫq for ll sufficiently lrge q, nd the desired result follows. We hve thus estblished tht the size of subset of [0, q 1] contining no proper solution to ny eqution from given set of cycle-governing equtions grows sub-linerly in q. This is disppointing result from the point of view of our strtegy of shortening rry codes to eliminte cycles. Indeed, strting with n rry code of column-weight r, length q 2 nd designed rte 1 r/q, if we shorten the code so s to eliminte cycles governed by n eqution of the form m i=1 c iu i 0 (mod q), the resulting shortened code cn hve rte no lrger thn 1 r/s(q), where s(q) is s defined in the sttement of Theorem 7. Since s(q)/q goes to 0 s q increses, the rte penlty ssocited with shortening is severe for lrge vlues of q (or equivlently, for lrge vlues of the length of the prent code). However, from prcticl stndpoint, this does not pper to be problem, s for the moderte vlues of q useful in prcticl code constructions, the rte penlty incurred by shortening remins within resonble limits. Consequently, it is possible to construct, for exmple, designed rte-1/2 codes of girth eight nd ten tht perform much better thn the comprble codes in the existing literture, s we shll see in Section 6. A precise estimte of the rte t which s(q)/q goes to zero for vrious types of cycle-governing equtions cn be very useful for the purpose of prcticl code design, s this provides us with n understnding of how the rte penlty incurred in shortening n rry code chnges with the modulus q. More generlly, given collection, Ω, of homogeneous liner equtions over Z q of the form (15), let s(q; Ω) be the size of the lrgest subset of [0, q 1] tht does not contin proper solution over Z q to ny of the equtions in Ω. From the result of Theorem 7, it is cler tht s(q; Ω) grows sub-linerly with q. In the rest of this section, we provide upper nd lower bounds on s(q; Ω) for vrious choices of Ω. A. Upper bounds on s(q; Ω) Explicit upper bounds for s(q; Ω) cn be obtined for ny Ω contining n eqution (over Z q ) of the form 2x y z = 0 or x + y z u = 0. These equtions hve been extensively studied in other contexts, nd in such cses, there re good estimtes vilble for the growth rte of sequences voiding solutions to these equtions.
MILENKOVIC, KASHYAP AND LEYBA: SHORTENED ARRAY CODES 7 Recll from Definition 1 tht sequences voiding proper solutions to 2x y z = 0 re clled non-verging sequences. Correspondingly, sequences voiding proper solutions to the eqution x + y z u = 0 re clled Sidon sequences (see e.g. [23]), s mde precise by the definition below. Definition 3. A Sidon sequence is sequence of distinct integers n 1, n 2, n 3,... with the property tht for ll i, j, k, l such tht i j, k l, n i + n j = n k + n l if nd only if {i, j} = {k, l}. Similrly, given n N Z +, Sidon sequence over Z N is sequence of distinct integers n 1, n 2, n 3,... in [0, N 1] such tht for ll i, j, k, l with i j, k l, n i + n j = n k + n l (mod N) if nd only if {i, j} = {k, l}. Upper bounds on the sizes of non-verging sequences nd Sidon sequences over Z N re given in the next lemm. Observe tht for ny N Z +, non-verging sequence over Z N is utomticlly non-verging sequence (over Z + ). The result of prt () of the lemm is thus strightforwrd ppliction of the clssicl upper bound, due to Roth [9, Section 4.3, Theorem 8], on the crdinlity of the lrgest nonverging sequence in [0, N 1]. Lemm 8. () (Roth s theorem) The crdinlity of ny nonverging sequence over Z N is bounded from bove by c 0 N/ loglog N, for some fixed constnt c 0 > 0. (b) For ny odd integer N > 0, the crdinlity of Sidon sequence over Z N is bounded from bove by N 3/4+1/2. We defer the proof of prt (b) of the bove lemm to the Appendix. In terms of the quntity s(q; Ω), the lemm cn be re-stted s: () If Ω contins the eqution 2x y z 0 (mod q), then s(q; Ω) c 0 q/ loglog q, for some fixed constnt c 0 > 0. (b) If Ω contins the eqution x + y z u 0 (mod q), then s(q; Ω) q 3/4 + 1/2. In PAC with modulus q nd column-weight r 3, the eqution 2x y z 0 (mod q) lwys governs six-cycles, s cn be seen by setting r 1 = 0, r 2 = 1 nd r 2 = 2 in (5). So, if shortened PAC hs girth eight, then its sequence of blockcolumn lbels must not contin solutions to 2x y z 0 (mod q), i.e., must be non-verging over Z q. Hence by Lemm 8(), the number of block-columns in the prity-check mtrix of the shortened PAC cnnot exceed c 0 q/ loglog q. Similrly, in n rry code with modulus q, the eqution x + y z u 0 (mod q) lwys governs eight-cycles tht pss through ny two distinct block-rows nd four distinct block-columns (see, for exmple, the cycles on the bottomright of Figure 1). So, if n rry code is shortened to obtin girth ten, then the sequence of block-column lbels retined in the shortened code must be Sidon sequence over Z q, nd therefore, Lemm 8(b) pplies. We hve thus proved the following theorem. Theorem 9. () The number of block-columns in the pritycheck mtrix of shortened PAC with modulus q, columnweight r 3 nd girth eight cnnot exceed c 0 q/ loglog q. (b) The number of block-columns in the prity-check mtrix of shortened rry code with modulusq, column-weightr 2 nd girth ten is t most q 3/4 + 1/2. Roughly speking, the bove theorem sys tht the rte of shortened PAC with modulus q, column-weight r 3 nd log log q girth eight cnnot be more thn 1 c 0q r. Similrly, the rte of shortened rry code with modulus q, column-weight r 2 nd girth ten is, s rough estimte, bounded from bove by 1 r q. It is nturl to wnt to compre the bounds of Theorem 9 to those obtined from the ppliction of the Moore bound to the Tnner grphs of rry codes. The Moore bound 3 for biprtite grph [11] bounds the number of vertices in the grph in terms of the girth nd the verge left nd right degrees. Consider biprtite grph with n L left vertices, n R right vertices, m edges nd girth g. Let d L = m n L be the verge left degree, d R = m n R the verge right degree. Then, n L n R g/2 1 i=0 g/2 1 i=0 (d R 1) i/2 (d L 1) i/2 (16) (d L 1) i/2 (d R 1) i/2. (17) The bove bounds re esily proved for bi-regulr biprtite grphs, i.e., grphs in which ech left (resp. right) vertex hs degree d L (resp. d R ). Now, the Tnner grph of n rry code of modulus q, column-weight r nd hving s block-columns is bi-regulr with n L = qs, n R = qr, d L = r nd d R = s. So, for such Tnner grph of girth eight, the bound in (16) becomes qs 1 + (s 1) + (s 1)(r 1) + (s 1) 2 (r 1) = s [1 + (s 1)(r 1)], which yields the bound s 1 + q 1 r 1. (18) The bound in (17) lso gives exctly the sme result. Note tht this bound is, symptoticlly in q, looser thn the bound in Theorem 9(). But for prcticl purposes, this is more useful bound thn tht of the theorem becuse the c 0 in the theorem is not explicitly specified. On the other hnd, pplying (16) to the Tnner grph of n rry code of girth ten, we get qs s [1 + (s 1)(r 1)] + (s 1) 2 (r 1) 2, which upon re-rrngement becomes r(r 1)s 2 [r(2r 3) + q] s + (r 1) 2 0. Solving for s now yields s q + r(2r 3) + (q + r(2r 3)) 2 4r(r 1) 3. 2r(r 1) (19) For q r 2 q, this upper bound is roughly r(r 1). It is cler tht in most cses of interest, this is not s good bound s tht of Theorem 9(b). We would like to remrk tht nother 3 To be correct, this should be clled Moore-type bound, s the originl Moore bound (see [3, p. 180]) only pplies to regulr grphs.
8 Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 upper bound cn be obtined vi (17), but this turns out to be looser thn the bound in (19). We summrize the bove bounds in the following theorem. Theorem 10. () The number of block-columns in the pritycheck mtrix of shortened rry code with modulus q, column-weight r nd girth eight cnnot exceed 1+(q 1)/(r 1). (b) The number of block-columns in the prity-check mtrix of shortened rry code with modulus q, column-weight r nd girth ten is t most q + r(2r 3) + (q + r(2r 3)) 2 4r(r 1) 3. 2r(r 1) B. Lower bounds on s(q; Ω) We next consider the converse problem of finding lower bounds on the size of integer sequences voiding solutions to collection of cycle-governing equtions. The problem of constructing long sequences of integers tht do not contin solutions to certin kinds of homogeneous liner equtions hs long history. For exmple, lrge non-verging subsets of [1, N] were described or constructed by Behrend [1], Moser [21] nd Rnkin [25], using geometricl rguments. We will generlize some of these results to cover certin clsses of equtions of the form given in (15). We strt with lower bound on the mximum length of sequences tht re c i -non-verging over Z q, for l distinct integers c i [2, q 2]. The proof of this bound is provided in the Appendix. Theorem 11. Let l 1, nd let Ω be the collection of equtions x + c i y = (c i + 1)z, i = 1, 2,..., l, for some constnts c i [1, q 2] such tht c i c j for i j. Then, ( ) 3q 2 1/3 s(q; Ω). l(q 1) The lower bound derived in the theorem bove is quite loose. For exmple, for q = 241, greedy lgorithm (to be described in Section 5) produces sequence of 15 integers tht is simultneously non-verging nd 2-non-verging over Z q. However, the theorem pplied with l = 2, c 1 = 1 nd c 2 = 2 gives lower bound of 8 for the crdinlity of such sequence. A more generl lower bound cn be derived by extending result of Behrend [1] derived originlly for non-verging sequences. Consider the following system, Ω, of l equtions in the vribles u 1, u 2,...,u m, v: m j=1 c 1,j u j = b 1 v Ω :... m j=1 c l,j u j = b l v, (20) where the coefficients c i,j, b i re non-negtive integers such tht for ech i [1, l], t lest two of the c i,j s re nonzero, nd m j=1 c i,j = b i > 0. Theorem 12. Given system, Ω, s in (20), let D = mx 1 i l b i. Then, for q > D 2, s(q; Ω) γ 1 q e γ2 log q 1 2 log log q (1 + o(1)) 1 where log denotes the nturl logrithm, γ 1 = D 2 2 log D, γ 2 = 2 2 logd, nd o(1) denotes correction fctor tht vnishes s q. We postpone the proof of the theorem to the Appendix. The bove result cn be compred directly to the result of Theorem 11 since the system of equtions x+c i y = (c i +1)z, i = 1, 2,...,l, is of the form given in (20). Therefore, the result of Theorem 12 pplies to this system of equtions Ω, with D = 1 + mx i c i. It is esily seen tht by the bound of Theorem 12, s(q; Ω) lim q q 1 ǫ = for ny ǫ > 0. Since ǫ cn be chosen to be rbitrrily smll, this is much stronger, symptoticlly in q, thn the result of Theorem 11, which only shows tht s(q; Ω) C q 1/3 for some constnt C > 0 independent of q. However, for smll vlues of q, prticulrly for the vlues of the modulus q typiclly used in prcticl rry code design, the bound of Theorem 11 is better thn tht of Theorem 12. For instnce, when pplied to the system, Ω, consisting of the pir of equtions x + y = 2z nd x + 2y = 3z, the bound of Theorem 12, for q = 241, evlutes to 0.66, which just shows tht s(q; Ω) 1. As stted erlier, the bound of Theorem 11 yields s(q; Ω) 8 in this cse. To conclude this section, we remrk tht while the problem of precisely estimting the growth rte of s(q; Ω) with q is one of considerble interest nd vlue, finding provbly good estimtes is notoriously difficult problem. For exmple, the current best lower bound for the growth rte of the crdinlity of non-verging sequences is tht due to Behrend (Theorem 12 for the specil cse of Ω consisting of the single eqution x + y = 2z), but it is still not known whether this is the best possible such bound. V. CONSTRUCTION METHODS The simplest nd computtionlly lest expensive methods for generting integer sequences stisfying given set of constrints re greedy serch strtegies nd vritions thereof. A typicl greedy serch lgorithm strts with n initil seed sequence tht trivilly stisfies the given constrints, nd progressively extends the sequence by dding new terms tht continue to mintin the constrints. As n exmple, to construct non-negtive integer sequence tht contins no solutions to ny eqution within system, Ω, of cycle-governing equtions of the form (15), we strt with seed sequence of m 1 non-negtive integers, n 1 < n 2 <... < n m 1, where m is the lest number of vribles mong ny of the equtions in Ω. For ech j m, we tke n j to be the lest integer greter thn n j 1 such tht {n 1, n 2,..., n j } contins no solutions to ny eqution in Ω. The rte of growth of elements in sequence generted by such greedy serch procedure is influenced by the choice of the seed sequence
MILENKOVIC, KASHYAP AND LEYBA: SHORTENED ARRAY CODES 9 [24]. The serch needs to be performed only once to generte sequence of integers voiding solutions to ny eqution in Ω, nd it is esily seen tht the lgorithm hs complexity O(L q M+1 ), where L denotes the number of equtions in Ω, M is the mximum number of vribles mong ll these equtions, nd q is the prime modulus. Tbles II V list the output of the greedy serch procedure, initilized by different seed sequences, for finding sequences tht void solutions to vrious cycle-governing equtions in PAC s nd IAC s. The first two terms of ech sequence listed in the tbles form the seed sequence for the greedy serch lgorithm. There is n lterntive procedure tht often genertes sequences with more terms thn simple greedy serch routine. The ide is to strt with some construction of dense sequence voiding solutions to some subset of the cycle-governing equtions in the set Ω, nd then to sequentilly expurgte elements of tht sequence tht violte ny of the remining constrints. After the expurgtion procedure is completed, dditionl elements my be dded to the sequence s long s they jointly void solutions to ll cycle-governing equtions in Ω. A good sequence with which to strt this lterntive procedure cn be constructed ccording to method outlined by Boszny [4]. The construction proceeds through the following steps. First, prime q is chosen, nd long with it the smllest integer t such tht q t 4. Let n j = j t 3 j(j + 1) +, j = 1, 2,..., t 1, 2 nd let S = {n 1, n 2,...,n t 1 } [0, q 1]. It cn be shown tht the sequence S does not contin proper solutions over Z q to ny eqution of the form m c i u i = b v i=1 where c 1, c 2,..., c m, b re positive integers such tht m i=1 c i = b. Next, one uses simple greedy lgorithm to find the lrgest subset S S tht does not contin proper solutions to cycle-governing equtions in Ω tht re not of the bove form. The lst step in the procedure is to check whether there exist integers in [0, q 1] tht cn be dded to S without creting proper solution within S to some cycle-governing eqution. If such integers exist, they re sequentilly dded to the set S. As illustrtive exmples, we list three sequences constructed using the dpttion of Boszny s method described bove. The sequence 1, 4, 8, 23, 40, 126, 253, 352, 381, 495 constructed by this method does not contin solutions to ny of the equtions listed in Tble III tht govern cycles of length six nd eight in PAC with modulus q = 911 nd column-weight r = 4. In comprison, the greedy lgorithm initilized by the seed sequence 0, 1 produces 0, 1, 5, 18, 25, 62, 95, 148, 207. The sequences 6, 8, 165, 217, 435, 654, 1095 nd 0, 1, 7, 29, 64, 111, 753, generted by the modified Boszny construction nd the greedy lgorithm with seed sequence 0, 1, respectively, void solutions to ny of the equtions listed in Tble IV. Finlly, in the cse of the equtions in Tble V, the sequences produced by the two methods re 2, 4, 28, 217, 255, 435, 654 nd 0, 1, 9, 20, 46, 51. Observe tht the sequences produced by the modified Boszny construction contin terms tht re lrger in generl thn the terms in the corresponding greedy sequences where lmost ll elements re much smller thn the prime q. VI. SIMULATION RESULTS In this section, we present the bit-error-rte (BER) curves over n AWGN chnnel for vrious (shortened) PAC s nd IAC s, nd lso provide comprisons with other codes of similr rtes nd lengths from the existing literture. All rry codes considered in this section were itertively decoded using sum-product/belief-propgtion (BP) decoder. Figures 3 nd 4 show the performnce curves, fter mximum of 30 rounds of itertive decoding, for rry codes of column-weight 3 nd row-weight 6; thus ll these codes hve designed rte 1/2. The prime modulus used for the construction of these codes is q = 1213, which yields codes with length 7278. The sets of block-column lbels used in the codes PACr3g6, PACr3g8 nd PACr3g8+ in Figure 3 re {0, 1, 2, 3, 4, 5}, {0, 1, 3, 4, 9, 10} nd {0, 1, 4, 11, 27, 39}, which correspond to PAC of girth six, shortened PAC of girth eight, nd shortened PAC of girth eight but without eight-cycles governed by the equtions in Tble II, respectively. The codes IACr3g8, IACr3g10 nd IACr3g12, whose performnce is plotted in Figure 4, re of girth eight, ten nd twelve, respectively. The respective sets of blockcolumn lbels re {0, 1, 2, 5, 7, 8}, {0, 1, 5, 14, 25, 57}, nd {0, 1, 7, 29, 64, 111}. All the IAC s in the figure hve blockrow lbels {0, 1, 3}. Figures 5 nd 6 show the results, fter mximum of 30 decoding itertions, for codes with designed rte 1/2 nd column-weight r = 4. The rry codes in Figure 5 re shortened PAC s with modulus q = 911 nd length 7288. The sequences used for the block-column lbels in the codes PACr4g6, PACr4g8 nd PACr4g8+ re {0, 1, 2, 3, 4, 5, 6, 7}, {0, 3, 4, 7, 16, 17, 20, 22} nd {0, 1, 5, 18, 25, 62, 95,148}, respectively. The codes PACr4g6 nd PACr4g8 re of girth six nd eight, respectively, while PACr4g8+ is code of girth eight with no eight-cycles governed by the equtions in Tble III. The codes IACr4g8 nd IACr4g10 in Figure 6 re IAC s of girth eight nd ten, respectively, tht use the set of block-row lbels {0, 1, 3, 7}, but differ in the modulus nd block-column lbels used. The code of girth eight hs modulus q = 911, hence length 7288, nd block-column lbels {0, 1, 2, 5, 9, 10, 18, 42}. The girth-ten code, on the other hnd, uses the modulus q = 1307, so tht it hs length 10456, nd block-column lbels {317, 344, 689, 1035, 1178,1251,1297, 1303} The reson for not choosing q to be 911 in the girth-ten code is tht none of the construction methods discussed in Section 5 produces sequence of length eight without solutions over Z 911 to ny of the equtions listed in Tble V. The smllest choice for the prime q which does produce sequence of eight blockcolumn lbels stisfying the eight-cycle constrints turns out to be 1307. For comprison purposes, ech of Figures 3 6 lso contins the BER curves for two other codes: designed rte-
10 Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 TABLE II CYCLE-GOVERNING EQUATIONS OVER Z q FOR PAC S WITH MODULUS q AND COLUMN-WEIGHT r = 3, AND GREEDY SEQUENCES AVOIDING SOLUTIONS OVER Z 1213 TO THEM. Six-cycle eqution 2i j k = 0 Eight-cycle equtions 2i + j k 2l = 0 i + j k l = 0 3i j 2k = 0 2i j k = 0 Greedy sequences voiding the six-cycle eqution 0, 1, 3,4, 9, 10,12, 13, 27, 28, 30, 38,... 0, 2, 3,5, 9, 11,12, 14, 27, 29, 30, 39,... 0, 3, 4,7, 9, 12,13, 16, 27, 30, 35, 36,... Greedy sequences voiding ll six- nd eight-cycle equtions 0,1, 4, 11, 27, 39, 48,84, 134, 163, 223, 284, 333,... 0,2, 5, 13, 20, 37, 58,91, 135, 160, 220, 292, 354,... 0,3, 4, 13, 25, 32, 65,92, 139, 174, 225, 318, 341,... TABLE III CYCLE-GOVERNING EQUATIONS OVER Z q FOR PAC S WITH MODULUS q AND COLUMN-WEIGHT r = 4, AND GREEDY SEQUENCES AVOIDING SOLUTIONS OVER Z 911 TO THEM. Six-cycle equtions 2i j k = 0 3i j 2k = 0 Eight-cycle equtions 3i j k l = 0 3i 2j 2k + l = 0 2i 2j k + l = 0 3i 3j + k l = 0 3i 3j + 2k 2l = 0 i + j k l = 0 2i j k = 0 4i 3j k = 0 3i 2j k = 0 5i 3j 2k = 0 Greedy sequences voiding ll six-cycle equtions 0, 1,4, 5, 11, 19, 20,... 0, 2,5, 7, 13, 18, 20,... 0, 3,4, 7, 16, 17, 20,... Greedy sequences voiding ll six- nd eight-cycle equtions 0,1, 5,18, 25, 62, 95, 148, 207,... 0,2, 7,20, 45, 68, 123, 160, 216,... 0,3, 7,22, 39, 68, 123, 154, 244,... 10 1 10 2 PACr3g6 (n=7278, q=1213, g=6) PACr3g8 (n=7278, q=1213, g=8) PACr3g8+ (n=7278, q=1213, g=8, few 8 cycles) Rndom lbel PAC (n=7278, q=1213, g=8) Mcky Dvey (n=8000, (3,6) regulr) 10 1 10 2 IACr3g8 (n=7278, q=1213, g=8) IACr3g10 (n=7278, q=1213, g=10) IACr3g12 (n=7278, q=1213, g=12) Rndom lbel IAC (n=7278, q=1213) Mcky Dvey (n=8000, (3,6) regulr) Bit Error Rte 10 3 10 4 Bit Error Rte 10 3 10 4 10 5 10 5 10 6 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 E b /N 0 (db) 10 6 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 E b /N 0 (db) Fig. 3. BER versus E b /N 0 (db) for designed rte-1/2 PAC s with r = 3. Fig. 4. BER versus E b /N 0 (db) for designed rte-1/2 IAC s with r = 3. 1/2, regulr LDPC code of length 8000 with rndom-like structure, s constructed by McKy nd Dvey in [17], nd rndom-lbel rry code in which the block-row nd blockcolumn lbels re rndomly chosen. The McKy-Dvey code in Figures 3 nd 4 is (3, 6)-regulr code, while tht in Figures 4 nd 6 is (4, 8)-regulr code. The rndom-lbel code in Figure 3 is PAC with q = 1213, r = 3 nd set of block-column lbels {24, 460, 610, 826, 1009, 1012}. Among the equtions in Tble II, this lbel set contins solutions over Z 1213 to only one eqution, nmely, 3i 2j k = 0; the solution is (i, j, k) = (826, 1009, 460). Thus, this PAC contins no six-cycles nd reltively few eight-cycles. The rndom-lbel code in Figure 4 is n IAC with the sme choices of q, r nd block-column lbels s in the rndom-lbel PAC