Quasi-Cyclic LDPC Codes on Latin Squares and the Ranks of their Parity-Check Matrices

Transcription

1 uas-cyclc LDPC Codes on Latn quares the Rans of ther Party-Chec Matrces L Zhang, n Huang, hu Ln, K Abdel-Ghaffar Department of Electrcal Computer Eng nversty of Calforna Davs, CA 95616, A Emal: Ian F Blae Department of Electrcal Computer Eng nversty of Brtsh Columba ancouver, BC, 6 1Z4, CANADA Emal: fblae@eceubcca Abstract uas-cyclc codes are the most promsng class of structured LDPC codes due to ther ease of mplementaton excellent performance over nosy channels when decoded wth message-passng algorthms as extensve smulaton studes have shown An approach for constructng quas-cyclc LDPC codes based on Latn squares over fnte felds s presented By analyzng the party-chec matrces of these codes, expressons for ther rans are derved Expermental results show that, wth teratve decodng algorthms, the constructed codes perform very well over the AWGN the bnary erasure channels I INRODCION LDPC codes have attracted wdespread nterest because of ther remarable performance that can be acheved by effcent decodng algorthms hese codes, frst dscovered by Gallager n ], lad dormant for about 35 years untl ther redscovery n the late 1990 s 2], 3] nce then a great deal of research effort has been expended n desgn, constructon, structural analyss, encodng, decodng, generalzatons applcatons of LDPC codes Many LDPC codes have been chosen as the stard codes for varous next generatons of communcaton systems A regular bnary LDPC code 1] s gven by the null space of a sparse party-chec matrx over GF(2) that has constant column weght constant row weght, where are small compared to the code length uch an LDPC code s sad to be ( )-regular If the columns /or rows of the party-chec matrx have varyng weghts, then the null space of gves an rregular LDPC code In almost all of the proposed constructons of LDPC codes, the followng constrant on the rows columns of the partychec matrx s mposed: no two rows (or two columns) can have more than one poston where they both have 1- components hs constrant on the rows columns of s referred to as the row-column (RC)-constrant he RCconstrant on ensures that the anner graph 4] of the LDPC code gven by the null space of has a grth of at least 6 5], 6], 7] It also ensures that the mnmum dstance of a -regular LDPC code s at least hs dstance bound s tght for regular LDPC codes whose party-chec matrces have large column weghts, such as fnte geometry LDPC codes 5] fnte feld LDPC codes constructed n 8], ths paper If the party-chec matrx of an LDPC code s an array (or a bloc) of sparse crculants of the same sze over GF(2), then the null space of gves a quas-cyclc (C)-LDPC code Many of the algebrac constructons of LDPC codes result n C-codes Extensve smulaton studes have shown that LDPC C-codes have very good performance over nosy channels when decoded wth message-passng algorthms, see 6], 7] the references theren Well desgned algebrac C-LDPC codes can perform close to the hannon lmt ust as well as (or even better than) ther correspondng rom or pseudo-rom C-LDPC codes constructed usng computer-based methods over the AWGN the bnary erasure channels, as demonstrated n 8], 9] In addton, a maor advantage of C-LDPC codes s that they can be effcently encoded usng smple shft-regsters 10] hs paper s concerned wth constructons of C-LDPC codes ran analyss of ther party-chec matrces We consder a general class of C-LDPC codes then a subclass constructed based on Latn squares he paper s organzed as follows ecton II presents a general algebrac method for constructng C-LDPC codes A general framewor to determne the ran of the party-chec matrces s developed n ecton III Although the general algebrac method for constructng C-LDPC codes presented n ecton II was proposed n 8], 9]; these papers dd not provde any analyss of the rans of the party-chec matrces of the constructed codes In ths paper, such an analyss s provded hs analyss generalzes smplfes the ran analyss of a class of C-LDPC codes constructed based on cyclc MD (or R) codes wth two nformaton symbols 11] In ecton I, we present a large class of algebrac C-LDPC codes constructed based on Latn squares over fnte felds, called C-LDPC codes on Latn squares he constructon of C-LDPC codes on Latn squares follows the general method explaned n ecton II (he constructon method as well as the constructed codes sgnfcantly dffer from those based on BIBD desgns devsed usng Latn squares 12], 13], 14]) We analyze, n ecton, the rans of the party-chec matrces of several subclasses of C-LDPC codes on Latn squares derve combnatoral expressons for these rans he paper s concluded n ecton I

2 O R R x II A GENERAL ALGEBRAIC CONRCION OF C-LDPC CODE Consder the Galos feld GF( ) Let be a prmtve element of GF( ) hen, the powers of, gve all the elements of GF( ) Let be a crculant permutaton matrx (CPM) whose top row s gven by the -tuple! over GF(2) where the components are labeled from 0 to #" the sngle 1-component s located at the 1st poston hen conssts of the ( )-tuple $% ts &" rght cyclc shfts as rows For ('),+-, let / wth tself ) tmes, called the ) th power of hen, ( s also a 4 4 CPM whose top row has a sngle 1-component at the ) th poston For )5, 26708, the 9, dentty matrx Let (: 26;<8 hen the set =?>@ ( 26BA of CPMs forms a cyclc group of order / under matrx multplcaton over GF(" ) wth / as the multplcatve nverse of ( as the dentty element be the product of For C'D)2+E /, we represent the nonzero element F of GF( ) by the $ G $ CPM 2 hs matrx representaton s referred to as the 7 -fold bnary matrx dsperson (or smply bnary matrx dsperson) of nce there are nonzero elements n GF( ) there are exactly H dfferent CPMs over GF(" ) of sze H I H, there s a one-to-one correspondence between a nonzero element of GF( ) a CPM of sze & 4 2 For a nonzero element J n GF( ), we use the notaton K J to denote ts bnary matrx dsperson If J20F, then K J,L( For the 0-element of GF( ), ts bnary matrx dsperson s defned as the 4 M 7 zero matrx (ZM), denoted by Consder a N$PO matrx over GF( ), XZ ] 6] ] ^^^ 6] ^^^ ] ` ] ` W H ] H6] ^^^ H6] a (1) whose rows satsfy the followng constrant: for ;'b) dc-+ N e)f c G'-g h+-, the Hammng dstance between the two -ary O -tuples, &, s at least Ol, (e, F & dffer n at least O places) he above constrant on the rows of matrx gven n (1) s called the row-dstance (RD)-constrant s called an RD-constraned matrx For m';),+nn &'oc3+-o, dspersng each nonzero entry of p] nto a CPM K p] 9:qK ) rc over GF(2) each 0-entry nto a s ZM, we obtan the followng NttO array of CPMs /or ZMs over GF(2) of sze 9 : uwv K yx p] zp{ ] z { (2) s called the bnary, -fold array dsperson of (or smply bnary array dsperson of ) t s a N 9 matrx over GF(2) he matrx s called the base XZ matrx Based on the RD-constrant on the rows of the bnary matrx dspersons of the entres of, t was proved n 8], 9] that, as a N s H/O matrx over GF(2), satsfes the RC-constrant Hence, ts assocated anner graph has a grth of at least 6 he total number of -entres n s at most Ǹ O 2, whle the total number of entres of s Ǹ O %" herefore, for a relatvely large, s a sparse matrx that satsfes the RC-constrant Hence, the null space of gves a C-LDPC code of length O 3 wth rate at least O; N }~O, whose anner graph has a grth of at least 6 he subscrpt g sts for quas-cyclc If has constant column row weghts, e s a regular C-LDPC code, otherwse t s an rregular C-LDPC code he constructon presented above s a smplfed verson of the constructon gven n 8], 9] where several classes of RD-constraned matrces over fnte felds were gven By array dspersons of these classes of RD-constraned matrces, several classes of C-LDPC codes were constructed he codes gven n the examples of 8], 9] decoded wth teratve decodng usng the sum-product algorthm (PA) dsplayed excellent performance over the AWGN bnary erasure channels n terms of error-rate, error-floor rate of decodng convergence However, no analyss of the ran of the array gven by (2) was provded n 8], 9] In the followng secton, such analyss s presented for the specal case n whch (L"@ III RANK ANALI Let be a prmtve element of GF(" ) Let be the "@ 1, "@ 1 Fourer ransform (F) matrx over GF(" ) ts nverse whch are defned as follows 15]: nwv x zƒ{ ] z { M qv zp{ ] z { hen L y 1L8, a " matrx "@ P dentty For C'E)/+b", applyng to the " " 0 CPM ( over GF(2), we obtan the followng "@ n 9 " L matrx over GF("@ ): H 0 ( H ( H e ( ˆ d ( (3) Let Š m ( Expng ( based on (3), we fnd that Š9 s a dagonal matrx over GF(" ) wth Œ,, Ž p,,, as entres on ts man dagonal zeros elsewhere For smplcty, we express the above dagonal matrx as follows: &, Ž ƒ Recall that n the last secton, we represent the feld element by the CPM 2 Wth the above transformng process, we map nto a dagonal matrx ŠM n frequency doman he mappng s one-to-one Š may be vewed as the matrx dsperson of n frequency doman Defne two matrces over GF("@ ):

3 R R H n H H H s a N /N dagonal array wth s on ts man dagonal H s an O7sO dagonal array wth H s on ts man dagonal Applyng H to the array v K yx ƒ] zp{ ] z { of CPMs /or ZMs gven by (2), we obtan the followng N$PO array of "@ n 9 "@ n square submatrces over GF(" ): H wv K p] x zp{ ] z { (4) If K ƒ] wth 5' h+ "@ b, then K p] n Š,2 ), Ž e If K s a ZM, then ƒ] K p] s also a zero matrx can be vewed as the frequency doman representaton (or the F) of array dsperson of n frequency doman) Defne the followng ndex sets: (or as the Dv " 6" " N& " x Dv " C" x Dv " 6" " O " x Dv " o" x We permute the rows columns of based on the ndex sets, respectvely hus, the ) th row after permutaton s gven by the row n whose ndex s the ) th element n, smlarly, the c th column after permutaton s gven by the column n whose ndex s the c th element n he permutatons result n a " L " q dagonal array wth "@ matrces, H!, each of sze N77O, on ts man dagonal, where denotes the Hadamard product 16] of wth tself h tmes, ('1h+n"@ he Hadamard product of two matrces Dv yx p] K5wv yx of the same p] sze s defned as ther element-wse product BK5qv ƒ] x ƒ] he superscrpt of sts for Hadamard product herefore, 0 Ž e (5) Hereafter, we refer to as the Hadamard product of to the h th power Let BO N denote the ran of a matrx over a feld nce s the frequency doman representaton of s obtaned by permutng the rows columns of, we must have s s~ BO N (6) It follows from (5) (6) that we have the followng theorem on the ran of heorem 1: he ran of the N O array of CPMs /or ZMs over GF(2) gven by (2) s equal to ~ BO N s I C-LDPC CODE ON LAIN ARE We present a new class of RD-constraned matrces whose constructons are based on Latn squares over fnte felds By array dspersons of ths class of RD-constraned matrces, a large class of C-LDPC codes s constructed A A Class of RD-constraned Latn quares over Fnte Felds Defnton 1: An array s called a Latn square of order O f each row each column contans every element of a set of O obects exactly once Latn squares form a specal type of combnatoral desgns 17] here are numerous constructons of Latn squares usng fnte felds 17], 18] In the followng, a large class of Latn squares s constructed based on fnte felds usng a very smple method Latn squares n ths class satsfy the RD-constrant Consder the feld GF( ) Let be a prmtve element of GF( ) Defne the ndex set P " he ndces n represent the powers of, G F, Let be a nonzero element of GF( ) For any )!", the elements, F#4 #4 #4 F6 #tl,, are all dstnct they form all the elements of GF( ) wth < ; Form the followng /# matrx over GF( ): X & & ^^^ &C $GC $GC ^^^ %&C, &C &C ^^^ F6&&, s'&c s'&c ^^^,(G, (7) Label the rows columns of wth 4C" From the structure of dsplayed by (7), we can readly see that has the followng structural propertes: 1) the entres of each row are all dfferent they form the elements of GF( ); 2) the entres of each column are all dfferent they form the elements of GF( ); 3) any two rows dffer n every poston; 4) any two columns dffer n every poston; 5) there are exactly zero entres located at dfferent rows dfferent columns It follows from propertes 1) 2) that s a Latn square of order For 30, we can construct % Latn squares from (7) heorem 2: For any nonzero element ) GF( ), satsfes the RD-constrant X Proof: Let be two dfferent rows n hen ) f ˆc For any two ntegers g h wth ' g h&+ M, consder the two -tuples over GF( ), s We need to prove that & cannot have more than one poston where they have dentcal components

4 R R ) ) ) uppose that & have dentcal components at two dfferent postons (Cf ) hen, we have the followng two equaltes: GC 0 G &C ˆ H G From these two equaltes, we obtan the equalty t,n hs equalty mples ether ),1c or ( whch contradcts the facts that )f lc mf herefore, F cannot have more than one poston where they have dentcal components hs proves that satsfes the RD-constrant For ml, has the smplest form,! he entres on the man dagonal of GF( ) B C-LDPC Codes on Latn quares By array dsperson of the Latn square obtan the followngmt array of 7 ZMs over GF(2): X (8) are the 0-element of gven by (7), we ( CPMs XZ K ] K ] ^^^ K ] K ] K ] K 6] ^^^ K 6] K ] D K 6y] K ] ^^^ K ] 6 K ] K m] K G] ^^^ K G] 6 K G] (9) where K ƒ] LK #H4 s the matrx dsperson of the entry #MW n wth ) c n the ndex set has ZMs of sze ( ( If we set #, has the form gven by (8) hen, the zero matrces of are on the man dagonal of s a ` s ` matrx over GF(2) wth both column row weghts equal to nce satsfes the RD-constrant, satsfes the RC-constrant Furthermore, snce all the nonzero entres n a row (or a column) of the Latn square gven by (7) are dfferent, all the CPMs n a row (or a column) of are dstnct For any par of ntegers wth ' W'-, let be a 4 subarray of s a ` matrx over GF(2) whch also satsfes the RC-constrant he null space of gves a bnary C-LDPC code e of length wth rate at least }, whose anner graph has a grth of at least 6 For a gven fnte feld GF( ), the above constructon gves a famly of bnary C-LDPC codes on Latn squares In the followng, we use two examples to llustrate the constructon of C-LDPC codes gven above o compute the error performances of the constructed codes over the bnarynput AWGN channel n these two examples, we assume BPK sgnalng use the PA (or mn-sum algorthm (MA)) for decodng he maxmum number of decodng teratons s set to 50 he codes gven n these two examples are constructed specfcally to show that they can perform Bt/bloc error rate C(992,750), BER, 50 teratons PA C(992,750), BLER, 50 teratons PA C(992,750), BER, 10 teratons PA C(992,750), BLER, 10 teratons PA C(992,750), BER, 5 teratons PA C(992,750), BLER, 5 teratons PA C(992,750), BER, 50 teratons FPGA Mn um C(992,750), BLER, 50 teratons FPGA Mn um phere Pacng Bound hannon Lmt E b /N 0 (db) Fg 1(a) he error performance of the ("#"#$&%'( ) C-LDPC code gven n Example 1 over the AWGN channel down to a very low error rate wthout error-floor Low errorfloor s a specfc feature of algebrac LDPC codes Example 1: Let GF(" ) be the feld for code constructon Let be a prmtve element of GF(" ) Frst, we construct a "$ " Latn square over GF(" ) of the form gven by (8) Dspersng each nonzero entry of nto a CPM each 0-entry on the man dagonal of nto a / zero matrx, we obtan a "/ " array of CPMs ZMs of sze 3 s a +,+B"$-+,+B" matrx over GF(2) wth both column row weghts 31 he null space of gves a ( )-regular (++ " /10@ ) C-LDPC code of rate 2/10,3 he error performances of ths code decoded wth 5, teratons of the PA are shown n Fgure 1(a) We see that the decodng of ths code converges very fast At the BLER (bloc error rate) of Œ 54, the code decoded wth 50 teratons of PA performs only % +60 db from the sphere pacng bound Also ncluded n Fgure 1(a) s the error performance of the code computed by an FPGA mn-sum decoder We see that the code performs down to the BER of Œ% wthout error-floor In Fgure 1(b) we show the unresolved erasure bt rate (EBR) the unresolved erasure bloc rate (EBLR) of the code wth teratve decodng over the bnary-erasure channel (BEC) As the rate of the code s 7 %8/,013, the hannon lmt for the BEC s 97ˆE ";:,: bts per channel usage From Fgure 1(b), we see that at the EBR of Œ%54, the code performs &< from the hannon lmt For such a short code, t performs very well over both the AWGN channel the BEC =>= Example 2: In ths example, we construct a long hghrate code he feld for code constructon s the prme feld GF(?< ) Based on ths feld, we can construct a?< / &< Latn square over GF(?< ) of the form gven by (8) Dspersng ths Latn square, we obtan a?<?< array of CPMs ZMs of sze &<@( &<@ ae a 3/@+@ subarray 3 A+ from, avodng the ZMs on the man dagonal

5 R nresolved erasure bt rate C(992,750), EBR C(992,750), EBLR hannon Lmt Channel erasure probablty p Fg 1(b) he error performance of the ("#"#$&% '( ) C-LDPC code gven n Example 1 over the BEC Bt/bloc error rate C(16200,15125), BER, 50 teratons PA C(16200,15125), BLER, 50 teratons PA C (16200,15125),BER, 50 teratons M C (16200,15125), BLER, 50 teratons M PEG(16200,15125), BER, 50 teratons PA PEG(16200,15125), BLER, 50 teratons PA phere Pacng Bound hannon Lmt E b /N 0 (db) Fg 2 n Example 2 over the AWGN channel he error performance of the ( $ # % ( $#( ) C-LDPC code gven 3 A+ s a,< $?3B" matrx over GF(2) of hen wth column row weghts 6 90, respectvely he null space of ths matrx gves a (3 A+ )-regular 0Œ"0 ) C- LDPC code of rate % +, 3 he error performances of ths code decoded usng the PA an FPGA mn-sum decoder wth 50 teratons are shown n Fgure 2 We see that the performance curves computed wth the PA the FPGA mn-sum decoder overlap wth each other he code performs down to a BER of almost e wthout error-floor At the BER of ŒaH!, t performs db from the hannon lmt Also ncluded n Fgure 2 s the error performance of a pseudorom 0&0 " ) C-LDPC code constructed wth the PEG-algorthm 19] usng lftng wth crculant permutaton (equvalent to a protograph-based code) We see that the performance curves of the algebrac the pseudo-rom codes overlap wth each other down to the BER of I =>= RANK ANALI OF C-LDPC CODE ON LAIN ARE In ths secton, we analyze the rans of the party-chec matrces of the C-LDPC codes that are constructed based on the Latn squares of the form gven by (7) for the specal case wthw5"@ (e, Latn squares over GF(" )) wth w" nce the characterstc of GF(" ) s 2, the subtracton n (7) can be replaced by modulo-2 addton For smplcty, we use the Latn square gven by (8) for analyss hs results n no loss of generalty Consder the array of CPMs ZMs over GF(2) gven by (9) obtaned by array dsperson of the Latn square over GF(" ) gven by (8) he CPMs ZMs n are of sze " w 4 " For ' o' "@, let be a l subarray taen from the upper left corner of, e the frst rows the frst columns of ang a subarray from ths way s ust for the smplcty of notaton expressons wth no loss of generalty Let be the submatrx taen from the upper left corner of the Latn # square, e, the frst rows the frst columns of It s clear that the subarray B of s the array dsperson of can be expressed as follows: X F! ^^^ H It follows from heorem 1 that the ran of by s H BO N s gven (10) herefore, to determne the ran of, we need to determne the ran of the Hadamard product of to the h th power for ' h + "@ Note that, ˆ he followng theorem gves an Its lengthy proof s omtted expresson for BO N In the theorem, we use to denote the number of odd ntegers n the h th row of the Pascal s trangle 20] heorem 3: For n' h#+ "@, the ran of upper bounded as follows: For ml"@, we have ~ BO N ' " he followng result on the ran of (10) heorem 3 s follows from heorem 4: For 1' w' "@, the ran of the n subarray of the array of CPMs ZMs obtaned by the array dsperson of the RD-constraned Latn square gven by (8) s upper bounded as follows: BO N 9' y

6 " Bt/bloc error rate C(4032,3304), BER, 50 teratons PA C(4032,3304), BLER, 50 teratons PA C(4032,3304), BER, 10 teratons PA C(4032,3304), BLER, 10 teratons PA C(4032,3304), BER, 5 teratons PA C(4032,3304), BLER, 5 teratons PA phere Pacng Bound hannon Lmt nresolved erasure bt/bloc rate C(4032, 3304), EBR C(4032, 3304), EBLR hannon Lmt E b /N 0 (db) Channel erasure probablty p Fg 3(a) he error performance of the ( $&% # ) C-LDPC code gven n Example 3 over the AWGN channel Fg 3(b) he error performance of the ( $&% # ) C-LDPC code gven n Example 3 over the BEC For G ", H ~ BO N s In case ml", a combnatoral expresson for the ran of the 2 " subarray "@ of the array gven by (9) can be derved as stated below he techncal proof of ths result s omtted heorem 5: For (0"@, (' ' +n" satsfyng " 6", In case #L"@, we have hen " " 6", '1"@, let be the $o" Example 3: Let GF(",4 ) be the code constructon feld Based on ths feld, we construct a 31:7 3,: Latn square of the form gven by (8) Dspersng, we obtan a 31:F 3,: array of CPMs ZMs of sze of 3 3 It s a : "(: matrx over GF(2) wth both column row weghts 3 It follows from heorem 5 that the ran of s 4M 4 / ",< he null space of gves a (3 A3 )-regular (: ", 1: ) C- LDPC code of rate % <?+1: he error performances of ths code decoded usng the PA wth 5, teratons are shown n Fgure 3(a) We see that the PA decodng of ths code converges very fast At the BLER of I54, the code performs db from the sphere pacng bound Its performance over the BEC s shown n Fgure 3(b) =>= I CONCLION In ths paper, we studed the ran of party-chec matrces of C-LDPC codes constructed based on the array dsperson of RD-constraned matrces over fnte felds hen, we presented a class of RD-constraned Latn squares over fnte felds Based on these Latn squares, we constructed a large class of C-LDPC codes, called C-LDPC codes on Latn squares We presented combnatoral expressons for the rans of the party chec matrces of C-LDPC codes on Latn squares ACKNOWLEDGMEN hs research was supported by NAA under the Grant NNX09AI21G, NF under the Grant CCF , gft grants from Intel Northrop Grumman pace echnology REFERENCE 1] RG Gallager, Low densty party chec codes, IRE rans Inform heory, I-8, pp 21 28, Jan ] D J C MacKay R M Neal, Near hannon lmt performance of low densty party-chec codes, Electro Lett, vol 32, no 18, pp , Aug ] D J C MacKay, Good error-correctng codes based on very sparse matrces, IEEE rans Inform heory, vol 45, no 2, pp , Mar ] R M anner, A recursve approach to low complexty codes, IEEE rans Inform heory, vol 27, no 5, pp , ept ] Kou, Ln M P C Fossorer, Low densty party chec codes based on fnte geometres: a redscovery new results, IEEE rans Inform heory, vol 47, no 7, pp , Nov ] Ln D J Costello, Jr, Error Control Codng: Fundamentals Applcatons, 2nd edton Prentce Hall, pper addle Rver, NJ, ] W E Ryan Ln, Channel Codes: Classcal Modern New or, N: Cambrdge nversty Press, ] L Lan, L Zeng, a, L Chen, Ln, K Abdel-Ghaffar, Constructon of quas-cyclc LDPC codes for AWGN bnary erasure channels: a fnte feld approach, IEEE rans Inform heory, vol 53, no 7, pp , Jul ] ong, B Zhou, Ln, K Abdel-Ghaffar, A unfed approach to the constructon of bnary nonbnary uas-cyclc LDPC codes based on fnte felds, IEEE rans Commun, vol 57, no1, pp 84 93, Jan ] Z L, L Chen, L Zeng, Ln W H Fong, Effcent encodng of quas-cyclc low-densty party-chec codes, IEEE rans Commun, vol 54, no1, pp 71 81, Jul ] N Kamya E asa, Effcent encodng of C-LDPC codes related to cyclc MD codes, IEEE J elec Areas Commun, vol 27, no 6, pp , Aug 2009

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