Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices

(1)

Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices

Marc P. C. Fossorier, Senior Member, IEEE

Abstract—In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than12, and a relatively mild necessary and sufficient condition for the code to have a girth of6 8 10 or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.

Index Terms—Iterative decoding, low-density parity-check (LDPC) codes, quasi-cyclic (QC) codes.

I. INTRODUCTION

Recently, several methods for constructing good families of low-density parity-check (LDPC) codes have been proposed. These methods can be decomposed into two main classes: random or pseudorandom constructions, and algebraic constructions. For long code lengths, random constructions [1]–[4] or pseudorandom constructions [5]–[7] of irregular LDPC codes have been shown to closely approach the theoretical limits for the additive white Gaussian noise (AWGN) channel. Generally, these codes outperform algebraically constructed LDPC codes. On the other hand, for medium-length LDPC codes (say, up to a few thousand bits long for rate1=2), the situation is quite different. For these lengths, irregular constructions are generally not better than regular ones, and graph-based or algebraic constructions can outperform random ones[8].

Algebraic constructions of LDPC codes can be decomposed into two main categories. The first category is based on finite geometries [9]–[12], while the second category is based on circulant permutation matrices.¹ This second approach was initially proposed by Gal- lager [13, Appendix C] (although in thisoriginal construction, the permutation matrices are not restricted to circulants). A special class of these codes was later analyzed in [14] and several recent works consider structured ways to design such codes [15]–[20]. In fact, these two methodsare interrelated and, for example, many of the code construc- tionsbased on finite geometrieshave an equivalent circulant permutation matrix representation [21, p. 286].

A(J; L)-regular LDPC code isdefined asa code represented by a parity-check matrixH in which each column hasweight J and each row hasweightL[13]. Hence, to construct the parity-check matrix H of a(J; L)-regular LDPC code of length N = Lp with the second method,J rowsof L circulant permutation matricesof size p 2 p

Manuscript received August 13, 2003; revised January 22, 2004. This work wassupported by the National Science Foundation under Grant CCR-0098029.

The material in this correspondence was presented at the IEEE International Symposium on Information Theory, Yokohama, Japan, June/July 2003.

The author iswith the Department of Electrical Engineering, University of Hawaii, Honolulu, HI 96822 USA (e-mail: marc@spectra.eng.hawaii.edu).

Communicated by S. Litsyn, Associate Editor for Coding Theory.

Digital Object Identifier 10.1109/TIT.2004.831841

1A permutation matrix is any square matrix with constant row and colum weight of one; a circulant permutation matrix isa permutation matrix which is cyclic.

can be judiciously adjoined. The code obtained is quasi-cyclic (QC) and therefore, can be encoded in linear time with shift registers [22, pp. 256–261]. Furthermore, since by row and column permutations, an equivalent code with only identity matricesin the first row block and the first column block ofH can be obtained, at most (J 0 1)(L 0 1) integers suffice to entirely specify the code. In this correspondence, we derive a simple necessary and sufficient condition for the Tanner graph [23] of these QC LDPC codes to have a given girth. In fact, we show that these QC LDPC codes have a girthg of at most 12, which generalizesthe result of [16]. Forg = 6, the condition isvery loose and, therefore, it is very easy is construct QC LDPC codes which perform quite well when iteratively decoded with the belief propagation (BP) algorithm [1]. The conditionsforg = 8; 10; or 12 are also quite easy to satisfy. This suggests that for LDPC codes of moderate lengths, additional constraints other than the girth need to be considered.

Based on a result of [25], it directly follows that the minimum Ham- ming distancedHof a(J; L)-regular QC LDPC code satisfies dH (J + 1)!. A set of (J + 1)! columnsin H summing to zero is explicitly determined in this correspondence. A necessary condition to have all columns in this set distinct is then proposed.

The correspondence is organized as follows. The necessary and sufficient condition for a given girth isderived in Section II. Applicationsof this condition to construct families of QC LDPC codes are presented in Section III. Code searches and simulation results are discussed in Sec- tion IV. In Section V, the necessary condition for a QC LDPC code to reach the upper bound on itsminimum distance isdeveloped. Finally, concluding remarksare given in Section VI.

II. GIRTH OFQC LDPC GRAPHREPRESENTATIONS

A. Preliminaries

The parity-check matrixH of a (J; L)-regular QC LDPC code of lengthN = pL can be represented by

H =

I(0) I(0) 1 1 1 I(0)

I(0) I(p1;1) 1 1 1 I(p1;L01)

... . .. ...

I(0) I(pJ01;1) 1 1 1 I(pJ01;L01)

(1)

where for1 j J 0 1, 1 l L 0 1, I(pj;l) represents the circulant permutation matrix with a one at column-(r + pj;l) mod p for row-r, 0 r p 0 1, and zero elsewhere. It follows that I(0) represents thep 2 p identity matrix. Also, since the p rowsof each of theJ submatrices [I(0)I(pj;1) 1 1 1 I(pj;L01)], 0 j J 0 1, in (1) sum to the all-1 vector, the rank ofH isat most Jp 0 J + 1.

A cycle of length2i in H =[hx;y] isdefined by 2i positions hx;y=1 such that: 1) two consecutive positions are obtained by changing alter- natively of row or column only; and 2) all positions are distinct, except the first and last ones. It follows that two consecutive positions in any cycle belong to distinct circulant permutation matrices which are either in the same row, or in the same column. Hence, a cycle of length2i can be associated with an ordered series of circulant permutation matrices I(pj ;l ); I(pj ;l ); I(pj ;l );

. . . ; I(pj ;l ); I(pj ;l ); I(pj ;l ) with for1 k i, jk 6= jk01andlk 6= lk01. With the conven- tion of going fromI(pj ;l ) to I(pj ;l ) via I(pj ;l ) (i.e., of changing first of row and then of column), any cycle of length2i in H can be represented by the ordered series

(2)

for1 k i, jk 6= jk01, andlk6= lk01We note that (2) doesnot necessarily define a unique cycle of length2i in H, but thisisnot an issue for the following results. Defining

1j ;j (l) = pj ;l0 pj ;l (3) the matrixH containsa cycle of length 2i given by (2) if and only if

i01 k=0

1j ;j (lk) = 0 mod p (4)

withj0 = ji,jk 6= jk+1, andlk6= lk+1. This simple necessary and sufficient condition can be rewritten in the following theorem.

Theorem 2.1: A necessary and sufficient condition for the Tanner graph representation of the matrixH defined in (1) to have a girth at least2(i + 1) is

m01 k=0

1j ;j (lk) 6= 0 mod p (5)

for allm, 2mi, all jk,0jk J 01, all jk+1,0jk+1J 01, and all0lk L01, with j0=jm,jk6=jk+1, andlk6=lk+1.

Note that an equivalent condition with respect to row index differ- encesrather than column index differencesasin Theorem 2.1 can also be obtained based onH^t, the transpose ofH. From Theorem 2.1, the next corollary follows.

Corollary 2.1: For QC LDPC codeswithJ = 2, g = 4i only is possible.

This result directly follows from the series given in (2), in whichj0

andj1have to alternate.

In the following two subsections, a lower bound on the minimum value ofp for which g 6 and g 8, respectively, is determined. Un- fortunately, for larger girth values, no meaningful bound was obtained.

B. Girthg 6

A simple necessary condition forg 6 isgiven by the following theorem

Theorem 2.2: A necessary condition to haveg 6 is pj ;l 6=pj ;l

forj16= j2, andpj ;l 6= pj ;l forl16= l2.

Proof: The first part of the theorem directly follows from (3) and (4) withi = 2. For g 6, Theorem 2.1 for rows 0 and j1and columns l1 andl2inH becomes

10;j (l1) + 1j ;0(l2) 6= 0 mod p (6) with10;j (l1) + 1j ;0(l2) = 0pj ;l + pj ;l , which completesthe proof.

A lower bound on the minimum value ofp for which g 6 isgiven by the following.

Corollary 2.2: A necessary condition to haveg 6 in the Tanner graph representation of a(J; L)-regular QC LDPC code is p L, or N L².

Theorem 2.2 and its corollary suggest that finding a(J; L)-regular QC LDPC code withg 6 should not be necessarily difficult. Corol- lary 2.2 can be refined depending on whetherL isodd or even asfol- lows.

Theorem 2.3: A necessary condition to haveg 6 in the Tanner graph representation of a(J; L)-regular QC LDPC code is p L, or N L²ifL isodd, and p L + 1, or N L(L + 1) if L iseven.

Proof: Assumep = L and g 6. Without loss of generality, we can choosep0;l = 0 and p1;l = l in (1) for 0 l L 0 1 based on Corollary 2.2. Forl > 0, define e and o asthe number of even and odd valuesp1;l, respectively. IfL = p iseven, then e = p=2 0 1 and o = p=2, so that o 0 e = 1.

We can choosep2;0 = 0 and, from Theorem 2.2, p2;l 6= p2;l for l1 6= l2. Forl > 0, define o1 ando2 asthe numbersof odd values p1;l corresponding to odd and even valuesp2;l, respectively, so that o1+ o2 = o. Similarly, for l > 0, define e1ande2 asthe numbers of even valuesp1;lcorresponding to odd and even valuesp2;l, respectively, so thate1+ e2= e. We also have o1+ e1= o and o2+ e2= e, which impliese1 = o2ando1 = e2.

Ifo1isodd, thene2isalso odd. Ife iseven, then e1and, hence,o2

are odd too, ando iseven; else if e isodd, then e1and, hence,o2are even, ando isodd. Similarly, if o1iseven, thene2isalso even. Ife is even, thene1and henceo2are even too, and so iso; else if e isodd, then e1and, hence,o2are odd, and so iso. As a result, in each case, e and o are either both odd or both even, which is impossible sinceo 0 e = 1.

It followsthat forL even, p L + 1, which completesthe proof.

C. Girthg 8

The previousapproach isextended to the caseg 8.

Theorem 2.4: ForJ 3 and L 3, a necessary condition to have g 8 is pj ;l 6= pj ;l for0 < j1 < j2and0 < l1< l2.

Proof: Assumepj ;l = pj ;l , with0 < j1< j2and0 < l1< l2, which requiresJ 3 and L 3, respectively. Then

1j ;j (0) + 1j ;0(l2) + 10;j (l1) = 0 + pj ;l 0 pj ;l = 0:

Based on Theorem 2.1, this indexing defines a cycle of length6 with respect to rows0, j1, andj2 and columns0, l1, andl2ofH, which contradictsg 8.

A lower bound on the minimum value ofp for which g 8 isgiven by the following.

Corollary 2.3: A necessary condition to have g 8 in the Tanner graph representation of a (J; L)-regular QC LDPC code is p > (J 0 1)(L 0 1) or N > (J 0 1)(L 0 1)L.

D. Girthg 10

The next theorem shows that a(J; L)-regular QC LDPC code necessarily hasg 12.

Theorem 2.5: For any (J; L)-regular QC LDPC code, we have g 12.

Proof: ForJ 3, thisresult directly followsfrom Theorem 2.1 withj1 = j4,j2 = j5, andj3 = j6, aswell asi1 = i3 = i5 and i2 = i4 = i6. ForJ = 2, the result follows in a straightforward way from the equivalent expression of Theorem 2.1 with respect toH^tand row index differences, which completes the proof.

Theorem 2.5 generalizesthe result of [16] to any(J; L)-regular QC LDPC code. Based on the distance bounds presented in [23], it impor- tantly indicatesthat for a given code rate, the only way of increasing the guaranteed minimum Hamming distance of a QC LDPC code is to increase bothJ and L.

We finally note that Theorem 2.1 can be viewed asa simplified formulation of [15, Theorem 2] in the case of circulant permutation matrices. A formulation of some results derived in this section with the notationsof [15] hasbeen given in [24].

III. FAMILIES OFQC LDPC CODES

A. Random Constructions

For given valuesof the code lengthN, the code dimension K, and the desired girthg, the most straightforward approach is to determine corresponding values ofJ, L, and p, and then randomly generate (J 0 1)(L 0 1) integers until Theorem 2.1 is satisfied. For given values of J and L, the smallest value of p for which a (J; L)-regular QC LDPC

(3)

TABLE I

SMALLESTVALUE OFp^FOR^W^{HICH A}(J; L)^-RÊGULAR^{QC LDPC C}ÔDE WITHGIRTHg 6^WÂS^FÔUND^WÎTH^CÔMPUTER^SÊARCH

TABLE II

SMALLESTVALUE OFp^FORWHICH A(J; L)-REGULARQC LDPC CODE WITHGIRTHg 8^WÂS^FÔUND^WÎTH^CÔMPUTER^SÊARCH

code with girthg = 6 and g = 8 wasfound with computer search are recorded in Tables I and II, respectively. Note that although the search for smaller values ofp failed, there isno guarantee that such codes do not exist, except for the values ofp which meet the lower bound of Theorem 2.3 or of Corollary 2.3. Forg = 6, the optimum value wasfound in each case, except forL = 9 and J 4 (represented in italics in Table I). As suggested from Theorem 2.3, we observe that the smallest values ofp remain the same as J increases in Table I.

Forg = 8, none of the valuesrecorded in Table II correspondsto the smallest possible valuep = (J 01)(L01)+1 given in Corollary 2.3.

It should be noted that some of these values, as well as those forg = 6, J = 5, and L 8 were found after relatively long computer searches, which suggests to search for structured values ofpj;lin the matrixH of (1). Several such constructions are discussed in the next section.

B. Structured Constructions

In order to speed up the code search based on Theorem 2.1, a particular structure on the(J 0 1)(L 0 1) integers necessary to specify the matrixH of (1) can be imposed. As a result, each structure deter- minesa particular family of QC LDPC codes. In the following, exam- ples of such families are given. These families generally correspond to previous works which are therefore now proposed within the same framework and often generalized.

1) Sumpj;l = jq1+ lq2mod p: In thiscase, we compute 1j ;j (l) = (j10 j2)q1

so that1j ;j (l1) + 1j ;j (l2) = 0. It followsfrom Theorem 2.1 that g 4 for thisconstruction.

2) Productpj;l= jl mod p: For thisfamily, we compute 1j ;j (l) = (j10 j2)l:

Forg 6, Theorem 2.1 becomes (j10j2)(l10l2) 6= 0 mod p for all j16= j2and alll16= l2. This condition is always satisfied forp prime, but depending onJ and L, other valuesof p are also valid. For example, for(J; L) = (3; 6) and p 15, p 2 f7; 9; 11; 12; 13; 14; 15g isa valid choice, while forJ 4, L = 6, and p 15, p 2 f7; 11; 13; 14g works. It is finally interesting to point out that this form defines array codes[15] and that forp prime and J = L = p, the p²2 p²matrixH of (1) definesthe Euclidean geometry plane EG(p; 2). For example, forp = J = L = 5, we obtain a 25 2 25 matrix H of rank 21 corresponding to EG(5; 2).

Forg 8, Theorem 2.1 becomes

(j10 j2)(l10 l3) + (j20 j3)(l20 l3) 6= 0 mod p:

This is impossible forl1 = j3,l2 = j1, andl3 = j2. It followsthat g 6 only with thisconstruction.

3) Powerpj;l = q₁^jq^l₂ mod p: It should first be noted that while the formpj;l = q^j₁q2^l mod p isthat used in [16], an equivalent repre-

sentation which satisfies the form (1) ispj;l= (q^j₁01)(q2^l01) mod p.

For thisfamily, we compute

1j ;j (l) = (q₁^j 0 q^j₁ )q₂^l

and after a little elementary algebra, (5) can be rewritten as

m02 k=0

(q₁^j 0 q^j₁ )(q₂^l 0 q₂^l ) 6= 0 mod p: (7) This equation can be satisfied forg 12. In fact, thisconstruction is a generalization of [16] in whichp waschosen asa prime and q1and q2astwo nonzero distinct elementsof GF(p) with order o1= J and o2= L, respectively. However, for p prime, only o1 J and o2 L is necessary to haveg 6. For example, for p = 7, J = 3, and L = 5, we can choose q1 2 f2; 3; 4; 5g and q2 2 f3; 5g, not necessarily distinct. Note also that for this construction, we haveJ L p 0 1.

Finally, forq1 = q2,g > 6 is impossible as (7) is not satisfied for l0 = j2,l1 = j0, andl2 = j1.

Theorem 2.3 can be refined for this construction as follows.

Theorem 3.1: Forpj;l = q^j₁q2^l mod p, a necessary condition to haveg 6 in the Tanner graph representation of a (J; L)-regular QC LDPC code isp L + 1, and p prime.

Proof: Forl16= l2,pj;l 6= pj;l isequivalent to

q₂^l 6= q₂^l mod p: (8)

Forp prime, it follows o2 L and since o2 p01, we have p L+1.

For thisconstruction and given valuesofJ and L, the smallest value ofp for which a (J; L)-regular QC LDPC code with girth g = 6 and g = 8 wasfound with computer search are recorded in TablesIII and IV, respectively. We observe that, in general, the values found are larger than those given in Tables I and II. By structuring the search, not only can much larger valuesofJ and L be considered, but most impor- tantly, the optimum valuesofp for thisconstruction can be determined.

Forg = 6 (note that g 8 for L = 2 based on Corollary 2.1), we observe that the optimum value ofp given in Table III corresponds to that of Theorem 3.1. On the other hand, asalready noted in [16], an an- alytical derivation of the optimum value ofp for g 8 seems quite difficult. Thisisfurther confirmed by the fact that the same value ofp can be found for the same value ofJ and different valuesof L. We also notice that the optimum value ofp doesnot have to be prime.

A(J; L)-regular LDPC code obtained by thisconstruction can be extended to either a(J +1; L)-regular LDPC code, or a (J; L+1)-regular LDPC code by appending toH a row of L I(0)’s, or a column of J I(0)’s. Similarly, a (J+1; L+1)-regular LDPC code can be obtained by appending toH both a row of L I(0)’sand a column of J I(0)’s.

Sinceq^j₁q₂^l 6= 0, and for p prime, o1 J and o2 L, these extensions preserveg = 6 and can be applied to any code given in Table III.

In the preceding sections, the main motivation was to find the smallest value ofp for which a (J; L)-regular QC LDPC code exists.

However, the search can be conducted with additional constraints such asa large number of dependent check sumsin the matrixH of (1).

IV. SEARCH ANDSIMULATIONRESULTS

Comparative error performance studies of LDPC codes constructed with variousmethodshave been conducted for given valuesofN and K. For example, two QC LDPC codeswith N = 1053, K = 812, J = 3, L = 13, p = 81, and g = 6 and 8, respectively, and one QC LDPC code withN = 1062, K = 819, J = 4, L = 18, p = 59, and g = 6 have been generated randomly based on Section III-A for g = 6 andg = 8, respectively, and compared in Fig. 1 to other LDPC codes with similar parameters. We notice that these codes slightly outperform their LDPC counterpartsconstructed from [1] withJ = 3 and L = 12 or13, and J = 4 and L = 17 or 18 [8]. They are easier to design,

(4)

TABLE III

SMALLESTVALUE OFp^FOR^W^{HICH A}(J; L)^-RÊGULAR^{QC LDPC C}ÔDE^WÎTH^GÎRTHg 6^WÂS^FÔUND^WÎTH^CÔMPUTER^SEARCH FOR THE CONSTRUCTIONpj;l = q^j₁q₂^l mod p

TABLE IV

SMALLESTVALUE OFp^FOR^W^{HICH A}(J; L)^-RÊGULAR^{QC LDPC C}ÔDE^WÎTH^GÎRTHg 8^WÂS^FÔUND^WÎTH^CÔMPUTER^SEARCH FOR THE CONSTRUCTIONpj;l = q^j₁q₂^l mod p

Fig. 1. BP decoding of different LDPC codesof about rate0:77^{and length}1050(maximum of 200 iterations).

represent, and encode than random Gallager codes. We also notice that increasing g from 6 to 8 had little effect on the error performance.

On the other hand, the QC LDPC codesdo not perform aswell asthe (1057; 813) projective geometry (PG) code for which J = L = 33 and the matrixH used for decoding has size 1057 2 1057. However, they have a much lower decoding complexity than the PG code. Note also that, in general, QC LDPC codes can be constructed in a more flexible manner than PG codesfor given valuesofN and K. However, while the PG code hasa minimum distance of34, that guaranteed from the boundsof [23] for the three QC LDPC codesare much smaller.

In Fig. 2, a similar comparison has been represented for longer codes of lower rate. Based on Section III-A, two QC LDPC codes withN =

4104, K = 2283, J = 4, L = 9, p = 456, and g = 6 and 8, respectively, and one QC LDPC code withN = 4104, K = 2287, J = 8, L = 18, p = 228, and g = 6 have been generated randomly.

A random(4; 9)-regular LDPC code with N = 4104 and N = 2281 and the(4096; 2238) with J = L = 16 and g = 8 constructed in [12] (referred to asfinite-geometry (FG) code in the figure) are also considered. For these longer codes, compared with that of the previous example, we observe that the three LDPC codes withJ = 4, L = 9 have similar error performance. Hence, at the word error rates represented, increasing the girth has no great influence. These codes also outperform that of [12] constructed from finite geometry. We finally notice that the QC LDPC code withJ = 8 performsquite poorly.

(5)

Fig. 2. BP decoding of different LDPC codesof about rate0:55and length4100(maximum of 200 iterations).

V. MINIMUMDISTANCE OFQC LDPC CODES

A result of [25] implies that the minimum Hamming distancedHof a(J; L)-regular QC LDPC code isupper-bounded by dH (J + 1)!.

Consequently, for a(J; L)-regular QC LDPC code, dHcannot grow withN, which suggests that QC LDPC codes compare favorably to random LDPC codes only for short to medium code lengths. Obser- vationsalong the same lineswere made in [16]. In the following, we derive for(3; L)-regular QC LDPC codes a necessary condition for dH= 24. Extensions to larger values of J are then discussed.

A. (3; L)-Regular QC LDPC Codes

Let usconsider four columnsof the matrixH given in (1). With respect to Theorem 2.1, the indexespj;lcan be normalized(mod p) as

[H0H1H2H3] =

I(0) I(0) I(0) I(0) I(0) I(i1) I(i2) I(i3) I(0) I(j1) I(j2) I(j3)

: (9)

In each3p 2 p submatrix Hk,k = 0; . . . ; 3, define l asthe position of thelth column of Hkforl = 0; . . . ; p 0 1, and define Skas a subset of column positions inHk. Then it isreadily seen that the following 24 columnssum to zero:

S0= f0i20 j3; 0i10 j2; 0i30 j1; 0i30 j2; 0 i10 j3; 0i20 j1g S1= f0i20 j3; 0j2; 0i3; 0i30 j2; 0j3; 0i2g S2= f0i10 j3; 0j1; 0i3; 0i30 j1; 0j3; 0i1g S3= f0i2; 0i10 j2; 0j1; 0j2; 0i20 j1; 0i1g

where each value in each subsetSkistaken modulo-p. Note that for k = 0; . . . ; 3, Sk is composed of the negative sums of all pairs of elements(ix; jy) of (9) indexed by x 6= y, x 6= k, and y 6= k. A necessary condition fordH = 24 is, therefore, that all six columns in each setSiare distinct.

Considering the matrixH given in (1), thiscan be jointly achieved for allL columnsof submatricesby the following procedure.

1. Construct theL 2 L table with the L columnsand the L rows labeled by the indexesof the second row and of the third row ofH in (1), respectively.

2. For all nondiagonal elementsof the table, insert the modulo-p sum of the corresponding row and column labels.

3. For any three distinct valuesx, y, and z in f0; . . . ; L 0 1g, check that no two of the six nondiagonal elements in thexth, yth, and zth rowsand columnsof the table are the same.

Note that any pair of similar indexes found at step-(3) decreases the designed distance by two.

Example: Consider a(3; 5)-regular code of length 155 and dimen- sion64 obtained with p = 31 and represented by the matrix

H =

I(0) I(0) I(0) I(0) I(0) I(0) I(4) I(24) I(1) I(5) I(0) I(12) I(10) I(3) I(15)

: (10)

The corresponding table is depicted in Table V. Since15 appearstwice when considering the zeroth, second, and fourth rows and columns, and 8 appears twice when considering the seond, third, and fouth rows and columns, we obtaindH 20. We also notice that 5 appearstwice in Table V, but not in the same three row and column positions.

(6)

TABLE V

THE(3,5)-REGULAR(155,64) QC LDPC CODE REPRESENTED BY(10)

If we consider now the(3; 5)-regular code of the same length and dimension constructed in [16] as described in Section III-B3 forq1= 5, q2 = 2, and p = 31, itsmatrix H isgiven by

H =

I(0) I(0) I(0) I(0) I(0) I(0) I(4) I(12) I(28) I(29) I(0) I(24) I(10) I(13) I(19)

: (11)

AlthoughdH= 20 for thiscode [16], no two elementsare the same in the table obtained by the proposed procedure. This confirms that this method only represents a necessary condition fordH = 24.

We finally mention that Theorem 2.3 can be further refined in order to havedH = 24. For example, it isreadily verified by contradiction that all nonzero valuesin (9) have to be distinct, which impliesp 2(L 0 1) + 1 for (3; L)-regular QC LDPC codeswith dH = 24.

Further consideration of this interesting problem is beyond the scope of thiscorrespondence.

B. (J; L)-Regular QC LDPC Codes

The results derived in Section V-A can be extended to any (J; L)-regular QC LDPC code but the procedure becomesquite tediousas(J + 1)! columnshave to be identified. However, thiscan be achieved by consideringJ + 1 submatrices Hkof sizeJp 2 p in (9). Each of theJ + 1 corresponding sets Skisthen composed of all J! possible negative sums indexed on f0; . . . ; Jgnfkg.

VI. CONCLUSION

In this correspondence, a simple necessary and sufficient condition to determine QC LDPC codeswith a given girth hasbeen derived. This condition impliesthat such codescannot have a girth larger than12.

Consequently, for a given code rate, their minimum distance cannot be increased by increasing the code length and thus, the girth as for random constructions. In fact, an upper bound on the minimum Ham- ming distance of QC LDPC codes was derived in [25], and a necessary condition to reach thisbound hasbeen proposed.

These simple results suggest that when constructing families of LDPC codes, either relatively large girth (i.e.,g > 12), or additional constraints such as large minimum distance, a large number of redundant check sums, or appropriate coset weight distribution (see [26]) have to be considered.

ACKNOWLEDGMENT

The author wishes to thank the two reviewers for their constructive comments which greatly improve the presentation of these results, and Norifumi Kamiya for interesting discussions.

REFERENCES

[1] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399–431, Mar. 1999.

[2] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity check codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 619–637, Feb. 2001.

[3] M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman,

“Improved low-density parity check codes using irregular graphs,” IEEE Trans. Inform. Theory, vol. 47, pp. 585–598, Feb. 2001.

[4] S. Y. Chung, G. D. Forney, T. J. Richardson, and R. L. Urbanke, “On the design of low-density parity check codes within 0.0045 dB of the Shannon limit,” IEEE Commun. Lett., vol. 5, pp. 58–60, Feb. 2001.

[5] X.-Y. Hu, “Low-delay low-complexity error correcting codes on sparse graphs,” Doctoral dissertation, Swiss Federal Institute of Technology Lausanne (EPFL), Lausanne, Switzerland, 2002.

[6] X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, “Irregular progressive-edge growth (PEG) Tanner graphs,” in Proc. IEEE Int. Symp. Information Theory, Lausanne, Switzerland, July 2002, p. 480.

[7] P. O. Vontobel and H. A. Loeliger, “Irregular codesfrom regular graphs,”

in Proc. IEEE Int. Symp. Information Theory, Lausanne, Switzerland, July 2002, p. 284.

[8] R. Lucas, M. Fossorier, Y. Kou, and S. Lin, “Iterative decoding of one-step majority logic decodable codes based on belief propagation,”

IEEE Trans. Commun., vol. 48, pp. 931–937, June 2000.

[9] Y. Kou, S. Lin, and M. Fossorier, “Low-density parity-check codes based on finite geometries: A Rediscovery and new results,” IEEE Trans. In- form. Theory, vol. 47, pp. 2711–2736, Nov. 2001.

[10] P. O. Vontobel and R. M. Tanner, “Construction of codes based on fi- nite generalized quadranglesfor iterative decoding,” in Proc. IEEE Int.

Symp. Information Theory, Washington, DC, June 2001, p. 223.

[11] S. J. Johnson and S. R. Weller, “Codes for iterative decoding from partial geometries,” in Proc. IEEE Int. Symp. InformationTheory, Lausanne, Switzerland, July 2002, p. 310.

[12] J. L. Kim, U. N. Peled, I. Perepelitsa, and V. Pless, “Explicit construction of families of LDPC codes with girth at least six,” in Proc. 40th Annu.

Allerton Conf. Communication, Control and Computing, Monticello, IL, 2002.

[13] R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA:

MIT. Press, 1963.

[14] R. Townsend and E. J. Weldon, “Self orthogonal quasicyclic codes,”

IEEE Trans. Inform. Theory, vol. IT-13, pp. 183–195, Mar. 1967.

[15] J. L. Fan, “Array codes as low-density parity-check codes,” in Proc. 2nd Int. Symp. Turbo Codes, Brest, France, Sept. 2000, pp. 545–546.

[16] R. M. Tanner, D. Sridhara, and T. Fuja, “A class of group-structured LDPC codes,” in Proc. ISTA, Ambleside, England, 2001.

[17] B. Vasic, “Combinatorial constructions of low-density parity check codesfor iterative decoding,” in Proc. IEEE Int. Symp. Information Theory, Lausanne, Switzerland, July 2002, p. 312.

[18] T. Mittleholzer, “Efficient encoding and minimum distance bounds of Reed-Solomon-type array codes,” in Proc. IEEE Int. Symp. Information Theory, Lausanne, Switzerland, July 2002, p. 282.

[19] I. Djurdjevic, J. Xu, K. Abdel-Ghaffar, and S. Lin, “A class of low-density parity-check codes constructed based on Reed-Solomon codes with two information symbols,” IEEE Commun. Lett., vol. 7, pp. 317–319, July 2003.

[20] T. Okamura, “Designing LDPC codes using cyclic shifts,” in Proc. IEEE Int. Symp. Information Theory, Yokohama, Japan, June/July 2003, p.

151.

[21] J. Denesand A. D. Keedwell, Latin Squares and their Applications.

New York: Academic Press, 1974.

[22] W. W. Peterson and E. J. Weldon Jr, Error-Correcting Codes, 2nd ed.

Cambridge, MA: MIT Press, 1972.

[23] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 533–547, Sept. 1981.

[24] N. Kamiya, “Recent results on the girth of a block matrix consisting of permutations,” Mitsubishi , Kanagawa, Japan, Tech. Rep., 2003.

[25] D. J. C. MacKay and M. Davey, “Evaluation of Gallager codesfor short block length and high rate applications,” in Proc. IMA Workshop Codes, Systems and Graphical Models, 1999.

[26] D. J. C. MacKay and M. S. Postol, “Weaknesses of Margulis and Ra- manujan-Margulis low-density parity-check codes,” Electron. Notes in Theor. Comput. Sci., vol. 74, 2003.