Construction of Regular and Irregular QC-LDPC Codes: a Finite Field Approach and Masking

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1 06 Iteratioal Symposium o Advaces i Electrical, Electroics ad Computer Egieerig (ISAEECE 06 Costructio of egular ad Irregular QC-LDPC Codes: a Fiite Field Approach ad Maskig Xiaopeg Che,a, Li Zhou,,, ui Zhao,c ad Yucheg He,,d Xiame Key Laoratory of Moile Multimedia Commuicatios, Natioal Huaqiao Uiversity, Xiame, Chia State Key Laoratory of Itegrated Services Netorks, Xidia Uiversity, Xi a, Chia a @qq.co lihou@hqu.edu.c, c rhao@hqu.edu.c, d yucheg.he@hqu.edu.c Keyords: LDPC; maskig; Quasi-Cyclic Astract. Quasi-Cyclic Lo-Desity Parity-Check (QC-LDPC codes require less hardare resources due to the quasi-cyclic structure of parity-check matrices. I this paper, to improved methods for costructig QC-LDPC codes are proposed. Firstly, i order to make sure the masked LDPC codes have flexile code rate ad code legth, a improved costructio of maskig matrix is preseted. The proposed regular maskig matrix of QC-LDPC codes have the girth at least 6. The, the irregular codes are costructed y usig improved fiite field approach ad PEG algorith hich result i large girth ad less short cycles. Simulatios demostrate that the proposed QC-LDPC codes have oth loer error floor ad good aterfall performace. Itroductio Lo-desity parity-check (LDPC codes ere discovered y Gallager i 96[], ad the rediscovered i the late 990s. They ere proved to e chael-capacity-approachig codes[]. Sice the, various methods for costructig LDPC codes have ee proposed. The costructio of LDPC codes ca e divided ito to categories: radom codes[3] ad structured codes[4,5]. adom codes have good performace, ut it s too complex to applicatio. Structured codes ith optimied desig ca ot oly performace as ell as radom codes, ut also get loer complexity i practical applicatios. ecetly, algeraic method have ecame a effective method to costruct structured codes. This method ca otai the codes ith good error performace more easily. Based o the algeraic costructio of LDPC codes that have quasi-cyclic structure, called QC-LDPC codes[6,7]. I order to improve the it error rate(be performace of QC-LDPC codes, may methods have ee proposed to optimie the girth, degree distriutio, the trappig set ad so o. To improve the error floor performace, maskig techique[8] as proposed to costructio QC-LDPC codes. I this paper, e preset costructios of oth regular ad irregular QC-LDPC codes ith large girth. First, a geeral costructio ased o fiite field ad maskig techology is itroduced. Secod, a improved costructio method of maskig matrix of QC-LDPC codes is proposed. This method ca otai codes ith a ider rage of rate ad also have a good performace. The, e propose a improved method for desigig ase matrix y usig to susets i the fiite field. Based o this ase matrix, e ca otai irregular codes via maskig y PEG algorithm[9]. At last, the simulatio results prove that our methods have good error performace. QC-LDPC Codes Based o Fiite Field ad Maskig Costructio of ase matrix. I geeral, QC-LDPC codes ased o fiite field eed to costruct a m ase matrix B. It is sho i Eq. : 06. The authors - Pulished y Atlatis Press 06

2 06 Iteratioal Symposium o Advaces i Electrical, Electroics ad Computer Egieerig (ISAEECE 06 B m,,,,, ( here i,j ( i j is a elemet of Galois field GF(q. Matrix B satisfied the ro-colum costrait: ay to ros(or to colums have at most oe same o-ero elemets at the same place. This costrait esures that the costructed codes have girth at least 6. Let A ( i,j e a (q- (q- circulat permutatio matrix if i,j 0, ad it s a (q- (q- ero matrix if i,j = 0. By replacig each i,j ith A ( i,j, e otaied a m(q- (q- parity matrix H. Costructio via maskig. Let Z = [ i,j ] e a m maskig matrix, here i j, i,j {0,}, the masked matrix M is sho i Eq. : M ZH (,,,,,,, here i,j A ( i,j =A ( i,j if i,j =, i,j A ( i,j = 0 if i,j = 0. The masked matrix M gets the same degree distriutio as the maskig matrix Z. So the maskig matrix plays a sigificat part i the performace of the masked LDPC codes. Costructio of egular ad Irregular Codes via the Proposed Method The improved Costructio of regular codes. I [0], a (3,6-regular maskig matrix is sho i Eq. 3: Z (4,8 ( The girth of the regular QC-LDPC codes icreased y his maskig matrix from 6 to 8, ad result i a masked matrix M ith ro eight 6 ad colum eight 3, respectively. It is very effective i decreasig short cycles ad icreasig the girth, so it has a good BE performace ad lo error floor. But this maskig matrix Z(4,8 oly suit coditio i fixed sie ad code rate of ase matrix. To solve the deficiecy of this maskig matrix, e proposed a improved maskig matrix as follos: repeat the first pair of the colums ad third pair of colums i Z(4,8 p times, otai Z(4,8p ith 4 8k sie. The doard expad Z(4,8p t times, otai Z(4t,8p ith 4t 8p sie. Let = Z(4,8, improved maskig matrix Z(4t,8p is sho i Eq. 4: Z(4t,8 p t,,, t,, p, p t, p (4 06. The authors - Pulished y Atlatis Press 07

3 06 Iteratioal Symposium o Advaces i Electrical, Electroics ad Computer Egieerig (ISAEECE 06 here t, p. Notice that the maskig matrix Z(4t,8p also satisfied the 3 3 SM-costrait i [0], so the masked matrices have the girth-8 at least, hich result i good BE performace ad lo error floor. The improved Costructio of irregular codes. To costruct ase matrix, let α e the primitive k0 k km l0 l l elemet of GF(q. For q, let S {,,..., }, S {,,..., } e to susets of GF(q, e ca otai matrix elemet i,j i ase matrix B, hich is sho i Eq. 5: k l i j ( i, j (5 here 0 k i m-, 0 l j -. With differet choices of to susets S ad S, e ca otai differet ase matrix B ith girth 6 or 8. I order to otai irregular codes, e use PEG algorithm to costruct maskig matrix. By usig degree distriutio optimiatio, a proper maskig matrix ca e otaied. This method has etter BE performace compared ith the traditioal costructio y usig PEG algorithm. Simulatio esults The performace of differet codes costructed y our methods are verified for additive hite Gaussia oise chael. Codes are modulated y BPSK ad decoded y sum-product algorithm. Maximum 50 iteratios are cosidered for iterative decodig. Example : First, e costruct ase matrix B(6, over GF(8. The maskig matrix Z(6, as costructed accordig to Eq.4. By maskig B(6, ith Z(6,, e otaied a masked matrix M(6,. Every elemets i M(6, ca e replaced y 7 7 permutatio matrices. Fially, e otaied a (6,-regular (5367,4605 QC-LDPC code C ith rate The BE performace of this code is sho i Fig.. It shos that the performace of the code C is much etter tha the code C (60,5345 give i [0]. The code C outperforms the code C y aout 0.5dB at BE=. Example : I this example, let GF(8 e the field for code costructio. Based o this field, e costruct a 4 8 ase matrix B(4,8. Maskig matrix Z(4,8 is give accordig to Eq. 4. By usig maskig techology, e ca get the masked matrix M(4,8 ith ro eight ad colum eight 3. Hece the ull space of Z(4,8 gives a (3556,3048 regular QC-LDPC code C 3 ith rate For compariso, a correspodig regular (3584,307 code C 4 is give i []. The error performace is sho i Fig.. We ca see that the error performace of C 3 is etter tha that of the code C 4 at all SN regio. It shos that the code costruct ith improved maskig matrix has good BE performace ad lo error floor. 0 - C (5367,4605 C (60, C 3 (3556,3048 C 4 (3584,307 BE BE E /N (db 0 Fig.. The error performace of the codes C give i Example ad C give i [0] E /N (db 0 Fig.. The error performace of the codes C 3 give i Example ad C 4 give i []. 06. The authors - Pulished y Atlatis Press 08

4 06 Iteratioal Symposium o Advaces i Electrical, Electroics ad Computer Egieerig (ISAEECE 06 Example 3: I this example, e costruct a irregular QC-LDPC code y usig to fiite field susets ad PEG maskig matrix. Let α e a primitive elemet of GF(64, e choose S {,,..., } ad S {,,..., } e to susets of elemets, form a 5 30 ase matrix B(5,30. Next, the 5 30 maskig matrix Z(5,30 as costructed y PEG algorithm ith degree distriutio ( x 0.46x 0.x 0.x 0.3x 0.07x, the e otaied a masked matrix M(5,30. The ull space of M(5,30 gives a irregular (890,945 code C 5 ith rate 0.5. For compariso, a irregular (890,945 code C 6 is give i[]. The BE performace is sho i Fig. 3. At the BE of 0-6, code C 5 outperforms the code C 6 y aout 0.dB. The code costructed y to susets performs slightly etter tha fiite field approach i []. 6 Example 4: Agai, cosider the field GF(64 for code costructio. Set S {,,..., }, S {,,..., } e to susets, form a 6 3 ase matrix B(6,3. The maskig matrix Z(6,3 as costructed y PEG algorithm ith degree distriutio ( x 0.46x 0.x 0.x 0.3x 0.07x. The ull space of masked matrix M(6,3 gives a irregular (06,008 code C 7 ith rate 0.5. The code performace is sho i Fig. 4. Also icluded i Fig. 4 is the BE performace of a (06,008 code C 8 give i [3]. It shos that the to codes perform the same efore SN of.0, ut the code C 7 outperforms the C 8 startig from the SN of C 5 (890,945 C 6 (890, C 7 (06,008 C 8 (06, BE BE E /N (db 0 Fig.3. The error performace of the codes C 5 give i Example 3 ad C 6 give i [] E /N (db 0 Fig.4. The error performace of the codes C 7 give i Example 4 ad C 8 give i [3]. Coclusios I this paper, e firstly proposed a improved method for costruct maskig matrix ith large girth ad lo error floor. Based o the improved maskig matrix hich satisfies the 3 3 SM-costrait, the regular QC-LDPC codes ith girth-8 ere costructed, it has flexile code rate ad good error performace compared to method i [0]. Moreover, e proposed to elemet susets i fiite field to costruct the irregular QC-LDPC codes via PEG algorithm. The costructio oeys the ro-colum costraits, hich esures that the girth is at least 6. Simulatio results demostrate that the proposed costructio approaches of QC-LDPC codes have etter BE performace ad lo error floor tha some existig methods. Ackoledgemets This ork as supported i part y the grats from the Natioal Natural Sciece Foudatio of Chia (630095, 64065, the Natural Sciece Foudatio of Fujia Provice of Chia (04J05076, 04J043, 05J06, the Sciece Foudatio of Natioal Huaqiao Uiversity (BS9, 06. The authors - Pulished y Atlatis Press 09

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