Families of LDPC Codes Derived from Nonprimitive BCH Codes and Cyclotomic Cosets

F amilie s of LDPC Codes Der i v e d from Nonprimiti v e BCH Codes and Cyclotomic Cosets Salah A. Aly Departmen t of Com puter Science T exas A&M Univ ersity College Station, TX 77843 , USA Email: salah@cs.tam u.edu Abstract —Low-density parity check (LDPC) codes are an important class of codes with many app lications. T wo algebraic methods f or constructing r egular LDPC codes ar e d eriv ed – one based on nonprimitive narro w-sense BCH codes and th e other directly based on cyclotomic cosets. The constructed codes ha ve high rates and are free of cycles of length four; consequently , th ey can be decoded using standard iterative decoding algorithms. The exact dimension and bounds f or the minimum distance and stopping d istance are derived. Th ese constructed codes can b e used to deriv e qu antum error -correcting codes. Index T erms —LDPC Codes, BCH Codes, Ch annel Coding, Perf ormance and iterative decoding, q uantum BCH codes. I . I N T R O D U C T I O N Bose-Chaudhu ri-Hoch queng hem (BCH) cod es are an in ter- esting class of linear codes that has been in vestigated for nearly half of a cen tury . These types of co des h av e a rich algebraic structure. BC H co des with p arameters [ n, k , d ≥ δ ] q are interesting b ecause one can ch oose their dimension k and minimu m d istance d once given their de sign distance δ and length n over a fin ite field with q elem ents. A linear code defin ed by a gen erator polyn omial g ( x ) has dim ension k = n − deg ( g ( x )) an d rate k /n . It is no t easy to show the dimension of nonprimitive BCH codes ov er higher finite fi elds. In [3], [4], we have given a n e xplicit formula for the dimension of these codes if their deigned distance δ is less than a constant δ max . Low-density parity check (L DPC) codes are a capacity - approa ching class of code s that were first described in a seminal work by Gallager [ 9]. T anner in [20 ] r ediscovered LDPC codes using a grap hical in terpretation . A regular ( ρ, λ ) LDPC co de is measured b y th e weigh ts of its co lumns ρ and rows λ . Iter ativ e deco ding algorithm s of LDPC and turbo codes high lighted the imp ortance of th ese classes of codes for com munication and storage chan nels. Further more, these codes are practical and h av e been u sed in many be neficial applications [ 6], [12 ]. In contrast to BCH and Reed-Solo mon (RS) cyclic codes, LDPC cyclic codes with sparse parity check matrices ar e custom arily constru cted by a comp uter search. I n practice, LDPC co des can achieve highe r perfo rmance and better error cor rection c apabilities than many other codes, because they have efficient iterative decodin g algor ithms, such as the pro duct-sum algorithm [12] –[14 ], [21]. Some BCH codes turned out to be LDPC c yclic codes a s well; for example, a [1 5 , 7] BCH code is also an L DPC cod e with a min imum distance five. Regular a nd irregular LDPC c odes have b een construc ted based o n algebraic and random approaches [1 2], [18], and referenc es therein. Liv a et al. [13] presented a survey of the previous work done on a lgebraic co nstruction s of LDPC cod es based on finite geom etry , elements of finite fields, and RS codes. Y i et al. [22] g av e a con struction for LD PC codes, based on binary narrow-sense primitive BCH cod es, and their method is free o f cycles o f len gth fo ur . Furthermo re, a go od construction of LDPC codes should ha ve a girth of th e T anner graph, of at least 6 [12], [13]. One m ight wonder how do the rates an d minimu m distance of BCH cod es compar e to LDPC codes? Do self-orth ogona l BCH codes g iv e raise to self- orthog onal LDPC codes as well under the con dition δ ≤ δ max . W e show th at how to d erive LDPC cod es fr om no nprimitive BCH codes. One way to measure th e d ecoding p erform ance of linea r codes is by com puting their min imum distan ce d min . Th e perfor mance of lo w-density p arity check cod es u nder itera ti ve decodin g can a lso be gaug ed by measuring th eir stopping sets S and stop ping distance s , which is the size of th e smallest stopping set [16 ], [17]. For any given parity check m atrix H of an LDPC code C , one ca n o btain the T anner graph G of this code an d co mputes th e stopping sets. H ence, s is a p roper ty of H , while d min is a pr operty of C . The m inimum distance is also bo unded by d min ≥ s . BCH codes are decoded invertible matrices suc h as Berkcamp e messay method , LDPC co des ar decoded using iterativ e decoding an d Belief propagatio n (BP) algorithm s. In this paper, we give a series of regular LDPC and Quasi- cyclic (QC)- LDPC code constructions based on non-primitive narrow-sense BCH codes a nd elemen ts of cyclotom ic cosets. The con structions are called T ype-I and T ype-II regular LDPC codes. The algebraic structures of these codes help us to pr edict additional p roperties of th ese cod es. Hence, Th e constructed codes h av e the following characte ristics: i) T w o classes of r egular LDPC cod es are co nstructed that have hig h rates and free of cycles of len gth fou r . Their proper ties can b e ana lyzed easily . ii) The exact dime nsion is compu ted and the minimum distance is bou nded for the c onstructed cod es. Also, the stopping sets an d stoppin g d istance can b e determine d 2 from th e structure of the parity check matrices. The mo tiv ation f or our work is to con struct Algebraic regular LDPC codes that can be used to derive quantum error-correcting co des. A lternatively , they can also be used f or wireless com munication chan nels. Someone will argue abou t the perfo rmance and usefu lness of the constructed regular LDPC codes in comparison to irregular LDPC cod es. Our first mo tiv ation is to der iv e qu antum LDPC co des based o n nonpr imitiv e BCH codes. Hence, the co nstructed LDPC-BCH codes can be u sed to der iv e classes of sym metric quantu m codes [1 ]–[3] , [5 ], [15] an d asym metric quan tum codes [8], [19]. The literature lacks many con structions o f algebraic quantum LDPC codes, s ee for e xample [1], [15] and re ferences therein. I I . C O N S T RU C T I N G L D P C C O D E S Let F q denote a finite field of characteristic p with q elements. Recall tha t the set F ∗ q = F q \ { 0 } of n onzero field elements is a m ultiplicative cyclic gr oup of order q − 1 . A generato r of this cyclic group is called a pr imitiv e element of the finite field F q . A. Definitions Let n be a positiv e integer su ch that gcd( n, q ) = 1 and q ⌊ m/ 2 ⌋ < n ≤ µ = q m − 1 , wher e m = or d n ( q ) is the multipicative order of q modulo n . Let α den ote a fixed p rimitive element o f F q m . Define a map z from F ∗ q m to F µ 2 such that all entries of z ( α i ) are equal to 0 except at position i , where it is e qual to 1. For example, z ( α 2 ) = (0 , 1 , 0 , . . . , 0) . W e call z ( α k ) the lo cation (or character istic) vector o f α k . W e can define th e location vector z ( α i + j + 1 ) as the right cyclic shift o f th e location vector z ( α i + j ) , for 0 ≤ j ≤ µ − 1 , an d th e power is taken module µ . Definition 1 : W e can define a map A that associates to an element F ∗ q m a circulant matrix in F µ × µ 2 by A ( α i ) =      z ( α i ) z ( α i +1 ) . . . z ( α i + µ − 1 )      . (1) By con struction, A ( α k ) co ntains a 1 in ev ery row and c olumn. For in stance, A ( α 1 ) is the identity matrix of size µ × µ , and A ( α 2 ) is th e shift matrix A ( α 2 ) =      0 1 0 . . . 0 0 0 1 . . . 0 . . . . . . . . . . . . . . . 1 0 0 . . . 0      . (2) W e will use the map A to a ssociate to a par ity ch eck matrix H = ( h ij ) in ( F ∗ q m ) a × b the ( larger and binar y) parity check matrix H = ( A ( h ij )) in F µa × µb 2 . The matrices A ( h ij ) ′ s are µ × µ circulant permutation matrice s ba sed on some primitive elements h ij as sh own in Defin ition 1. B. Re gular LDPC Codes A low-density p arity check code (or LPDC short) is a binar y block code that has a parity check matrix H in which each row (and each column) is sparse. An LDPC code is called r egular with par ameters ( ρ, λ ) if it has a sparse parity check m atrix H in which ea ch row has ρ no nzero entries and eac h column has λ no nzero entries. A regular L DPC c ode defined by a par ity ch eck m atrix H is said to satisfy th e r ow-column cond ition if and only if any two rows ( or, equivalently , any two co lumns) of H have at mo st one po sition of a n onzero entr y in com mon. The row-column condition ensures that the T anne r graph d oes not have cycles of len gth four . A T anner gr aph of a binary code with a parity ch eck matrix H = ( h ij ) is a grap h with vertex set V . ∪ C that h as one vertex in V for each co lumn of H and on e vertex in C for each row in H , and there is an ed ge betwee n two vertices i and j if and on ly if h ij 6 = 0 . Thus, the T an ner graph is a bipartite graph. T he vertices in V ar e c alled the variable no des, an d the vertices in C are called th e chec k nodes. W e refer to d ( v i ) and d ( c j ) as the d egrees of variable nod e v i and check no de c j respectively . T wo values used to measure the performa nce of the deco d- ing algorith ms of LDPC codes are: girth o f a T ann er graph and stopp ing sets. T he minimu m stopping set is analog ous to the min imum Ham ming distan ce of linea r blo ck code s. Definition 2 (Grith of a T a nner graph): The g irth g of the T ann er g raph is the length of its shortest cycle (minimum cycle). A T anne r gra ph with large girth is desirable, as iterative decodin g con verges faster for g raphs with large girth. Definition 3 (Sto pping set): A stopping set S o f a T anne r graph is a subset of the variable nodes V such that ea ch vertex in the neighbors of S is conn ected to at least two nodes in S . The stop ping distance is the size of the smallest stopping set. The stopping distance d etermines the nu mber of co r- rectable erasures by an iter ativ e decod ing alg orithm, see [7] , [16], [ 17]. Definition 4 (Sto pping distance ): T he stopping distance of the parity ch eck matrix H can be defined as the largest in teger s ( H ) such that every set o f at most ( s ( H ) − 1) colum ns of H contains at least one row of weight on e, see [17 ]. The sto pping r atio σ of the T an ner gra ph of a code of length n is defin ed b y s over the code len gth. The minim um Hamming distance is a pro perty of th e code used to m easure its per forman ce for m aximum- likelihood decodin g, while the stopping distance is a property of the parity check matr ix H or the T anner g raph G of a spec ific code. Hence, it varies for different choices of H f or the same code C . Th e stopping distance s ( H ) g iv es a lower bound of the min imum d istance of the code C defined b y H , na mely s ( H ) ≤ d min (3) It has b een shown that finding the stopping sets of minimum cardinality is an NP- hard p roblem , since the minimum- set vertex covering problem can be reduced to it [11] . 3 I I I . L D P C C O D E S B A S E D O N B C H C O D E S In this section we give two co nstruction s of LDPC codes derived fro m nonprimitive BCH cod es, and from elemen ts of cyclotomic co sets. In [22 ], th e authors derived a class o f regular LDPC codes fro m prim iti ve BCH c odes but they d id not prove that the construction has free of cycles of length fou r in the T anner graph. In fact, we will show that no t all primitive BCH co des can b e used to co nstruct LDPC with cycles gr eater than or equal to six in the ir T a nner g raphs. Our constru ction is free of cycles o f len gth four if the BCH co des are chosen with prim e lengthes as proved in Lemma 7; in addition the stopping distance is compu ted. Fu rthermo re, W e are able to derive a f ormula fo r the dim ension of th e constru cted LDPC codes as given in Theorem 9. W e also in fer the dimension and cyclotom ic coset structure of the BCH codes b ased on our previous results in [3], [4]. W e keep the definitions of the previous section. Let q be a power of a p rime and n a positive integer such that gcd( q , n ) = 1 . Recall that the cycloto mic coset C x modulo n is defined as C x = { xq i mo d n | i ∈ Z , i ≥ 0 } . (4) Let m be the m ultiplicative order of q modulo n . Let α be a prim iti ve element in F q m . A nonp rimitive narrow-sense BCH code C of designed d istance δ and len gth n over F q is a cyclic code with a gener ator mon ic po lynomial g ( x ) that has α, α 2 , . . . , α δ − 1 as zeros, g ( x ) = δ − 1 Y i =1 ( x − α i ) . (5) Thus, c is a codeword in C if an d only if c ( α ) = c ( α 2 ) = . . . = c ( α δ − 1 ) = 0 . The parity check matrix of this code can be d efined as H bch =      1 α α 2 · · · α n − 1 1 α 2 α 4 · · · α 2( n − 1) . . . . . . . . . . . . . . . 1 α δ − 1 α 2( δ − 1) · · · α ( δ − 1) ( n − 1)      . (6) W e n ote the following fact abou t the car dinality of cyclo- tomic c osets. Lemma 5 : Let n be a positiv e integer an d q b e a power of a pr ime, such that g cd( n, q ) = 1 an d q ⌊ m/ 2 ⌋ < n ≤ q m − 1 , where m = or d n ( q ) . The cyclotom ic coset C x = { xq j mo d n | 0 ≤ j < m } ha s a cardinality of m for all x in the range 1 ≤ x ≤ nq ⌈ m/ 2 ⌉ / ( q m − 1) . Pr o of: See [3 , Lem ma 8 ]. Therefo re, all cyclotomic cosets have the same size m if their ra nge is bounded by a c ertain value. This lemma enables one to determine the dimension in clo sed form for BC H code of small designed distance [ 3], [4]. In fact, we show the dimension of no nprimitve BCH codes over F q . Theorem 6 : Let q be a pr ime power and gcd( n, q ) = 1 , with ord n ( q ) = m . Then a narr ow-sense BCH code of length q ⌊ m/ 2 ⌋ < n ≤ q m − 1 over F q with design ed distance δ in the ra nge 2 ≤ δ ≤ δ max = min {⌊ nq ⌈ m/ 2 ⌉ / ( q m − 1) ⌋ , n } , has dimension of k = n − m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ . (7) Pr o of: See [3 , The orem 10 ]. Based on these two observations, we can construct regular LDPC codes fr om BCH codes with a kn own dimension and cyclotomic coset size. A. T ype-I Con struction In this constru ction, we use the parity check matrix of a nonpr imitiv e narrow-sense BCH code over F q to d efine the parity check matrix of a r egular LDPC over F 2 . Consider the n arrow-sense BCH code of prim e length q ⌊ m/ 2 ⌋ < n ≤ q m − 1 ov er F q with de signed distance δ and or d n ( q ) = m . W e use th e fact that there must be some primes in the integer range ( q ⌊ m/ 2 ⌋ , q m − 1) . In fact, the re must exist a prime b etween x and 2 x for some integer x, in wh ich it ensures existence prim es in the given inte rval. A parity ch eck matrix H o f an LDPC co de can be obtained b y applying the map A in Eq uation ( 1) to each entry of the parity check matr ix (6) of this BCH c ode, H = (8)      A (1) A ( α ) A ( α 2 ) · · · A ( α n − 1 ) A (1) A ( α 2 ) A ( α 4 ) · · · A ( α 2( n − 1) ) . . . . . . . . . . . . . . . A (1) A ( α δ − 1 ) A ( α 2( δ − 1) ) · · · A ( α ( δ − 1) ( n − 1) )      . The matrix H is of size ( δ − 1) µ × nµ and by construction it has the fo llowing properties: • Every column has a weigh t of δ − 1 . • Every row has a weight of n . The matrix H of size ( δ − 1) µ × nµ has a weight of ρ = δ − 1 in ev ery colu mn, and a we ight of λ = n in every row . The null space of the matrix H defines a ( ρ, λ ) LDPC code with a hig h rate for a sm all designed distance δ as we will show . The m inimum d istance o f the BCH cod e is boun ded by d min ≥  δ + 1 , odd δ ; δ + 2 , ev en δ . (9) Also, the minimum distanc e of the L DPC code s is b ound ed by d min . No w , we will sho w that in g eneral regular ( ρ, λ ) LDPC codes derived from primitive BCH codes of len gth n are not free o f cycles of length fo ur a s claim ed in [22] . Lemma 7 : The T anner g raph of LDPC codes constru cted in T ype-I are free of cycles of leng th four for a p rime length n . Pr o of: Consider the bloc k-colum n indexed by n − j for 1 ≤ j ≤ n − 1 an d let r i and r ′ i be two different b lock-rows for 1 ≤ r i , r ′ i ≤ ( δ − 1) . Assume by co ntradiction tha t we have A ( α r i ( n − j ) ) = A ( α r ′ j ( n − j ) ) . Thu s r i ( n − j ) mo d n = r ′ i ( n − j ) mo d n or n ( r i − r ′ i ) mo d n = ( r i − r ′ i ) j mo d n = 0 . This co ntradicts the assumption that n > j ≥ 1 and r i 6 = r ′ i . 4 Hence p rimitive BCH codes o f com posite length n can not b e used to deriv e LDPC codes that are cycles-free of length four using our co nstruction . The proof of the f ollowing lemma is straight for ward by exchanging , adding, and permu ting a b lock-r ow . Lemma 8 : Let ( . . . , 1 ℓ , . . . ) be a vector of leng th µ that has 1 at position ℓ . Under the cyclic shift, the following two blocks h a and h b of size µ × µ are eq uiv alent, where h a and h b are gen erated by the rows  1 . . . 1 i . . .  and  1 . . . 1 j . . .  and their cyclic shifts, r espectively . One might im agine that the ran k of the parity check matrix H in (1 0) is g iv en by ( δ − 1) µ since rows of ev ery block-r ow h a is linea rly ind epend ent. A computer pr ogram has been written to check the exact f ormula and then we d rove a fo rmula to giv e th e ran k o f the matr ix H . Theorem 9 : Let n be a prime in the range q ⌊ m/ 2 ⌋ < n ≤ µ = q m − 1 and δ be an integer in the ran ge 2 ≤ δ < n for some prime p ower q an d m = ord q ( n ) . T he rank o f the parity check matrix H given by H =      A o A 1 A 2 · · · A n − 1 A 0 A 2 A 4 · · · A 2( n − 2) . . . . . . . . . . . . . . . A 0 A δ − 1 A δ − 1 ) · · · A ( δ − 1) ( n − 1)      (10) is ( δ − 1) µ − ( δ − 2) , where A i = A ( α i ) . Pr o of: The pr oof of this theo rem can be sh own by mathematical induction fo r 1 , 2 , . . . , δ ≤ n . W e know that ev ery blo ck-row is lin early ind ependen t. i) Case i. Let δ = 2 , th e statement is true since ever block- row has only 1 in e very column , the first n co lumns represent th e identity matrix . ii) Case ii-1. Assume th e statement is true for δ − 2 . In this case, the matrix G has a full rank given by ( δ − 2 ) µ − ( δ − 3) . So, we h av e G =        h 11 h 12 h 13 . . . . . . h 1 n 0 h 22 h 23 . . . . . . h 2 n 0 0 h 33 . . . . . . h 3 n 0 0 0 . . . . . . h in 0 0 . . . h ( δ − 2) ( δ − 2) . . . h ( δ − 2) n        . The elements h ′ ii s hav e 1’ s in the diag onal an d zeros ev erywher e using simp le Gauss elimination me thod and Lemma 8. iii) Case iii-1. W e can form the sub-matrix H 2 of size ( δ − 1) µ × ( δ − 1 ) µ by adding o ne b lock-row to the matrix G . The last block -row is gen erated b y ( A ( α 0 ) , A ( α δ − 1 ) , A ( α 2( δ − 1) ) , . . . , A ( α n − 1( δ − 1) )) . All µ − 1 r ows o f the last block- row are lin early in depen- dent a nd can not b e g enerated from the previous δ − 2 blocks-row . N ow , in o rder to obtain the last row-block to be zero at positions h ( δ − 1) 1 , h ( δ − 1) 2 , . . . , h ( δ − 1) ( δ − 2) , we can add the e lement h j j to th e element h ( δ − 1) j . In addition, the last r ow (row ind exed by ( δ − 1) µ ) of block - row δ − 1 can be gen erated b y adding all ele ments of the first block-r ow to the first µ − 1 rows of the last block-row . G =        h 11 h 12 h 13 . . . . . . h 1 n 0 h 22 h 23 . . . . . . h 2 n 0 0 h 33 . . . . . . h 3 n 0 0 0 . . . . . . h in 0 0 . . . h ( δ − 1) ( δ − 1) . . . h ( δ − 1) n        . Therefo re, the matrix G h as rank of ( δ − 2) µ − ( δ − 3) + µ − 1 = ( δ − 1) µ − ( δ − 2) . W e n otice that th e matr ix H has th e same r ank as the matrix G , hence th e pr oof is completed . The pro of can also be shown by d ropp ing the last r ow of ev ery block-r ow except at the last row in the first block- row . Hence, the re maining m atrix has a full rank. Obtaining a fo rmula for r ank of the par ity ch eck m atrix H allows us to com pute ra te of the constru cted LDPC cod es. Now , we can ded uce the relationship between n onpr imi- ti ve narrow-sense BCH c odes and LDPC co des constru cted in T ype-I . Theorem 1 0 ( LDPC-BCH Theor em): Let n be a prime and q be a power of a prime, such that gcd( n, q ) = 1 and q ⌊ m/ 2 ⌋ < n ≤ q m − 1 , wh ere m = or d n ( q ) . A n onprim iti ve narrow- sense BCH co de with param eters [ n, k , d min ] q giv es a ( δ − 1 , n ) L DPC code with ra te ( nµ − [( δ − 1 ) µ − ( δ − 2)]) /nµ , where k = n − m ⌈ ( δ − 1)(1 − 1 /q ) ⌉ and 2 ≤ δ ≤ δ max . The constructed codes ar e free of cycles with length four . Pr o of: By T ype-I con struction of LDPC co des derived from n onprim iti ve BC H codes using Equation (1 0), we know that every element α i in H bch is a cir culant matrix A ( α i ) in H . Therefo re, there is a parity check matrix H with size ( δ − 1) µ × nµ . H has a r ow we ight of n and a colu mn weig ht of δ − 1 . He nce, th e null space of the matrix H define s an LDPC co de w ith the given rate using L emma 9. The con structed code is free of cycles of length fou r , because the matr ix H bch has no two rows with th e same value in th e same co lumn, except in th e first colum n. Hen ce, the matrix H h as, at m ost, one p osition in common b etween two rows d ue to circulant pr operty and Lemma 7 . Conseq uently , they have a T ann er gra ph with gir th greater than or equ al to six. Based on T y pe-I construction o f regular LDPC cod es, we notice that every variable node has a d egree δ − 1 and every check nodes ha s a degree n . Also, the maximu m num ber of columns that do not ha ve one in comm on is n . Therefore, the following Lemm a counts the stopp ing d istance of the T anner graph d efined by H . Lemma 1 1: T he cardin ality of the smallest stopping set of the T ann er graph of T ype-I co nstruction of regular LDPC codes is µ + 1 . Example 1 2: L et n = µ = q m − 1 , with m = 7 and q = 2 . Consider a BCH co de with δ = 5 an d length n . Assum e α to be a primitiv e elem ent in F q m . The matrix H c an be written 5 as H =     1 α α 2 . . . α 126 1 α 2 α 4 . . . α 125 1 α 3 α 6 . . . α 124 1 α 4 α 8 . . . α 123     , (11) and the m atrix H has size 50 8 × 161 29 . Therefor e, we constructed a (4 , 127) regular LDPC with a rate of 12 3 / 12 7 , see Fig . 1. T ABLE I P A R A M E T E R S O F L D P C C O D E S D E R I V E D F R O M N P B C H C O D E S q µ BCH Codes LDPC code rank of H size of H 2 31 [23 , 12 , 4] (93,713) 91 3 26 [23 , 12 , 5] (104,598) 101 2 31 [31 , 26 , 3] (62,961) 61 2 31 [31 , 21 , 5] (124,961,) 1 21 2 31 [31 , 26 , 6] (155, 961) 151 2 31 [31 , 16 , 7] (186,961) 181 2 63 [47 , 24 , 4] (189 ,1961) 187 2 63 [61 , 21 , 6] (315, 3843) 311 2 63 [61 , 11 , 10] (567,3843) 559 2 127 [127 , 113 , 15] (1778 ,16129) 1765 2 127 [127 , 103 , 25] (3048 ,16129) 3025 I V . L D P C C O D E S B A S E D O N C Y C L OT O M I C C O S E T S In this section we will constru ct regular LDPC codes based on the struc ture o f cyclotom ic cosets. Assume th at we use the same notation as shown in Section II. Let C x be a cyclotomic coset modu lo prime integer n , d efined as C x = { xq i mo d n | i ∈ Z , 1 ≤ x < n } . W e can also define the lo cation vector y of a cyclotomic coset C x , instead of the location vector z of an elem ent α i . Definition 1 3: The location vector y ( C x ) d efined over a cyclotomic coset C x is the vector y ( C x ) = ( z 0 , z 1 , . . . , z n ) , where all po sitions are zeros excep t at p ositions corr espondin g to elements o f C x . Let ℓ be the number of different cyclotomic cosets C i x ’ s that are used to constru ct the m atrices H i C j ’ s. W e can in dex the ℓ locatio n vectors correspond ing to C x 1 , C x 2 , . . . , C x ℓ , as y 1 , y 2 , . . . , y ℓ . Let y 1 ( γ C x ) be the cyclic shift of y 1 ( C x ) where ev ery eleme nt in C x is in cremented by 1. A. T ype- II Construction W e con struct the matrix H 1 C x from the cycloto mic C x as H 1 C x =      y 1 ( C x ) y 1 ( γ C x ) . . . y 1 ( γ n − 1 C x )      , (12) where y 1 ( γ j +1 C x ) is the cyclic shift of y 1 ( γ j C x ) f or 0 ≤ j ≤ n − 1 . From Lemm a 5, we know that all cycloto mic co sets C x ’ s have a size of m if 1 ≤ x ≤ nq ⌈ m/ 2 ⌉ / ( q m − 1) . W e can generate all rows of H C x , by shiftin g th e first row one position to the rig ht. Ou r construction of the matr ix H i c x has the fo llowing restrictions. • Let x ≤ Θ( √ n ) , this will g uarantee that all cyclotom ic cosets hav e the same size m . • Any tw o rows of H i c x have on ly one nonzero position in common . • Every row (colum n) in H i c x has a weig ht o f m . W e can co nstruct the m atrix H from d ifferent cyclotomic cosets as fo llows. H = h H 1 C 1 H 2 C 3 . . . H ℓ C j i (13) = 0 B B B @ y 1 ( C 1 ) y 2 ( C 2 ) . . . z ℓ ( C j ) y 1 ( γ C 1 ) y 2 ( γ C 2 ) . . . y ℓ ( γ C j ) . . . . . . . . . . . . y 1 ( γ n − 1 C 1 ) y 2 ( γ n − 1 C 2 ) . . . y ℓ ( γ n − 1 C j ) 1 C C C A , where we choo se the n umber ℓ o f d ifferent sub-matrices H C j . The n × ( ℓ ∗ n ) matrix H constructed in T ype- II has the following prope rties. i) Every column h as a weigh t of m and ev ery row has a weight of m ∗ ℓ , where ℓ is the number of matric es H ′ C j s . ii) For a large n, the m atrix H is a sparse lo w-density parity check matr ix. W e can also sh ow that the null space o f the m atrix H defines an ( m, mℓ ) LDPC code with rate ( ℓ − 1) /ℓ . Clearly , an incre ase in ℓ , incr eases the rate of th e code. Since all cyclotomic cosets C x 1 , C x 2 , . . . , C x ℓ used to con- struct H are dif ferent, th en the first co lumn in each sub -matrix H j C x is d ifferent from the first co lumn in all su b-matrices H i C x for j 6 = i an d 1 ≤ i ≤ ℓ . Now , we can giv e a lower bou nd in the stop ping distan ce of T ype-II LDPC codes. Lemma 1 4: T he stoppin g d istance o f LDPC cod es, that are in T ype-II constructio n, is at least ℓ + 1 . One can impr ove this boun d, by coun ting th e numb er of columns in each sub-matrix H i C x that do not have one in common in a ddition to all co lumns in the other su b-matrices. Example 1 5: Con sider n = q m − 1 with m = 5 , q = 2 , and δ = 5 . W e can com pute th e cycloto mic cosets C 1 , C 3 and C 5 as C 1 = { 1 , 2 , 4 , 8 , 16 } , C 3 = { 3 , 6 , 12 , 24 , 17 } and C 5 = { 5 , 10 , 20 , 9 , 18 } . The matrices H 1 C 1 , H 2 C 3 and H 3 C 5 can be d efined based on C 1 , C 3 and C 5 , respectiv ely . H 1 C 1 = 0 B B B B B B B B B B @ 1101 0001 0000 0001 0000 0000 0000 000 0110 1000 1000 0000 1000 0000 0000 000 0011 0100 0100 0000 0100 0000 0000 000 0001 1010 0010 0000 0010 0000 0000 000 0000 1101 0001 0000 0001 0000 0000 000 . . . . . . . . . . . . . . . . . . . . . . . . 0100 0100 0000 0100 0000 0000 0000 011 1010 0010 0000 0010 0000 0000 0000 001 1 C C C C C C C C C C A (14) The m atrix H of size (31,9 3) is g iv en b y H = h H 1 C 1 H 2 C 3 H 3 C 5 i , (15) therefor e, th e null space of H defines an (5 ,15) LDPC cod e with pa rameters (62 , 93) . W e note that T ype-I and T ype-II constructions can be used to d erive quantum codes, if the p arity check ma trix H is 6 1 1.5 2 2.5 3 3.5 4 10 −5 10 −4 10 −3 10 −2 10 −1 Bit Error Rate of LDPC BER Eb/No (dB) 4,31)−LDPC from BCH code with size (124,961) (4,23)−LDPC from BCH code with size (124,713) (5,61)−LDPC from BCH code with size (315,3843) Fig. 1. T ype I: The error perf ormance of an (4,31) LDP C code with rate 27 / 31 and H matrix with size (124 , 961) ba sed on a BCH code. modified to be self-orthog onal or using the nested p roper y of LDPC-BCH codes. Recall that quantu m error-correcting codes over F q can be constructed from self-or thogo nal classical codes over F q and F q 2 , see fo r example [2 ], [3 ], [5], [10], [15] an d r eferences th erein. V . S I M U L AT I O N R E S U LT S W e simu lated the per forman ce of the constru cted co des using stan dard iterati ve decodin g a lgorithms. Fig. 1 sho ws th e BER curve for an (4,31) LDPC code T ype I with a len gth of 961, d imension of 837 , and number of iterations of 50 . Th is perfor mance can also be improved f or various lengths and the designed distan ce of BCH codes. The perfor mance of the se constructed code s can b e impr oved f or large cod e length in compariso n to other LDPC codes co nstructed in [ 12], [13 ]. As shown in Fig. 1 at the 10 − 4 BER, the code perfo rms at 5 . 5 E b/ N o ( dB ) , which is 1 . 7 u nits from the Shann on limit. V I . C O N C L U S I O N W e in troduce d two families of regular LDPC code s b ased on nonpr imitiv e nar row-sense BCH co des an d structu res o f cyclotomic cosets. W e gave a systematic metho d to write e very element in th e parity check m atrix of BCH codes as vector of length µ . W e demonstrated that th ese con structed codes have high rates and a uniform stru cture that made it easy to co mpute their dimensions, stoppin g distance, and bou nd their minim um d istance. Furtherm ore, one can use standar d iterativ e d ecoding algo rithms to decode th ese cod es. One can easily der iv e irregular LDPC codes based on these cod es and possibly increase performance of the iterati ve deco ding. Also, in future research, these co nstructed codes can be u sed to derive quantum LDPC error-correctin g codes. A C K N O W L E D G M E N T S . 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