Spectral Graph Analysis of Quasi-Cyclic Codes

Spectral Graph Analysis of Quasi-Cyclic Codes Roxana Smarandac he Departmen t of M athematics and Statistics San Diego State Un iv ersity San Diego, CA 9 2182, USA Email: rsmaran d@sciences.sdsu.edu Mark F . Flanagan Departmen t of E lectronic Engin eering University Colle ge Dublin Belfield, Dublin 4, Irelan d Email: m ark.flanag an@ieee.org Abstract —In this paper we analyze the bound on the additive white Gaussian noise channel (A WGNC) pseudo-weight of a ( c, d ) -regular linear block code b ased on the two large st v alues λ 1 > λ 2 of the eigen values of H T H : w min p ( H ) > n 2 c − λ 2 λ 1 − λ 2 . [6]. In particular , we analyze ( c, d ) -regular quasi-cyclic (QC) codes of length r L d escribed b y J × L b lock parity-check matrices with circulant block entries of size r × r . W e proceed by showing how the problem of computi ng the eigen v alues of the r L × rL matrix H T H can be re duced to the problem of computi ng eigen v alues fo r r matrices of si ze L × L . W e also give a necessary conditi on fo r the b ound to be attain ed for a circulant matrix H and show a few classes of cyclic codes satisfying t his criterion. Index T erms —Low-density parity-check codes, pseudo- codewords, pseudo-weights, eigenv alues, eigenv ectors. I . I N T R O D U C T I O N Low-density parity-ch eck (LDPC) co des o ffer excellent tradeoffs between perfo rmance and complexity fo r err or c or- rection in comm unication systems. Quasi-cyclic (QC) LDPC codes in particular hav e prov ed extremely attractive due to their implementation advantages, both in encoding an d decod- ing [1], [ 2], [ 3]. Many analyses o f QC-L DPC co des ha ve been carried out based o n op timization of parameters such as the minimum H amming d istance of the cod e or the girth of th e T anner graph . Howe ver , it ha s been shown that an excellent first-order m easure of performance over the A WGNC is the minimum p seudo-weig ht of the cod e [4]. So far, fe w results exist in the literatu re o n th e minimum pseudo- weight of QC- LDPC a nd r elated codes. Spectral graph analysis was used in [5], and more recently , in [6], to obtain bounds on the minimum Hamming weight, and minimum A WGNC pseudo-weig ht, respectively , of a length- n ( c, d ) -regular code C over the b inary field F 2 : d min > w min p ( H ) > n 2 c − λ 2 λ 1 − λ 2 ; d min > n 2 d 2 c + d − 2 − λ 2 λ 1 − λ 2 , with λ 1 > · · · > λ s being the distinct ordered eigen v alues of H T H ∈ R n × n (where H is viewed as a matrix in R m × n ). Th ese boun ds are, f or mo st codes, loose. Howe ver , in particular cases, like the projectiv e ge ometry codes [7], [8], [9], th ey are attained. A curre nt problem with these boun ds is that for most LDPC code s, it is not pra ctical to e valuate the eigenv alues λ 1 , λ 2 due to th e size of the matrix H T H . In this paper we show how to compute the A WGN pseudo - weight lower bou nd for quasi-cyclic (QC) and related cod es by utilizing the A -submod ule structure of quasi-cyclic cod es, A = R [ X ] / ( X r − 1) [1 0], [11], [12]. In particu lar , we begin by showing h ow th e p olyno mial pa rity-check matr ix that describe s a cyclic co de can b e used to co mpute the required eig en values, and then g eneralize th is a pproach to compute the required eige n values for QC cod es. W e a lso define the class of “nested circulant” m atrices, and show that these have eigen values which are given by ev aluating a multiv ariate associated polynomial at points whose coordinates are particular ro ots of unity . Finally , we gi ve a necessary condition f or th e p seudo-weig ht lower bo und to be a ttained when H is circulant and show a few classes of cyclic codes satisfying this criterion. I I . B A S I C N OTA T I O N A N D D E FI N I T I O N S All codes in this pape r will be binary lin ear co des o f a certain leng th n specified thro ugh a ( scalar) p arity-chec k matrix H = ( h j,i ) ∈ F m × n 2 as the set of all vector s c ∈ F n 2 such tha t H · c T = 0 T , where T denotes transp osition. The minimum Hamm ing d istance of a code C will be deno ted b y d min ( C ) . The fund amental co ne K ( H ) of H is the set of all vectors ω ∈ R n that satisfy ω i > 0 for all i ∈ I ( H ) , (1) ω i 6 X i ′ ∈I j ( H ) \ i ω i ′ for all j ∈ J ( H ) , i ∈ I j ( H ) , (2) where J ( H ) and I ( H ) denote the sets of r ow and column indices o f H respectively , and I j ( H ) , { i ∈ I | h j,i = 1 } for each j ∈ J ( H ) . A vector ω ∈ K ( H ) is called a pseudo - codewor d . The A WGNC p seudo-weigh t of a pseudo-c odeword ω is defined to b e w p ( ω ) = w A W GNC p ( ω ) , k ω k 2 1 / k ω k 2 2 . (For a motiv ation of these d efinitions, see [1 4], [15]) . The minimum of the A WGNC pseudo-weigh t over all nonzer o pseudo- codewords is called the minimum A WGNC pseu do- weight and is denoted by w min p ( H ) . For any integer s ≥ 1 , let R s = { exp( ı 2 π r/ s ) : 0 ≤ r < s } denote the set of com plex s -th roots of unity , and let R − s = R s \{ 1 } . The symbol ∗ denotes com plex conjugation . Also, an r × r circulant matrix B , whose entries are square L × L matr ices, w ill be called an L -block c ir culant matrix ; we shall deno te this by B = circ( b 0 , b 1 , · · · , b r − 1 ) where the (square L × L ma trix) entr ies in the first column o f B are b 0 , b 1 , ... , b r − 1 respectively . Finally , Z , R , C , and F 2 will be th e ring o f integers, the field of real numb ers, the complex field, and the finite field of size 2 , respectively . For a po siti ve integer L , [ L ] will den ote the set of nonnegative in tegers smaller than L : [ L ] = { 0 , 1 , . . . , L − 1 } . I I I . C O M P U T I N G T H E E I G E N V A L U E S O F H T H F O R A Q C C O D E In this section we will show th at the polyno mial represen - tation of a QC code will prove very helpfu l in comp uting the eigenv alues of the large matr ix H T H , easing in this way th e computatio n o f th e lower boun d d min > w min p ( H ) > n 2 c − λ 2 λ 1 − λ 2 . (3) This section is o rganized in thr ee subsections. In Sec. I II-A and III-B we p rovide some backg roun d o n circ ulant m atrices and QC codes. Section III- C will contain the main result on the eigenv alues of H T H , wh ere H is the p arity-chec k matrix of a QC c ode. A. Eigenvalues o f a Circulant Matrix The eigenv alues of a square cir culant matrix ar e well known [16]. If B ∈ C n × n is a circu lant matrix and w ( X ) = b 0 + b 1 X + . . . + b n − 1 X n − 1 its (column) associated polynom ial, then the eigenvalues of B are gi ven by this polyno mial’ s ev aluation at the complex n -th roots o f u nity , i.e. w ( x ) for all x ∈ R n . The f ollowing gives a pr oof of this result based on the polyno mial represen tation of a circulant matrix. It may be seen as a sp ecial ca se of the meth od we present later f or QC co des. Let λ be an eigenv alue of B . Then there exists a nonzero vector v = ( v 0 , . . . , v n − 1 ) T ∈ C n such th at Bv = λ v . In polyno mial for m, this eq uation is equiv alent to (h ere v ( X ) = v 0 + v 1 X + . . . + v n − 1 X n − 1 ): w ( X ) v ( X ) = λv ( X ) mo d ( X n − 1) iff X n − 1 | w ( X ) v ( X ) − λv ( X ) in C iff w ( x ) v ( x ) = λv ( x ) , ∀ x ∈ R n iff ( w ( x ) − λ ) v ( x ) = 0 , ∀ x ∈ R n . For eac h x ∈ R n , λ = w ( x ) is a solutio n of the above equation, and ther efore it is a n eigenv alue for the matrix B . There are n such solution s, therefo re, these are all possible eigenv alues o f B . In th e next theorem we will consider an L -block circulant matrix instead of a circulant matrix. This th eorem m ay be found in [17]; we provide her e an a lternative proo f b ased on the polyn omial represen tation. Theorem 1 Let B = circ( b 0 , b 1 , · · · , b r − 1 ) ∈ C r L × r L be an L -block circulant matrix. Let W ( X ) = b 0 + b 1 X + . . . + b r − 1 X r − 1 its (co lumn) associated matrix polynomia l. Then the eigen values of B are g iven b y th e u nion of the eigen values of the L × L ma trices W ( x ) , for all x ∈ R r . Pr oof: Th e p roof fo llows th e reasoning in the theorem above. Let λ be an eigenv alue of B . Then there exists a nonzero vector v , ( v 0 , . . . , v r L − 1 ) T ∈ C r L such th at Bv = λ v . (4) Let p ( X ) ∈ C L [ X ] given by p ( X ) = ( v 0 , . . . , v L − 1 ) T + ( v L , . . . , v 2 L − 1 ) T X + . . . + ( v r ( L − 1) , . . . , v r L − 1 ) T X r − 1 . In polyno mial f orm, eq uation (4) is equiv alent to: B ( X ) p ( X ) = λ p ( X ) mo d ( X r − 1) iff X r − 1 | B ( X ) p ( X ) − λ p ( X ) in C iff B ( x ) p ( x ) = λ p ( x ) , ∀ x ∈ R r . The last equation is the equation fo r the eigenv alues of the matrix B ( x ) . Each such matrix has L eigenv alues, counting multiplicities, and there ar e r distinct complex numbers in R r ; this acco unts for the total num ber rL of eigenv alues of B . T he eigenv ectors can also be ded uced fro m th e above. B. Defi nition an d P r operties o f QC Cod es A lin ear QC-LDPC cod e C QC , C ( r ) QC of length n = rL can b e descr ibed by an r J × rL (scalar) parity-che ck matrix ¯ H ( r ) QC , ¯ H that is f ormed by a J × L array of r × r circulant matrices. ¯ H =      P 1 , 1 P 1 , 2 . . . P 1 ,L P 2 , 1 P 2 , 2 . . . P 2 ,L . . . . . . . . . . . . P J, 1 P J, 2 . . . P J,L      , (5) where the entr ies P i,j are r × r circulant matric es. Clear ly , by cho osing these circulan t matrices to be low-density , the parity-ch eck matrix will also be low-density . W ith the help of the well-kn own isomorp hism b etween the ring of r × r circu lant matrices and the ring of p olynom ials modulo X r − 1 , to each matrix P i,j we can associate a po lyno- mial p i,j ( X ) , and thus a QC-LDPC cod e can equivalently be described by a polyno mial parity -check m atrix P ( X ) of size J × L , with polyno mial oper ations performed m odulo X r − 1 : P ( X ) =      p 1 , 1 ( X ) p 1 , 2 ( X ) . . . p 1 ,L ( X ) p 2 , 1 ( X ) p 2 , 2 ( X ) . . . p 2 ,L ( X ) . . . . . . . . . . . . p J, 1 ( X ) p J, 2 ( X ) . . . p J,L ( X )      . (6) By p ermuting the ro ws and columns of the scalar parity- check matrix ¯ H , 1 we obtain an equiv alent p arity-che ck matrix representatio n H for the QC co de C ( r ) QC , H ,      H 0 H r − 1 · · · H 1 H 1 H 0 · · · H 2 . . . . . . . . . . . . H r − 1 H r − 2 · · · H 0      . (7) 1 i.e., by taking the first row in the first block of r rows, the first ro w in the second block of r ro ws, etc., then the second row in the first block, the second row in the second block, etc., and similarly for the col umns. where H 0 , H 1 , . . . , H r − 1 are scalar J × L matr ices. The connectio n b etween th e two rep resentations is H 0 + H 1 X + · · · + H r − 1 X r − 1 = P ( X ) . (8) C. Th e Eigenvalues o f the Ma trix H T · H of a QC Code Note th at fo r a fixed value of r ≥ 1 , (8) provid es a simple bijective correspondence between the set of poly nomial matrices P ( X ) ∈ ( R [ X ] / ( X r − 1 )) J × L and the set of par ity- check matrices o f the fo rm ( 7). Furthermo re, the pr oduct o f two such po lynomial m atrices, wh ere defined, y ields another which correspon ds via this bijection with the produ ct of the correspo nding par ity-check matr ices in the f orm (7). Also note that transpo sition of a po lynomial matrix in the f orm (8 ) correspo nds to tran sposition of the cor respond ing parity-check matrix in th e form (7), unde r this bijection. It f ollows that H T · H is an L -block circulant matrix; ap ply- ing Theorem 1 to this m atrix y ields the following co rollary . Corollary 2 The eigenvalues of H T · H ar e given by th e union of the eigenvalues of the L × L ma trices P T ( x ∗ ) · P ( x ) , for x ∈ R r . Pr oof: W e ap ply Th eorem 1 to the L -block circ ulant matrix H T · H , circ( b 0 , b 1 , · · · , b r − 1 ) ∈ C r L × r L and form the matrix W ( X ) = b 0 + b 1 X + . . . + b r − 1 X r − 1 . This is equal to the p rodu ct of th e two matrix polyno mials of H T and H , which ar e H T 0 + H T r − 1 X + · · · + H T 1 X r − 1 = X r P T (1 /X ) and H 0 + H 1 X + · · · + H r − 1 X r − 1 = P ( X ) , respectively . There- fore W ( X ) = ( X r P T (1 /X )) · P ( X ) and so the eigen v alues of H T · H are the eigenvalues of P T (1 /x ) · P ( x ) , f or all x ∈ R r ; these are then equ al to the eigenv alues of P T ( x ∗ ) · P ( x ) , f or all x ∈ R r (as x ∗ = 1 /x fo r all such x ). Example 3 Let r = 31 and consider the (3 , 5) -regular QC- LDPC c ode given by the scalar 93 × 155 m atrix 2 ¯ H =   I 1 I 2 I 4 I 8 I 16 I 5 I 10 I 20 I 9 I 18 I 25 I 19 I 7 I 14 I 28   . The polyn omial par ity-check matrix P ( X ) ∈ ( R [ X ] / ( X r − 1)) 3 × 5 is P ( X ) =   X X 2 X 4 X 8 X 16 X 5 X 10 X 20 X 9 X 18 X 25 X 19 X 7 X 14 X 28   . This code is the famous (3 , 5) -regular QC-LDPC code of length 155 presented in [18]. Note that the co de p arameters are [1 55 , 64 , 20] . The correspo nding matrix H in the form (7) is a 3 1 × 31 matrix with blo ck en tries H i , i ∈ [31] obtained by decomp osing P ( X ) accor ding to the p owers of X : P ( X ) = H 0 + H 1 X + · · · + H 30 X 30 . (9) Obviously only 15 matr ices among the H i are non zero, and all of these contain only one 1 , the oth er en tries bein g zero. 2 Here I ℓ denotes the 31 × 31 identity matrix wit h rows shifted cy clica lly to the left by ℓ positio ns. The matrix H T · H is a 5 -blo ck circu lant matrix. Coro llary 2 above tells us that in order to compu te its eigenv alues, we need to for m the m atrices P T ( ρ − i ) · P ( ρ i ) , for all i ∈ [3 1] (here ρ d enotes a primitiv e com plex 31 -th roo t of un ity). W e have tha t P T (1 /x ) =   x 30 x 29 x 27 x 23 x 15 x 26 x 21 x 11 x 22 x 13 x 6 x 12 x 24 x 17 x 3   T and P T (1 /x ) · P ( x ) =       3 a e ∗ c e ∗ a ∗ 3 b a ∗ d e b ∗ 3 c b ∗ c ∗ a c ∗ 3 d e d ∗ b d ∗ 3       , for all x ∈ R 31 , where a = x + x 5 + x 25 ; b = x 2 + x 10 + x 19 ; c = x 4 + x 7 + x 20 ; d = x 8 + x 9 + x 14 ; e = x 16 + x 18 + x 28 . Obviously fo r i ∈ [31] , each matrix P T ( ρ − i ) · P ( ρ i ) is Hermitian (in fact nonnegative definite), hence each ha s 5 real nonn egati ve eig en values, gi ving a total o f 31 · 5 = 1 55 nonnegative eigen values for H T · H . W e ob tain th at for each i ∈ [31 ] , i 6 = 0 , th e associated polyno mial of P T ( ρ − i ) · P ( ρ i ) may b e written as (using ρ 31 = 1 ) u ( λ ) = λ 2 ( λ 3 − 15 λ 2 + 62 λ − 62 ) = λ 2 ( λ − λ 2 )( λ − λ 3 )( λ − λ 4 ) where λ 2 = 8 . 6801 , λ 3 = 4 . 8 459 an d λ 4 = 1 . 4740 . Also, for i = 0 th e associated polyno mial of P T ( ρ − i ) · P ( ρ i ) may be written as u ( λ ) = λ 4 ( λ − λ 1 ) whe re λ 1 = 15 . This yields the no nzero eigenv alues of H T · H as { λ 1 , λ 2 , λ 3 , λ 4 } with multiplicities 1 , 3 0 , 3 0 an d 3 0 re spectiv ely .  I V . E I G E N V A L U E S O F N E S T E D C I R C U L A N T M A T R I C E S In this section we d efine the class of nested circulant matrices, an d show that they have eigen values which are g i ven by evaluating a multi variate associated polynomial at points whose c oordin ates are p articular ro ots of unity . Theorem 4 Let B = circ( b 0 , b 1 , · · · , b r − 1 ) ∈ C r L × r L be an L -blo ck circulant ma trix. Sup pose tha t ea ch subblo ck b i , i ∈ [ r ] , is a lso cir culant, with associated polynomia l p ( i ) ( X ) = P L − 1 j =0 b i,j X j . Defin e the associated p olynomial of B b y q ( X , Y ) = r − 1 X i =0 L − 1 X j =0 b i,j X i Y j . Then the set of eigenvalues of B is given by { q ( x, y ) : x ∈ R r , y ∈ R L } . Pr oof: For each j ∈ [ L ] define u ( j ) ( X ) = P r − 1 i =0 b i,j X i . By Th eorem 1, the eigenv alues of B ar e equ al to those of the matrices g iv en by W ( x ) for x ∈ R r ; each of th ese is circulant with associated polyn omial (in Y ) given b y L − 1 X j =0 u ( j ) ( x ) Y j = q ( x, Y ) . Thus the eige n values of each W ( x ) are equ al to q ( x, y ) for y ∈ R L , and the result follows. W e n ext define what is meant by a nested circulant m atrix. Definition 5 Let m ≥ 1 and let i t be a p ositive integer for each t = 1 , 2 , · · · , m . Also let B = cir c( b 0 , b 1 , · · · , b i 1 − 1 ) be a block-cir culant matrix such that for every t = 1 , 2 , · · · , m − 1 , j t ∈ [ i t ] b j 1 ,j 2 , ··· ,j t = circ( b j 1 ,j 2 , ··· ,j t , 0 , b j 1 ,j 2 , ··· ,j t , 1 , · · · , b j 1 ,j 2 , ··· ,j t ,i t +1 − 1 ) is also b lock-cir culant, and that b j 1 ,j 2 , ··· ,j m = b j 1 ,j 2 , ··· ,j m ar e scalars. Then B is said to b e an m - nested circulant ma trix (with dimension n = Q m t =1 i t ). The associa ted po lynomial o f B is d efined by q ( X 1 , X 2 , · · · , X m ) = i 1 − 1 X j 1 =0 i 2 − 1 X j 2 =0 · · · i m − 1 X j m =0 b j 1 ,j 2 , ··· ,j m m Y t =1 X j t t (10)  Note that the 1 -nested circulants are p recisely the circulan t matrices, and that the 2 -n ested circ ulants ar e p recisely th e i 2 - block-cir culant matrices with circulant subblocks. Also note that the associated polyn omial q ( X 1 , X 2 , · · · , X m ) provides a succinct description of the matrix B . A straightfor ward generalization of Theorem 4 is as follo ws. Theorem 6 Let B be an m -nested circulant matrix with as- sociated polynomial q ( X 1 , X 2 , · · · , X m ) given b y (10) a bove. Then the set of eigenvalues of B is given by { q ( x 1 , x 2 , · · · , x m ) : x t ∈ R i t ∀ t = 1 , 2 , · · · , m } Pr oof: The pro of uses indu ction, and follows the lines of the proo f o f Theor em 4 in a rather straightfo rward m anner . Example 7 Here we take an example o f a n 3 -nested c irculant (i.e. m = 3 ), where i t = 2 fo r t = 1 , 2 , 3 . Th e eigenv alues of B =             0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0             are eq ual to the eigenv alues of B ′ =     0 1 + x x x 1 + x 0 x x x x 0 1 + x x x 1 + x 0     for x ∈ {− 1 , 1 } , whic h are equ al to th e eige n values of B ′′ =  xy 1 + x + xy 1 + x + xy xy  for x ∈ {− 1 , 1 } an d y ∈ {− 1 , 1 } . Finally , th ese are equa l to the set { q ( x, y , z ) : x, y , z ∈ {− 1 , 1 }} where the associated polyno mial o f B is q ( x, y , z ) = xy + z (1 + x + xy ) . In this example b 0 , 0 , 0 = 0 , b 0 , 0 , 1 = 1 , b 0 , 1 , 0 = 0 , b 0 , 1 , 1 = 0 , b 1 , 0 , 0 = 0 , b 1 , 0 , 1 = 1 , b 1 , 1 , 0 = 1 , b 1 , 1 , 1 = 1 ; these may be easily ob tained by match ing the elem ents of the first column of B with the binary expansio n of the cor respondin g row position. This examp le may be g eneralized to the case wh ere n = 2 m and the circulan t is m -nested; the eigenv alues are real. No te that the choice of the first column in B determ ines which terms in { 1 , x, y , z , xy , y z , z x, xy z } ar e includ ed in the associated polyno mial, a nd hence co ntrols th e eigenv alues of B .  Theorem 8 I f H is a n m -nested cir culant matrix, then B = H T H is an m -nested cir culant matrix. Pr oof: It is straightfor ward to prove the stro nger result that if A and B are m -nested circulants with specified nested dimensions, then A T B is also m -nested circulant, with th e same ne sted dimension s. The pr oof proce eds by ind uction on m . Th e base case m = 1 is straightfo rward. Next, let A be block-cir culant with blo ck entries in the first column eq ual to some ( m − 1 ) -nested circula nts A i , and let B be block- circulant with blo ck entries in the first column eq ual to some ( m − 1) -n ested cir culants B j . The matrix A T B is then block - circulant, and each block entry is a sum of matrices of the form A T i B j . By the prin ciple of induction, each of these matrices is an ( m − 1) -nested circulant, and it is easy to show tha t a sum of t - nested circulants (o f th e same n ested dimensio ns) is another t -nested circulant (with th ese nested dimension s). V . C O N D I T I O N S F O R T H E P S E U D O - W E I G H T L O W E R B O U N D T O H O L D W I T H E Q U A L I T Y It is straightforward to sh ow th at a ne cessary co ndition for the bou nd of [13] to ho ld with equ ality is that the eige n values of B = H T H ∈ R n × n are λ 1 with mu ltiplicity 1 and λ 2 < λ 1 with mu ltiplicity n − 1 . If H is circulant with (r ow) associated po lynomial w ( X ) of degree k ≤ n , the eigenvalues of B are precisely {| w ( x ) | 2 : x ∈ R n } ; the refore the largest eigenv alue of B is λ 1 = | w (1) | 2 = d 2 where d is the n umber o f no nzero coefficients in w ( X ) (n oting that | w (1) | 2 > | w ( x ) | 2 for all x ∈ R − n ). Let ˜ w ( X ) = X k w (1 / X ) denote the recipr oca l polynomia l of w ( X ) w hich is obtain ed by reversing the order of co efficients in w ( X ) . No w assume that the b ound of [13] holds with equality . Then we m ust have | w ( x ) | 2 = w ( x ) w ∗ ( x ) = λ 2 ∀ x ∈ R − n for some positive rea l num ber λ 2 , i.e. w ( x ) w (1 /x ) = λ 2 ∀ x ∈ R − n . This is equiv alent to w ( x ) ˜ w ( x ) = λ 2 x k ∀ x ∈ R − n Thus R − n is a subset of the ro ots of the p olynom ial w ( X ) ˜ w ( X ) − λ 2 X k , and so w ( X ) ˜ w ( X ) − λ 2 X k = (1 + X + X 2 + · · · + X n − 1 ) r ( X ) (11) where r ( X ) is a polynomial of degree 2 k − n + 1 ≥ 0 with integer coefficients. In the fo llowing we giv e details of th is condition fo r so me codes which attain the boun d of [13] with equality . Example 9 The EG(2 , 2) code with q = 2 , n = 3 , k = 1 , d = 2 has w ( X ) = 1 + X . Here λ 1 = d 2 = 4 an d (1 1) ho lds in the form (1 + X ) 2 − X = 1 + X + X 2 so in this case λ 2 = 1 and r ( X ) = 1 . Here d min = w min p ( H ) = n  2 d − λ 2 d 2 − λ 2  = 3 = q + 1 .  Example 10 The PG(2 , 2) code with q = 2 , n = 7 , k = 3 , d = 3 h as w ( X ) = 1 + X + X 3 . Here λ 1 = d 2 = 9 and (11) holds in the form (1 + X + X 3 )(1 + X 2 + X 3 ) − 2 X 3 = 1 + X + · · · + X 6 so in this case λ 2 = 2 and r ( X ) = 1 . Here d min = w min p ( H ) = n  2 d − λ 2 d 2 − λ 2  = 4 = q + 2 .  Example 11 The PG(2 , 4) cod e with q = 2 , n = 21 , k = 11 , d = 5 has w ( X ) = 1 + X 2 + X 7 + X 8 + X 11 . Here λ 1 = d 2 = 25 and (1 1) holds in th e form (1 + X 2 + X 7 + X 8 + X 11 )(1 + X 3 + X 4 + X 9 + X 11 ) − 4 X 11 = (1 + X + X 2 + · · · + X 20 )(1 − X + X 2 ) so in this case λ 2 = 4 and r ( X ) = 1 − X + X 2 . Here d min = w min p ( H ) = n  2 d − λ 2 d 2 − λ 2  = 6 = q + 2 .  Note that for a general PG(2 , q ) c ode, for the bound to hold with equality we requir e w min p ( H ) = q + 1 = n  2 d − λ 2 d 2 − λ 2  = ( q 2 + q + 1)  2( q + 1) − λ 2 ( q + 1) 2 − λ 2  . and therefo re we must h av e λ 2 = q . Also, for a general EG(2 , q ) co de, for the bound to h old with equality we require w min p ( H ) = q + 1 = n  2 d − λ 2 d 2 − λ 2  = ( q 2 − 1)  2 q − λ 2 q 2 − λ 2  . and therefore we must have λ 2 = q if q > 2 , wh ereas for q = 2 , a ny λ 2 will achieve the bo und. V I . C O N C L U S I O N S A N D F U T U R E W O R K A m ethod has been presented for e valuation of the eigenv alue-based lower b ound on the A WGNC pseu do-weigh t based on spectral an alysis, for QC and related cod es. It was shown that the r elev ant eigenvalues may be found by com- puting the eigenv alues of a certain n umber of small matrices. W e also presented a n ecessary co ndition for the b ound to b e attained with eq uality a nd g av e a few examples of codes fo r which this happ ens. Future work inv olves o ptimization of QC code designs based on these bou nds. V I I . A C K N O W L E D G M E N T The first author was supported by NSF Grant DMS-070803 3 and TF-083 0608. R E F E R E N C E S [1] Z. L i, L. Chen, L. Zheng, S. Lin and W . H. Fong, “Ef ficient encodi ng of quasi-c yclic low-densi ty parity-che ck codes, ” IEE E T r ansactio ns on Communicat ions , vol. 54, no. 1, pp. 565–5 78, Jan. 2006. [2] Y . Dai, Z . Y an and N. Chen, “Optimal overla pped message-p assing decodin g of quasi -cyc lic LDPC code s, ” IEEE T ransactions on V ery Larg e Scale Inte grat ion Systems , vol. 16, no. 5, pp. 565–578, May 2008. [3] M. M. Mansour and N. R. Shanbhag, “High-thro ughput LDPC de- coders, ” IEEE T ransact ions on V ery Larg e Scale Inte gr ation Systems , vol. 11, no. 6, pp. 976–996, Dec. 2003. [4] N. Wi berg, Codes and Decoding on General Graphs. Ph. D. Thesis, Link ¨ oping Uni v ersity , Sweden, 1996. [5] R. M. T an ner , “Minimum-distanc e bound s by graph analysis, ” IEE E T r ans. on Inform. T heory , vol. IT –47, no. 2, pp. 808–821, 2001. [6] P . O. V ont obel and R. K oette r , “Lo wer bounds on the minimum pseudo- weight of line ar codes, ” in Proc. IEEE Inte rn. Symp. on Inform. Theory , (Chicag o, IL, USA), p. 70, June 27–July 2 2004. [7] P . O. V onto bel and R. Smarandache, “On minimal pseudo-code wo rds of T a nner graphs from projecti v e planes, ” in Pro c. 43rd Allerton Conf . on Communicat ions, Contr ol , and Computing , (Allerton House, Montice llo, Illi nois, USA), Sep. 28–30 2005. A v ail able online under http://www.arx iv.org/abs/cs.I T/0510043 . [8] R. Smarandac he and P . O. V ont obel, “Pseudo -code wo rd analysis of T anner graphs from projecti ve and E uclide an planes, ” IEEE T ran s. Inform. Theory , vol. 53, no. 7, pp. 2376–2393, 2007, av ailab le online under http://www .arxiv .org/abs/c s.IT/0602089 . [9] Y . Kou, S. Lin, and M. P . C. Fossorier , “Low-densi ty parity-c heck codes based on finite geo metries: a redisco ver y and new results, ” IEEE T r ans. on Info rm. Theory , vol. IT –47, pp. 2711–273 6, Nov . 2001. [10] K. Lally and P . Fitzpa trick, “ Algebraic structure of quasicycl ic codes, ” Disc. Appl. Math. T rans. Inform. Theory , vol. 111, pp. 157–175, 2001. [11] S. Ling and P . Sole, “On the algebraic structure of quasi-c yclic codes I: finite fields, ” IE EE T rans. Inform. Theory , vol. 47, pp. 2751–2760, 2001. [12] S. L ing and P . S ole, “On the algebraic structure of quasi-c ycl ic codes II: Chain rings, ” Designs, Codes and Crypto graphy , vol. 30, pp. 113–130, 2003. [13] P . O. V ontobe l and R. Koe tter , “Lo wer Bounds on the Minimum Pseudo- W eight of Linear Codes, ” Proc. IEEE Internati onal Symposium on Informatio n Theory (ISIT) , June/July 2004, pp. 67, Chic ago, USA. [14] P . O. V ont obel and R. Koetter , “Graph-cov er decodin g and finite- length analysis of message-passing ite rati v e decoding of LDPC cod es, ” Submitte d to IEEE T rans. Inform. Theory , available online under http://www.arx iv.org/abs/cs.I T/0512078 , Dec. 2005. [15] R. K oetter , W .-C. W . Li, P . O. V ontobel , and J. L. W al ker , “Characteri - zatio ns of pseudo -code wo rds of (low- density) parity-check codes, ” Adv . in Math. , vol. 213, Aug. 2007. [16] F . J. MacW ill iams and N. J. A. Sloane , The Theory of Err or-Corr ect ing Codes . New Y ork: North-Holl and, 1998. [17] G. J. T ee, “Eigen v ectors of block circulant and alterna ting ci rculant matrice s, ” Res. Lett. Inf . Mat h. Sci. , vol. 8, 2005. [18] R. M. T ann er , “ A [155,64,20] sparse graph (LDPC) code, ” present ed at the Recent Resul ts Session at the IEEE Internati onal Symposium on Information Theory , Sorrento , Italy , June 2000.