A method of constructing the half-rate QC-LDPC codes with linear encoder, maximum column weight three and inevitable girth 26

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1 Communications 20; 2(): 22-4 Publishe online January (htt:// oi: /j.com ISSN: (Print); ISSN: (Online) A metho of constructing the half-rate QC-LDPC coes with linear encoer maximum column weight three an inevitable girth Li Peng Wuhan National Laboratory for Otoelectronics School of Electronic Information an Communications in Huazhong University of Science an Technology Wuhan China aress: engli@hust.eu.cn To cite this article: Li Peng. A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth. Communications. Vol. 2 No oi: /j.com Abstract: This aer resents a metho of constructing the half-rate irregular quasi-cyclic low-ensity arity-check coes which can rovie linear encoing algorithm an their H-matrices may contain almost the least 1 elements comaring with H-matrices of all existing LDPC coes. This metho shows that three kins of secial structural matrices resectively name as S-matrix M-matrix an A-matrix are efine an constructe. With regar to the arbitrary large structural girth base on A-matrix its general attern is conceive an its basic rule is rove. A general metho of constructing M-matrix with the inevitable girth larger than 24 is introuce by using generalize block esign an treating A-matrix as its sub-matrix. S-matrix is generate by substituting secially circular-shift values for non-zero elements in M-matrix. Combining H -matrix generate from lifting the S-matrix an H -matrix with the aroximate lower triangular array structure forms the H-matrix i.e. H=[H H ] which efines a class of half-rate irregular QC-LDPC coes with maximum column weight an inevitable girth of length. Simulation tests show that the erformance of the resente QC-LDPC coe can achieve the signal-noise-ratio of below 2B at the bit-error-rate of 10-5 which is comarable with the erformance of the ractical QC-LDPC coes in inustrial Stanar but the comlication of the former owing to the least 1 elements in H-matrix is lower than that of the later as well as the storage requirement is smaller. Keywors: Quasi-Cycle Low-Density Parity-Check (QC-LDPC) Coe Sarse Parity-Check Matrix Girth Generalize Block Design 1. Introuction The irregular quasi-cyclic low-ensity arity-check (QC- LDPC) coes with linear encoable structure here calle the ractical coes are aote by several IEEE inustrial Stanars [1-2] because of their erfect erformance inherently arallelizable ecoing algorithm an linear encoing algorithm which are well suite for harware imlementation. Since the methos of constructing these ractical coes have not yet been ublishe u to now besies it is necessary to robe into whether there are better ractical coes (i.e. lower comlexity an/or better erformance as well as algebraic structural coes) the metho of esigning the structural QC-LDPC coes has been a research hotsot in the fiel of error correcting coes in recent years [ ]. The LDPC coe efine by artitione arity-check matrix H-matrix for short first aeare in aenix C of Gallager s Ph.D issertation in []. At the beginning a secial array structure name as the array coes in [4] was stuie. Later then the general array structure name as the quasi-cycle (QC)-LDPC coe ha receive enormous attention because each sub-matrix in H-matrix can be generate by simly circular-shifting an ientity matrix in [5-6] an the H-matrix can be imlemente by shift register in [7]. An imortant arameter in the rocess of stuying the LDPC coes is strongly consiere an it is girth. The erformance of a LDPC coe with iterative ecoing strongly eens on the girth which is efine as the length of the shortest cycle in a Tanner grah associate with the H-matrix. In [ ] the girth structure of full-element QC-LDPC coes is stuie an a significant conclusion is that the girth of length calle the evitable girth or the free girth can be eliminate by esigning

2 2 Li Peng: A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth circular-shift value an one kin of girth of length larger than at least can not be eliminate an this girth is calle the inevitable girth. Here the term inevitable girth introuce in [9] means that this girth can not be eliminate by esigning the circular-shift value or the inevitable girth is ineenent of the size of sub-matrix an the circular-shift value. For the convenience of elaboration girth or cycle in an H-matrix is ivie into two classes: the free girth (cycle) an the inevitable girth (cycle). If there are four 1 elements at four crossing ositions of any two rows an any two columns within H-matrix of efining a QC-LDPC coe like the tye 1 1 the aer [7] calle this case unsatisfying row-column 1 1 (RC) constraint then the Tanner grah associate with this H-matrix contains the free girth of length 4. If an H-matrix 1 1 oes not contain such submatrix as the tye then the 1 1 H-matrix is thought of as satisfying RC-constraint an there is no any free girth of length 4 in its Tanner grah. Many aers escribes that if there is no free girth of length 4 in Tanner grah of an H-matrix then the LDPC coes efine by this H-matrix erforms erfect. But this aer effectively observes that it is not enough just to eliminate the free 4-girth in Tanner grah of H-matrix esecially for those cases that the maximum column weigh of an H-matrix is or the maximum egree of its Tanner grah is only eliminating the free girth of length larger than 4 such as an so on an making the inevitable girth as larger as ossible can make the LDPC coe have goo erformance. In orer to eliminate the inevitable girth of length the moel matrix M-matrix for short corresoning to the H-matrix of QC-LDPC coes must be sarse matrix instea of full-one matrix or the H-matrix must consist of artial ermutation submatrices an artial full-zero submatrices or the shift matrix S-matrix for short is an sarse matrix rather than a full-element matrix. Here the efinitions involving M-matrix an S-matrix are available in section 2.2. From the alication oint of view the less the H-matrix contains 1 elements the lower the comlications of encoers an ecoers for the QC-LDPC coes are. Therefore the investigation of H-matrix with small column weight has been ai attention to [5-6 -1]. The aer [] stuie the regular QC-LDPC coes with column weight three. The aer [1] iscusse the irregular QC-LDPC coes with column weight three an two which gives erfect erformance. But there is less reort about the irregular QC-LDPC coes with column weight three an two uner the constraint of framework of the linear encoable structure. In this aer the author will ay attention to the algebraic metho of constructing the ractical irregular QC-LDPC coes with maximum column weight as well as their girth structures uner the constraint of framework available linear encoing algorithm. The aer [9] investigate the rotograh coe an built u the concet of the inevitable girth with length 2i i = for all subgrah atterns which are use to construct rotograh coes. In this aer the matrix corresoning to this subgrah attern which can form an inevitable girth is calle the atom matrix A-matrix for short. The girth of an A-matrix is exane from i = of the rotograh P 2i to i > 10 of A 2i. The general construction of A-matrix is efine an the general rule of its structural girth is rove. The author uses two A-matrices an the metho of generalize block esign to construct an M-matrix which is calle as the moel matrix or also the base matrix of H-matrix in some aers an guarantees that there is no 4-cycle in H-matrix with aroximate lower triangular array. In fact the esign of H-matrix is simly transforme into the esign of M-matrix an S-matrix an the esign of the M-matrix is ivie further on into the esign of A-matrix. The contributions of this aer are summarize as follows: 1) A-matrix with arbitrary large inevitable girth is efine an create an its general structural features are iscusse an rove. The inevitable girth in a (01)-matrix can be generate by means of algebraic metho rather than comuter searching metho an the size of inevitable girth is any ositive integer i > 6 rather than only restricte to i = [9]. 2) An half-rate irregular H-matrix with linear encoing algorithm column weight three an two an without 4-cycle is exactly constructe for the first time an its tanner exactly contains two inevitable girths of length or maybe even large. ) It is eclare for the first time that H-matrix without 4-cycle erforms baly over the aitive white Gaussian noise (AWGN) channel an with belief roagation (BP) iterative ecoing algorithm an binary hase-shift keying (BPSK) moulation an the erformance of H-matrix uner the same environment can be imrove only if the small free cycles such as an so on are eliminate at the greatest extent or comletely an the inevitable girth is as large as ossible in Tanner grah of this H-matrix. 4) A simulation result with ractical alication is first resente this it is that the grou of ifferent coe-length an half-rate irregular QC-LDPC coes efine by H-matrix with linear encoer base on aroximate lower triangular array matrix an maximum column weight three erform within 1.9B from the Shannon limit at the BER of 10-5 in aition they have the lowest comlication because of the least 1 elements in H-matrix an occuy the least memory because of maximum column weight three comaring with the existing ractical half-rate irregular QC-LDPC coes in multile inustrial stanars. The outline of the aer is as follows. Section 2 escribes the basic knowlege which will be use in this aer. Section investigates the general rule of arbitrary large girth in A-matrix. Section 4 resents a metho of constructing M-matrix by means of two A-matrices. Section 5 iscusses some consieration of S-matrix an Section 6 gives the results of the igital simulation for the resente QC-LDPC coes. Finally Section 7 conclues the aer.

3 Communications 20; 2(): Preliminaries 2.1. A Practical Framework of Irregular QC-LDPC Coes A ractical framework of the sarse arity check matrix H-matrix for short which is use to efine the irregular QC-LDPC coes is the following q ( q + t) array: Ia I 00 a I 01 a I 02 a I 0 j a I I t Ia I 10 a I 11 j a I a I 1 j a 0 I 1 t 1 0 I Ia I 20 a I 21 a I 22 a I 2 j a t 1 H = [ H H ] = I0 0 0 I0 I0 0 0 (1) Ia I i0 a I i1 a I i2 a I i j a i t 1 0 I0 I 0 Ia I q 10 a I q 11 a I q a I q 1 j a I q 1 t I0 where the H -matrix with the size M M = qn qn is an aroximate lower triangular array matrix which can rovie linear encoing algorithm an the H -matrix with the size M K = qn tn is an q t array; n is the size of sub-matrix I a i j an 0 I ; a i j I is either a full-zero matrix or a ermutation matrix an I 0 is an ientity matrix; the subscrit a i j is either the circular-left-shift value (CSV) of a ermutation matrix if Ia i j is a ermutation matrix or a symbol enote as z if Ia i j is a full-zero matrix aij Z { z} 0 i q 1 0 j t 1 { z } enotes that the set {} only contains a symbol z. In aition three matrices at three ositions such as the first q/2th an last ositions within the most-left column of H are simle ientity matrix instea of circular-shift ermutation matrix because CSVs either consume clock erio or occuy memory cells or even both. The meaning of what is calle ractical framework is that the aroximate lower triangular array of H in H-matrix can comlete the linear encoing oeration Definition of Several Secial Matrices Definition 1 [shift matrix]: All subscrit values ai js are extracte from all sub-matrix Ia i j s in H -matrix of (1) an form the following q t sarse integer matrix: a00 a01 a0 j a0 t 1 a10 a11 a1 j a1 t 1 S = S( H ) = ai0 ai1 ai j ai t 1 a q 10 aq 11 aq 1 j a q 1 t 1 then matrix of (2) is calle the shift matrix or subscrit matrix S-matrix for short. Definition 2 [moel matrix]: If one creates a matrix in such way that one makes all ermutation matrices in H -matrix of (1) be substitute by 1 s an all full-zero matrices by 0 s then this new matrix is calle the moel matrix of H -matrix of (1) M-matrix for short. M-matrix is a q t binary matrix. Definition [atom matrix]: For any ositive integer α an β let β = α + 1. If α α an α β two binary matrices (2) satisfy the following structural constraints resectively 1) For a α α matrix there must exist only one row which has maximum weight an each of the other α 1 rows has minimum weight 2; there must exist only one column which has maximum weight an each of the other α 1 columns has minimum weight 2; 2) For a α β matrix there must exist only one row which has maximum weight 4 an each of the other α 1 rows has minimum weight 2; the weight of each column among β columns is 2; then both α α an α β matrices are calle the atom matrix A-matrix for short enote as A α α an A α β. A manatory rovision for an A-matrix is that an A-matrix must contain at least a 0 element or an A-matrix is not a full-1 matrix. For the convenience of escrition the author 1 1 calls the matrix such as A 2 2 = or A 2 = the seuo atom matrix. If an H-matrix is comletely forme by a grou of ermutation matrices then the H-matrix is regular an the corresoning S-matrix is forme by urity integers which is calle full-element S-matrix (FS-matrix). If an H-matrix is forme by artial ermutation matrices an artial full-zero matrices then the H-matrix is either regular or irregular an the corresoning S-matrix is forme by artial integers an artial symbols enote as z in this aer which is calle the sarse S-matrix (SS-matrix). Because H of (1) is fixe the main task is to esign H - matrix in this aer. For this goal this aer is concerne with the structural esign of M-matrix corresoning to the H - matrix an it is require that this M-matrix satisfies the following constraint: 1) M-matrix itself satisfies the RC constraint; 2) There is no two ajacent 1 elements at each column of M-matrix; ) Any two of three ositions such as the first q/2th an last ositions on each of t columns in the M-matrix are not occuie by 1 elements at the same time. The above constraints 2) an ) avoi the 4-cycle between H an H. Therefore the above three constraints guarantee that the H-matrix of (1) oes not contain the free girth of length 4.

4 25 Li Peng: A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth 2.. Necessary an Sufficient Conition of Girth2g In Theorem 2.1 of the aer [6] Fossorier gave the necessary an sufficient conition for the Tanner grah of H-matrix efine in (1) to have a girth 2g g = This conition is also suite to M-matrix an A-matrix. For S-matrix of (2) the author makes the following moification. Theorem 1: If the ifference t ( ) k 1 t q k k = aq k t a + k + 1 qk tk (or q ( ) k 1 q t k k = aq k 1 t a + + k qk tk ) of element values of two ositions on any row (column) in the S-matrix efine by (2) exists then a necessary an sufficient conition for the Tanner grah reresentation of H -matrix forme through lifting the S-matrix to have a girth at least 2( g + 1) is m 1 m 1 t ( ) 0 k t q k + 1 k (or q k qk + 1 k = 0 k = 0 ( tk) 0) (mon) () for all m2 m g all q k an q k qk qk + 1 q 1 all t k an t k tk tk + 1 t 1 with t0 = tm qk q k + 1 an tk t k + 1. Note that because symbol z of S-matrix results in the ifference t ( ) k t q k 1 k = aq k t a + k qk tk + 1 not to exist exression () is only a sufficient conition when S-matrix is sarse. If one raws a horizontal line between any two elements on any one of q rows an a vertical line between any two elements on any one of t columns in S-matrix then the close ath forme by g horizontal lines an g vertical lines emonstrates the structure of a cycle or a girth in S-matrix. Through lifting the S-matrix by using a grou of n n ermutation matrices the corresoning H -matrix contains n such structural cycles or girths. From Theorem 1 the next corollary follows. Corollary 2: For any g rows in S-matrix of (2) if the mon sum of any g ifference values t ( ) k t q k + 1 k s equals zero then the g horizontal lines corresoning to the g ifference values must generate a close ath forme by 2g lines. Conversely if g horizontal lines can form a cycle of length 2g then the mon sum of the g ifference values must equal zero. From Theorem 1 an Corollary 2 the next corollary follows. Corollary : If there is a girth whose length is 2g in S-matrix then there is also a 2g-girth of same structure in corresoning M-matrix. Obviously the metho of rawing horizontal an vertical lines in M-matrix can be use to investigate the girth characteristic of M-matrix. Because the istribution of all 1 s in M-matrix is still comlex so the girth characteristic of A-matrix which can be use to construct M-matrix is consiere firstly. From a structural girth oint of view an A-matrix can be seen as a basic unit of an M-matrix (shown as in Fig. ). Note that interretation of terminology. The same-row horizontal lines mean that there are two or more horizontal lines on the same row which corresons to two or more subscrit ifferences on the same row. The ifferent-row horizontal lines mean that all horizontal lines are locate on ifferent rows. The horizontal lines in an A-matrix have the following corollary. Corollary 4: The following cases are naturally satisfie in an A-matrix: 1) On the row containing only two 1 elements there must exactly be two same-row horizontal lines whose length is same. 2) On the row containing only three 1 elements there must exactly be three same-row horizontal lines in which the sum of lengths of any two horizontal lines must be equal to the length of the remaining one. ) On the row containing only four 1 elements there must exactly be four same-row horizontal lines such that two cases aear: either the sum of length of three horizontal lines equals the length of the remaining one or the sum of length of any two horizontal lines equals the sum of length of the remaining two Structural Characters of Inevitable Girth 2g Researches show that the girth an 10 in a full-element S-matrix (FS-matrix) can be eliminate by esigning element values (circular-shift values) of S-matrix by means of algebraic metho an comuter searching metho. For examle in the aer [5] uner the strict constraint of structure arameters n q t that is that n must be rime an j 1 i 1 all shift values ai j { φ = ϕ : i = q 1 j = t 1} must satisfy the constraint of elements in a multilicative grou which means a set of integers moulo n j 1 i.e. φ < n i 1 ϕ t q < n φ = 1(mo n) ϕ = 1(mo n) an efficient arrangement j 1 i 1 of all elements φ ϕ generates a FS-matrix in which the cycles forme by lines can be eliminate an the girth is at least. Thus a significant conclusion is that the girth in a FS-matrix is inevitable. In other wors the moel matrix corresoning to a FS-matrix is a full-1 matrix an inclues many seuo A-matrices like the tye = A in which there must be inevitable girth. One can still infer that if a 2 sub-matrix in an M-matrix is fille to the full of 1 elements then there must be inevitable girth in this M-matrix. Furthermore if all elements on six crossing ositions of any two non-neighbor rows (columns) an any three non-neighbor columns (rows) in any an M-matrix are fille by six 1 elements then there exists inevitable girth in this M-matrix. If one wants to estroy the structure of the inevitable girth in a seuo A-matrix farther in an M-matrix incluing any number of seuo A-matrices as long as he/she substitutes one 0 element for any one of six 1 elements in the seuo A-matrix then the girth in this seuo A-matrix can be eliminate. The aer [9] gives some A-matrices with girth larger than by means of the searching metho of comuter. These A-matrices are seen as the subgrah atterns of the rotograh coes an inclue girth 2g g = Generally the structures of inevitable girth of length have a general attern as follows:

5 Communications 20; 2(): 22-4 A A = A = A = = The following makes an A-matrix A with inevitable girth as an examle in orer to exlain its structural characteristics. An A has the following structural characteristic: its size is ; the total number of non-zero elements is 7 the total number of zero elements is 2; the istribution of column weight is that any a column has three 1 s an the other two columns inclue two 1 s resectively; the istribution of row weight is that any a row has three 1 s an the other two rows inclue two 1 s resectively; in articular A contains at least two free 4-girths (observing all A-matrices in the following set of { A } ). For a A in Fig. 1 (a) shows the case of seven horizontal lines (b) shows the case of the inevitable girth of length (c) shows that a A contains two free girths of length 4 an () shows that a A contains one free cycle of length 6. { A } = Fig. 1. (a) shows 7 horizontal lines (b) shows an inevitable -girth in (c) shows two 4-cycles in A an () shows one 6-cycle in A. A All A-matrices with inevitable girth form a set { A }. Accoring to the efinition of A-matrix the author enumerate all 18 elements in this set as follows: Similarly one can analyze the structural characteristics of the other A-matrices with inevitable girth but it is ifficult for a erson to enumerate all A-matrices in the sets { A } { A } an { A }. One also observes that A contains two free 4-cycles an one free 8-cycle A 4 4 contains one free 4-cycle one free 6-cycle an one free 20 8-cycle an A 4 5 contains one free 4-cycle one free 6-cycle an one free 10-cycle. All these free cycles can easily be estroye by esigning circular-shift values.. Girth Structure of Atom Matrix Without loss of generality the author will euce the general structural characteristic with regar to arbitrarily large girth in an A-matrix with arbitrary size α. Theorem 5 [the structural rule of girth in an A-matrix]: Let α be an arbitrary ositive integer. 1. Let g be an o number an α α be the size of an A-matrix. If g = 2α + 1 then there exactly exists an 4 inevitable girth 2g excet those free girth an free cycles smaller than an equal to 2g in this A-matrix 2g which can be enote as A α α. 2. Let g be an even number α β or β α be the size of an A-matrix an β = α + 1. If g = 2β = 2( α + 1) then there exactly exists an inevitable girth 2g excet those free girth an free cycles smaller than an equal to 2g in this A-matrix or in its transose matrix which can be 2g 2g T 2g enote as A or [( A ) ] resectively. α β α β β α Proof: Accoring to the efinition of A-matrix an Corollary 4 there always exist 2( α 1) + = 2α + 1 2( α 1) + 4 = 2( α + 1) an 2β 1 elements which can generate 2α + 1 2( α + 1) an 2β horizontal lines in three α α α β an β α A-matrices resectively. Due to g = 2α + 1 for an o number g or g = 2β = 2( α + 1) for an even number g then there exist a close ath forme by g horizontal lines an g 2g 2g vertical lines in each of three A-matrices A A an 2g T α β β α [( A ) ] α α α β. Obviously this close ath of lenght 2g is

6 27 Li Peng: A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth ineenent of the subscrit value a i j an the size n of ermutation matrix an is only eenent of the size α of an A-matrix. Therefore this close ath of lenght 2g in each of three A-matrices is an inevitable cycle 2g. In each of three A-matrices all 1 elements take art in constructing this inevitable cycle 2g so there is no suerabunant 1 elements that can form an inevitable cycle larger than 2g. Furthermore if any one of g 1 elements oes not take art in constructing this inevitable cycle 2g then the remnant g 1 1 elements can not generate an inevitable cycle less than or equal to 2g esite the fact that they can form several free cycles an free girth less than or equal to 2g. So this inevitable cycle in each of three A-matrices is the minimum inevitable cycle 2g that is it is an inevitable girth 2g. Accoring to the rule of Theorem 5 one can esign an A-matrix with any large girth. Next the author will gave two examles about how to esign the A-matrix with inevitable girth 2g an g > 10. Examle 1: Let g = which is an even number. Accoring to the rule 2) of Theorem 5 one has α = g/2 1 = 5 an β = g/2 = 6. So he/she gets an 5 6 A-matrix in which the weight of each of six columns is 2; the istribution of row weight is that any row has four 1 elements an each of the other four rows has two 1 elements. This istribution of row weight can generate at least horizontal lines. Therefore one 24 can get an A-matrix A 5 6 with an inevitable girth of length 24 shown as in Fig. 2 (a). A = (a) A = (b) 24 Fig. 2. (a) shows the girth 24 of the A-matrix A 5 6 an (b) shows the girth of the A-matrix A. Examle 2: Let g = 25 which is an o number. Accoring to the rule 1) of Theorem 5 one has α = ( g 1)/2 =. So he/she gets a matrix in which the istribution of row weight is that any row has three 1 elements an each of the other eleven rows has two 1 elements. The istribution of column weight is the same as the istribution of row weight. This istribution of row weight can generate 25 horizontal 50 lines. Therefore one can get an A-matrix A with an inevitable girth of length 50 shown as in Fig. 2 (b) Obviously A-matrices A an A have a lot of atterns which can form the sets of { A 5 6} an { A } resectively. In other wors the roblem of constructing an A-matrix with any large girth base on Theorem 5 is a multi-solution roblem an the A-matrix with eterminate structural characteristic can form a solution set which hints that the istribution of all 1 elements in an A-matrix has a ranom-like characteristic uner the constraint of a fixe framework of A-matrix which is etermine by the arameter α an some eterministic istribution of row weight an column weight. For any ositive integer α a α α A-matrix (or α ( α + 1) A-matrix) must exactly contain an inevitable girth of length 4α + 2 (or 4α + 4) an three free girths of length less than 2α (or 2α + 2). 4. Design M-Matrix by Using A-Matrix This Section will escribe a metho how to construct an M-matrix by selecting several elements in the set forme by A-matrix. Firstly let α = 6 an g = 1 is an o number accoring to the efinition of atom matrix the author constructs an 6 6 A-matrix in which there are thirteen 1 elements an the istribution of row weight is that any row for examle the first row among six rows has three 1 s an each of the other five rows has two 1 s. Accoring to 1) an 2) of Corollary 4 thirteen 1 elements can generate thirteen horizontal lines. Accoring to the rule 1) of Theorem 5 the thirteen horizontal lines form an inevitable girth of length. The resente A-matrix A 6 6 in fact selecte from the set { A 6 6} has the following general attern: A 6 6 = (4) The requirement of selecting this A 6 6 is that there is no any 4-cycle an 6-cycle in it. Observe the istribution of thirteen 1 elements in A 6 6 of (4) an iscover that eight 1 elements with the subscrit 1 or 2 resectively form an 8-cycle an ten 1 elements with the subscrit form a 10-cycle. The A6 6-matrix with an inevitable -girth inee oes not contain 4-cycle an 6-cycle therefore its free girth is

7 Communications 20; 2(): at least 8. It's worth noting that the A-matrix A 6 6 of (4) whose meaningful characteristics are to contain an inevitable -girth two free 8-girths an one free 10-cycle an not to contain any free 4-cycle an free 6-cycle is not the only attern an here is constructe by using observation metho. In articular the author constructe A 6 6 of (4) by using the following constraints: i) there is no such column that contains two ajacent 1 elements in it; ii) it satisfies RC-constraint; iii) there is no any free 6-cycle in it. Seconly the author constructs an 6 m M-matrix by using the above A-matrix A 6 6 where m 6 is any ositive integer. In this 6 m M-matrix there only exists an inevitable -girth two free 8-girths an a free 10-cycle forme by an A-matrix A 6 6 an the new m 6 columns are either the full-0 columns or the full-1 columns. Those new columns of weight 1 are selecte ranomly an the osition of each 1 element is arrange ranomly but the istribution of all 1 elements in these new m 6 columns must guarantee not to generate any new cycle. That is the 6 m M-matrix oes not contain the free 4-cycle the free 6-cycle an the inevitable girth of length less than. Therefore its free girth is also at least 8 an its inevitable girth is also exactly. Let m = then the 6 M-matrix generate the new six columns in which there are a full-0 column an five full-1 columns. For examle this 6 M-matrix may have the following attern: girth Thirly the author constructe an M-matrix by stacking u two 6 M-matrices similarly to (5). This M-matrix is require in such way that it has the same row an column weights all of which are equal to three an oes exactly not contain 4-cycle. In orer to achieve the aim the ositions of ten ineterminate 1 elements in the M-matrix nee to be consiere carefully. As long as those columns of weight 1 in two 6 M-matrices are suitably lace an the same two A-matrices A 6 6 of (4) are roerly arrange at the to left corner an the bottom right corner resectively in the M-matrix then this new create M-matrix can guarantee not to contain 4-cycle. However the istribution of ten ineterminate 1 elements must generate 6-cycle. Here comes into being a research roosition that in an M-matrix with row an column weight whether or not there must be any free 6-cycle. The esign roceure of this M-matrix is shown in Fig.. For q = t = let the M-matrix corresoning to H - matrix in (1) be constructe by M in Fig.. In the sequence the author will give a metho to etermine the (5) ositions of ten ineterminate 1 elements in girth M. ile u the first 6 M-matrix on the to of the secon girth generate an M-matrix with inevitable girth M = Fig.. shows the rocess of constructing an M-matrix without 4-cycle an with two inevitable girths of length an the new cycles forme by ten 1 elements an inicate by the subscrits 1 2 resectively. Because its row an column weights are this M-matrix M contains thirty-six 1 elements in which the ositions of twenty-six 1 elements are given by two inevitable -girths an the ositions of ten 1 elements are not known. The author use tritule to enote row coorinates of three 1 elements on each column in M so one can get the below twelve tritules: (246)(15 a)(6 c)(25 )( e)(1 f) (810)(711 g)(9 h)(811 s)(710 u)(79 w) where ositive integer a c e f g h s u w enote the row (6)

8 29 Li Peng: A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth coorinates of unknown ositions of ten 1 elements in M. Consiering constraints 1) an 2) in sub-section 2.1 one can euce the range of ten unknown numbers a c e f g h s u w : a c e f {891011} a c e f g h s u w {2456} g h s u w. Further consiering constraints 2) an ) in sub-section 2.1 one can get a c e f {} an = g h u w {6} an s = 6. So (6) is change into the following attern by bringing = an s = 6 into (6): (246)(15 a)(6 c)(25)( e)(1 f) (810)(711 g)(9 h)(8116)(710 u)(79 w) From (7) one can get c 811 an h 25 in orer to avoi the 4-girth inm. The remaining issues is to etermine eight unknown numbers a e f {891011} c {910} h {4} an g u w {245}.The author borrows five arameters b v k r λ an some concets of BIBD base on combinatorial esign theory in [10] in orer to solve a c e f g h u w. Let the number of varieties v = q = enote the number of rows of M the number of blocks b = t = the number of columns k = the column weight r = the row weight an λ = 1 means that any air of v = varieties occurs exactly once in b = blocks which is equivalent that M oes not contain any free 4-cycle. Because v k r λ o not satisfy the constraint r( k 1) = λ( v 1) this ( b v k r λ ) block esign is not a BIBD an the author call it the generalization block esign. Uner the conition of satisfying the constraints 1) 2) an ) in the sub-section 2.1 form there are some limite collections of b = blocks with regar to( v k r λ ) = (1). The author selecte a = 11 c = 10 e = 9 f = 8 an g = 4 h = u = 2 w = 5 from a collection of enumerating all blocks which satisfy the above series of constraints an forme the following exact twelve tritules : (2 4 6)(1 511)( 610)(25)(1 4 9)(1 8) (810)(4 711)( 9)(6811)(2 710)(5 7 9) Fig. shows the structure of M -matrix in which many new cycles of length 6 8 an 10 are generate. For examle six 1 elements with the subscrit 1 show a free 6-cycle eight 1 elements with the subscrit 2 show a free 8-cycle an ten 1 elements with the subscrit shows a free 10-cycle. Although it is ifficult for one to etermine the number an the structure of all free 6 8 an 10-cycles in M one can conclue that there is no the free girth of length 4 in M an its girth is at least 6. Many aer reorte that if an H-matrix oes not contain any free girth of length 4 then the LDPC coe efine by it can rovie erfect erformance. But this is not always the case esecially when the maximum column weight is. If one regars M in Fig. as the M-matrix corresoning to H of (1) uses thirty-six n n ientity matrices to lift all 1 elements in M an twenty-five n n ientity matrices to lift all I 0-submatrices in H of (1) then one can create an H-matrix without the free girth of length 4. Nevertheless the (7) (8) QC-LDPC coe efine by this H-matrix erforms baly which can be seen from the simulation testing curves in Fig. 5 of Section 5 where there are three ot lines resectively with coe length n = which show the ba erformance in the signal-noise-rate (SNB) of below 7 B at the bit-error-rate (BER) of In consieration of the above simulation results researchers nee to use a grou of circular-shift ermutation matrix rather than ientity matrix to lift M-matrix corresoning to H of (1) in orer to eliminate the small free cycles such as free 6 8 an 10 cycles as more as ossible or comletely. Therefore the next section will iscuss how to esign the circular shift values in S-matrix. 5. Designing S-Matrix an Digital Simulation Once M-matrix is etermine by twelve tritules of (8) then the corresoning S-matrix ossesses the fixe structure an the remaining roblem is to etermine the value of each element in S-matrix. In fact esigning S-matrix is to fin out the CSVs which are lace the ositions etermine by twelve tritules of (8). There are many aers introucing the methos of esigning full-element shift (FS)-matrix for examle multilication grou [4-5 11] an aition grou [11]. But there are few aers investigating how to construct the sarse shift (SS) matrix. A viable metho is first to esign a FS-matrix without free 4-cycle by using the algebraic metho an then to use the reesign M-matrix without free 4-cycle to mask this FS-matrix so as to obtain a SS-matrix which is the esire result. In this aer the author choose a grou of the existing CSVs from the base moel matrix of the half-rate QC-LDPC coes in IEEE e Stanar to create a SS-matrix. This treatment metho is base on two thoughts. On the one han the half-rate irregular QC-LDPC coes with maximum column weight three in the framework of (1) has not been reorte by far to be able to rovie the goo erformance for examle below 2B at the BER of 10-5 ; on the other han the main goal of this aer is to emonstrate by means of the simulation tests whether or not there exists such the H-matrix with maximum column weight an the inevitable girth uner the constraint of the framework of (1) that it can efine the half-rate irregular QC-LDPC coes with the above erfect erformance. Therefore the author ha extracte the CSVs from the half-rate QC-LDPC coe of Stanar [1] an forme the following twelve tritules of circular-shift values accoring to the osition coorinates rovie by twelve tritules of (8): (614)(942711)(47957)(24565)(22468)(8466) (614)(99427)(554795)(25245)(722246)(84) Accoring to (1) (8) an (9) one can get an H-matrix of (10) at the to of the next age. All 6 shift values are etermine in such way that seven among the twelve triules of (9) is in 1-1 corresonence with the shift values of seven columns with column weight in the base moel matrix which corresons to the half-rate (9)

9 Communications 20; 2(): H -matrix in IEEE e an one of the other five triules is forme by selecting arbitrarily an resectively three values from each of five columns with column weight 6. Note that in H-matrix of (10) the shift values of the thirteen ositions in one inevitable girth of length are the same as those in another resectively. H = [ H H ] = I I22 56I I I I I I I I I I I I I 2 4I I I I I22 2 6I I66 I I I I I I I I I I I I I I I0 I I0 I I0 I I0 I I0 I I I0 I I0 I I0 I I 0 I I 0 I I0 I0 I I0 (10) As each column of H -matrix array of (10) contains only three circular shift ermutation matrixes so that the simlifie reresentation of H -matrix can be forme as follows. One can efine a number air with subscrit: ( µ η ) τ which is use to enote the osition of each shift value in SS-matrix corresoning to H -matrix where the subscrit value τ enotes the column inex of SS-matrix corresoning to H -matrix of (10) µ enotes the row inex of the shift values on the τth column an η enotes the shift values on the τth column an the µ th row in the SS-matrix. Then this SS-matrix can be reresente as the following 6 number airs: (261) (4) (64) (194) (527) (1111) (47) (695) (107) (224) (55) (65) (2) (446) (98) (181) (24) (866) (861) 7(10) 7(4) 7(4) 8(794) 8(17) 8(55) 9 (947) 9 (95) 9 (625) (824) (115) (272) (722) (1046) (52) (781) (924) The above simlifie reresentation of H -matrix rovies a kin of storage structure for H-matrix in memorizer. Remark 1: Uner the framework of (1) with maximum column weight an the inevitable girth at least the H-matrix of (10) is not the only one. If one reselects the A-matrix similar to (4) or reesigns the twelve tritules like (8) which has many selection schemes or uses the otimizing searching metho or the algebraic metho to construct the SS-matrix then he/she can obtain an H-matrix ifferent from (10) uner the same framework. The H-matrix obtaine by the above methos can still guarantee a grou of ifferent-length half-rate irregular QC-LDPC coes to erform below 2B at the BER of Remark 2: The metho of constructing the M-matrix introuces the ranomness from two asects. On the one han the set { A } 1 contains 6!/ 1! selectable 6 6 solutions from which the author chose a 6 6 A of (4). Although the A 6 6 of (4) is esigne by consiering the following constrains: first no 4-cycle an then satisfy three conitions in Section 2.2 through integrate into account the arrangement of two A matrices in M-matrix the asect of 6 6 A 6 6 similar to (4) still has thousans of ways. So the A 6 6 of (4) is a ranom-like atom matrix. On the other han in the rocess of constructing M -matrix the uncertain istribution of ten 1 elements introuces the ranomness. Therefore the metho of constructing M-matrix is a ranom-like metho. In this aer the SS-matrix is constructe by using ranom metho but in fact one can basically use the algebraic metho to construct the SS-matrix. For examle firstly the FS-matrix is constructe by means of the metho of aition grou an multilication grou in [11]; seconly the SS-matrix can be forme by alying M in Fig. to mask the above FS-matrix. So this metho of constructing SS-matrix is not comletely algebraic metho because of the ranom-like roerty of M. 6. Simulation Results For the 1/2 rate irregular QC-LDPC coes efine by H-matrix of (10) the author teste the erformance of the lowest oint of the characteristic curve in the coorinate system forme by the signal-noise-ratio (SNR) an the bit error rate (BER) for the ifferent coe lengths from N = 72 to N = 000 by the ergoic ositive integer of the circular shift ermutation submatrix size n from n = to n = 500 in orer to obtain a set of coes in which all coes of ifferent length erform below 2 B at the BER of If one moifies (8) an/or (9) uner the same framework then the coe length containe in this set is change. In the other wor the tritules ifferent from (8) an/or (9) can generate the ifferent coe set of various coe lengths.

10 1 Li Peng: A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth Table 1 gives one set of the half-rate irregular QC-LDPC coes with various coe lengths efine by H-matrix of (10) in which each coe length is less than 00 an the SNR of each coe is between 2B an B when the BER is Table 2 gives another coe set in which each coe length is larger than 00 an the SNR of each coe is below 2B when the BER is The first column of two Tables gives the inex of the coe number in the coe set. In the following the 1/2 coes of several length is selecte from the coe sets in Table 1 an 2 their error erformances over the AWGN channel with belief roagation (BP) iterative ecoing algorithm an binary hase-shift keying (BPSK) moulation are comute. The maximum number of ecoing iterations is 50. Fig. 4 shows the erformance comarisons between the half-rate short-length irregular QC-LDPC coes liste in Table 1 an those aote by IEEE e an IEEE n Stanar. Three coes selecte from Table 1 have coe length N = whose corresoning sizes of the circular shift ermutation matrices are n = resectively. Three coes selecte from IEEE e Stanar have coe lengths N = an their submatrix sizes n = which are aroximate or equal to the sizes of the resente coes resectively. IEEE n Stanar only aots a short coe of length N = 648 an its submatrix has the size n = 27. number Table 1. half-rate short-length coes below B. Coe length (N) Information length (K) Size of submatrix (n) From Fig. 4 it is seen that the selecte coes from Table 1 (three soli lines) outerform the Stanar coes (three ot lines for IEEE e an one ash line for IEEE n). In aition Fig. 4 also reveals the Stanar coes have the error floor (see the ot line with the circle corresoning to the coe of length N = 576 in IEEE e Stanar an the ash line with the nabla symbol corresoning to the coe of length N = 648 in IEEE n Stanar) an the us an owns change of BER within the small-range of SNR (see the ot line with triangular symbol corresoning to the coe of N = 768 in IEEE e Stanar). In aition the maximum column weight of the half-rate QC-LDPC coes is 6 in IEEE e Stanar an in IEEE n Stanar but the half-rate QC-LDPC coes efine by (10) has only the maximum column weight. The simulation tests in Fig. 4 inicates that for the case of short-length coes the half-rate irregular QC-LDPC coes efine by the H-matrix similar to (10) erform not only in slightly better erformance an but also in lower comlexity than those aote by the Stanars. Table 2. half-rate mile-length coes below 2B. Number Information Size of sub Coe length(n) length (K) matrix(n) Fig. 5 shows the erformance comarisons of the coes efine by (1) in which array H -matrix is constructe between by lifting the circular shift ermutation matrices for M in Fig. like (10) an by lifting the ientity matrix without circular shift values for the same M. Three soli lines within 2B at the BER of 10-5 which show the erformance of the half-rate irregular QC-LDPC coes with circular shift values outerform three ot lines near 7B at the BER of 10-5 which show the erformance of those neither circular shift values nor 4-girth in about 5B or more for the coe length N = corresoning to the submatrix sizes n = resectively. The reason of istinctive erformance between two grous of coes with the same M-matrix M but the ifferent lifting cases can be analyze as follows. Accoring to the necessary an sufficient conition of the existence of girth of S-matrix in Theorem 1 one can analyze the case of the cycles in H-matrix of (10). It is necessary for one to list the exressions of the algebraic sum of the circular shift values of those cycles emhasize by left suerscrits an left subscrits in H-matrix of (10). There are three tyes of cycles in H-matrix of (10). Tye one: Two 8-cycles an one 10-cycle forme by the subscrit 1 2 an in (4) may corresoningly be exhibite by the left suerscrit 1 2 an in H of (10) as the ossible formation of four 8-cycles an two 10-cycles an the algebraic sum of the circular shift values of these otential free (for simlification these two wors are omitte in the following) cycles are calculate as follows. For 8-cycle with left suerscrit 1 one can get: = 169. For 8-cycle with left suerscrit 2 one can get:

11 Communications 20; 2(): = 64 For 10-cycle with left suerscrit one can get: = 105 N=504 n=21 CSV N=720 n=0 CSV N=1056 n=44 CSV N=576 n=24 IEEE802.16e N=648 n=27 IEEE802.11n N=768 n=2 IEE802.16e N=1056 n=44 IEEE802.16e 10 - B E R E b / N 0 ( B ) Fig. 4. Performance comarison of the QC-LDPC coes for short coe length between in this aer an in IEEE stanar Shanno Limit N=804 n=46 CSV N=4704 n=196 CSV N=720 n=155 CSV N=804 n=46 no CSV N=4704 n=196 no CSV N=720 n=155 no CSV no coe BPSK B E R E b / N 0 ( B ) Fig. 5. Performance comarison of the 1/2 rate irregular QC-LDPC coes with length n = for M with CSV an without CSV. Tye two: One 6-cycle one 8-cycle an one 10-cycle inicate by the subscrit 1 2 an within M -matrix in Fig. unoubtely are exane into n 6-cycles n 8-cycles an n 10-cycles if M is lifte by ientity matrix. If M is lifte by circular shift ermutation matrix such as H -matrix of (10) then the three cycles within M -matrix in Fig. are exhibite by left subscrit 4 5 an 6 in H of (10) an the algebraic sum of the circular shift values of these otential free cycles are calculate as follows: = 80 for 6-cycle; = 8 for 8-cycle; = 7 for 10-cycle. Tye three: The first the sixth the seventh the tenth an the twelfth columns in H -matrix of (10) combining with the biiagonal matrix of H -matrix of (10) can generate the following eight free 6-cycles:

12 Li Peng: A Metho of Constructing the Half-Rate QC-LDPC Coes with Linear Encoer Maximum Column Weight Three an Inevitable Girth = 49 this 6-cycle aears two times; = 1 this 6-cycle aears two times; = 1 this 6-cycle aears one time; = 69 this 6-cycle aears one time; = 57 this 6-cycle aears two times. In aition the maximum free cycles forme by combining the secon an the forth columns of H with those corresoning columns of H are two otential free 22-cycles an their algebraic sums of circular shift values are = 8 an = 41 resectively. The other otential free cycles between H an H such as otential free cycles can also be observe an analyze. Although it is ifficult to enumerate all otential free cycles within H-matrix of (10) the author thought that the above three tyes shoul be able to exlain some roblems. One can observe from the above analyses with regar to the algebraic sums of circular shift values that the mon sun of each algebraic sum for each submatrix size n in Table 1 an 2 is not equal to zero in most situations that is to say all otential free cycles of the above three tyes have been elete to a large extent which results in that the coes base on circular-shifting ermutation matrix is suerior to the coes base on the ientity matrix in site of no 4-cycle. From the above analyses about the mon sun of circular shift values for otential free cycles in H-matrix the author can auntlessly conjecture without roof that there must exist such roosition that the girth in H-matrix of (1) is just the inevitable girth of length 2g each of those otential free cycles of length less than 2g is likely to be elete by the alicable circular shift values whose mon sun for each n for examle in Table 1 an 2 is not equal to zero. 7. Conclusion an Future Work Uner the framework of the QC-LDPC coes with linear encoable structure the maximum column weight three an the inevitable girth of length at least 2g ( g > 10) the esign of a sarse arity check matrix can be ecomose into the esign of the atom matrix the meal matrix an the sarse shift matrix. The atom matrix will be investigate with the ai of the mathematic tool of grah theory. The meal matrix is constructe by means of the block esign base on combinatorial mathematics. The esign of shift matrix nees to use the multilicative grou an the aition grou over the finite fiel as well as block esign base on combinatorial mathematics. The inevitable girth of the full-element shift matrix is exactly an that of the sarse shift matrix may be arbitrary size only if the size of the framework is not the limit. Uner the conition of eleting the free girth by means of circular shift values the size of girth for the QC-LDPC coes uner the resente framework eens on the size of the inevitable girth an generally the erformance of the coes can be imrove as the size of the inevitable girth increases. Uner the constraint of the framework resente in this aer the future research work has three oints. First is to fin out the fixe structural A-matrix with the inevitable girth 2g 2g as large as ossible from the sets { A } an { A }. α α α β Secon is to esign M-matrix without 4-cycle an 6-cycle. Thir is to investigate the algebraic metho how to construct sarse shift matrix in which all free cycles less than an inevitable girth can be elete by means of circular shift values. Acknowlegements This work was suorte by the National Natural Science Founation of China uner Grant No References [1] Part 16: Air Interface for Fixe an Mobile Broaban wireless Access Systems IEEE Stanar e [2] 2. Part 11: Wireless LAN Meium Access Control (MAC) an Physical Layer (PHY) secifications IEEE Stanar n [] Gallager R. G. Low-Density Parity-Check Coes. Cambrige MA: MIT Press 196. [4] J. L. Fan Array Coes as Low-Density Parity-Check Coes in Proc. 2n Int. Sym. Turbo Coes an Relate Toics Brest France Set [5] R. M. Tanner D. Srihara A. Sriharan T. E. Fuja an D. J. Costello LDPC Block an Convolutional Coes Base on Circular Matrices IEEE Trans. Inf. Theory vol. 50 no Dec [6] M. P. Fossorier Quasi-Cyclic Low-Density Parity- Check Coes From Circular Permutation Matrices IEEE Trans. Inf. Theory vol. 50 no Aug [7] Zongwang Li Lei Chen Lingqi Zeng Shu Lin an Wai H. Fong Efficient Encoing of Quasi-Cyclic Low- Density Parity-Check Coes IEEE Trans. on Commu. vol. 54 no Jan [8] Olgica Milenkovic Navin Kashya an Davi Leyba Shortene Array Coes of Large Girth IEEE Trans. Inf. Theory vol. 52 no Agu [9] Sunghwan Kim Habong Chung an Dong-Joon Shin Quasi-Cyclic Low-Density Parity-Check Coes With Girth Larger Than IEEE Trans. Inf. Theory vol. 5 no Aug [10] R. A. Bruali Introuctory Combinatorics Thir Eition Prentice Hall [11] Lan Lan L. Q. Zeng Y. Y. Tai L. Chen S. Lin an K. A-Ghaffar Construction of Quasi-Cyclic LDPC Coes for AWGN an Binary Erasure Channels: A Finite Fiel Aroach IEEE Trans. Inf. Theory vol. 5 no July [] M. Gholami M. Samaieh an G. Raeisi Column-Weight Three QC LDPC Coes with Girth 20 IEEE Commun. Lett. vol. 17 no July 201.