Algebraic constructions of LDPC codes with no short cycles

Algebraic constructions of LDPC co des with no short cycles. T ed Hurley ∗ P aul McEv o y † Jakub W en us ‡ Abstract An algebraic group ring metho d for constructing co des with no short cycles in the c heck matrix is deriv ed. It is sho wn that the matrix of a group ring element has no short cycles if and only if the collection of group differences of this elemen t has no rep eats. When applied to elements in the group ring with small supp ort this giv es a general method for constructing and analysing low densit y parit y chec k (LDPC) co des with no short cycles from group rings. Examples of LDPC codes with no short cycles are constructed from group ring elements and these are simulated and compared with known LDPC co des, including those adopted for wireless standards. 1 In tro duction An LDPC co de is a co de where the chec k matrix has only a small num ber of non-zero elements in eac h row and column. These were introduced by Gallager in [1], expanded further b y T anner in [14], and rediscov ered [8, 9] and expanded on by MacKa y and Neal; details can now b e found in [6]. Structured LDPC co des usually use v arious types of com binatorial ob jects suc h as designs or algebraic geometry . LDPC codes ha v e often b een pro duced b y randomised techniques, but there has b een recent activit y in the area of algebraic constructions [13, 12, 15]. Having no short cycles in the (T anner) graph of the c heck matrix of an LDPC co de has b een shown to dramatically improv e the p erformance of the co de. Short cycles in an LDPC co de deteriorate the p erformance of the deco ding algorithms and having no short cycles may in effect increase the distances of such co des. Here a group ring method for the construction and analysis of LDPC codes with no short cycles is presen ted. 1.1 Group ring metho d Cyclic and related co des ow e muc h of their structure and properties because they o ccur as ideals or mo dules within a cyclic group ring. So also general group rings may b e used to construct, analyse and give structure to man y other types of co des and may b e used to construct codes of a particular t yp e or with a particular prop ert y . Group ring mo dule co des are obtained from either zero-divisors or units in a group ring RG as describ ed in [5] or [4]. Thus elements u, v ∈ RG are considered with either uv = 0 (zero divisors) or uv = 1 (units), where 1 denotes the identit y of RG , and co des are derived therefrom. The unit-deriv ed group ring co de method is particularly useful and has great flexibility while still retaining m uch of the algebraic structure. This metho d is employ ed here to directly and algebraically construct low densit y parit y chec k (LDPC) co des with no short cycles in their (T anner) graphs. Zero-divisors may also b e used but the unit-deriv ed metho d has adv antages and has less theoretical complications. ∗ National Univ ersity of Ireland, Galwa y , Ireland. ted.hurley@n uigalwa y .ie † T ec hnologyF romIdeas, Old Kilmeaden Road, W aterford, Ireland. paul.mcevo y@technologyfromideas.com ‡ T ec hnologyF romIdeas, Old Kilmeaden Road, W aterford, Ireland. jakub.wen us@technologyfromideas.com 1 Using an injection φ : RG → R n × n , as for example in [3], from the group ring RG with | G | = n into the ring of n × n matrices ov er R , corresp onding matrix co des are obtained from the group ring co des. In this injection the notation φ ( v ) = V is used so that the capital letter V in the matrix ring corresp onds to the lo wer case letter v in the group ring. Th us we are lead to consider u, v ∈ R G with uv = 1 = v u from which the codes are defined. Certain rows are chosen from the matrix U of the unit u to form the generator matrix of a code and the corresp onding columns are deleted from the matrix V of the inv erse v of u to form the chec k matrix of this code. If v has short supp ort then its corresponding matrix V has only a small num b er of elemen ts in eac h ro w and column and is thus low density . ‘Short support’ of a group ring element v means that only a small n umber (compared to the size of the group) of the co efficien ts in v are non-zero. If the chec k matrix of a group ring code is derived from a group ring c hec k element v with small supp ort then the resulting co de will b e an LDPC co de. It is determined here where precisely the short cycles can o ccur in the matrix of a group ring element. It is then easy to construct group ring elements, and group ring elements of small supp ort, which will hav e no short cycles anywhere in their matrices. Using group ring elements with small supp ort and with no short cycles in their matrices, LDPC co des with no short cycles are constructed. Th us LDPC codes with no short cycles can be constructed by the following algebraic group ring metho d: 1. Construct u, v is a group ring R G with uv = 1 and such that v has small supp ort compared to the size | G | of the group G . These can b e constructed with some property in mind. Units in group rings ab ound and are easy to construct. 2. Decide on the rate r | G | of the co de required. This is often decided b y reference to u, v and their structures. 3. Cho ose r rows of U with which to construct the generator matrix and delete the corresp onding r columns of V to form the c hec k matrix. This gives a rate r | G | co de. 4. If V has no short cycles at all, which can be ensured by Theorem 2.1 below, then an y choice of columns of V and consequen t choice of rows of U will give an LDPC co de with no short cycles. 5. If V has short cycles, it may be p ossible to a void these b y deleting appropriate columns and still obtain an LDPC co de with no short cycles. It is of ob viously easier to ensure there are no short cycles in the resulting LDPC co des if the original (c heck) element v from whic h the co de is constructed has no short cycles at all in its matrix V . This can b e ensured in the construction of v by Theorem 2.1 b elo w. Another adv antage of codes deriv ed from units is that there is a h uge choice of columns from which to form a chec k matrix of a code. F or example supp ose from a unit of size 1000 × 1000 with no short cycles and low density a (1000 , 500) co de is required. There are  1000 500  , whic h is of the order of 2 500 , c hoices from the 1000 × 1000 matrix with which to form the co de and each co de is low density and has no short cycles. F rom the nature of the indep endence of any set of rows (or columns) in a unit (= non-singular) matrix each co de deriv ed is different. One could thus envisage a hybrid whereby a random construction is p erformed within the parameters of an algebraic construction. 1.2 Examples and simulations Examples, simulations and comparisons are giv en in Section 3. These compare extremely well with existing LDPC co des and in many cases outstrip them. An example is also given of a unit from which 10 random LDPC co des of rate 1 / 2 are constructed. These are then simulated and all of them p erformed w ell. The 2 p oten tial applications of having random LDPC co des with no short cycles deriv ed from a single unit and all p erforming w ell are ob vious. In addition comparisons of Bit Error Rate (BER) and Block Error Rate (BLER) p erformance of LDPC codes defined in the 802.11n & 802.16e standard with equiv alent co des generated by the present metho d are given in Section 3.6. BER is not everything and often fast and pow er-efficien t co ding is more imp ortan t than p erformance. The group ring metho d for LDPC co des needs only a relativ ely few initial parameters and can re-create the matrix line-by-line without the need to store the whole structure in memory . The metho d has thus in addition applications where lo w p o wer and low storage are requirements. 1.3 Notation RG denotes the group ring of the group G ov er the ring R ; when R is a field, RG is often referred to as a gr oup algebr a . No deep knowledge of group rings is required but familiarity with the ideas of units, zer o- divisors in rings is assumed. F or further information on group rings see [11]. C n denotes the cyclic group of order n and H × K denotes the direct product of groups H , K . The words ‘graph’ and ‘short cycles’ is used but as no w explained no knowledge of graph theory is required and the problem of a v oiding short cycles reduces to lo oking at a prop erty of matrices. F or any matrix H = ( h ij ) the T anner graph [14] of H is a bipartite graph K = V 1 ∪ V 2 where V 1 has one v ertex for each ro w of H and V 2 has one vertex for column in H and there is an e dge b et ween tw o vertices i, j exactly when h ij 6 = 0. A short cycle in the (T anner) graph of a matrix is a cycle of length 4. Th us a matrix has no short cycles in its graph if and only the intersection of p ositions in whic h tw o columns hav e non-zero v alues is at most 1. This definition is used when considering the absence or otherwise of short cycles and th us no deep graph theory is inv olv ed. 2 Av oiding short cycles Av oiding short cycles in the (T anner) graph of the chec k matrix of a co de is important, particularly for LDPC (low density parity chec k) co des. Sp ecifically here, necessary and sufficien t conditions are given on the group ring element v in terms of the group elements with non-zero coefficients o ccurring in it so that its corresp onding matrix V has no short cycles. A mathematical pro of is giv en. Some sp ecial cases, suc h as when G is cyclic or ab elian, of the general result are easier to describ e and useful in practice and these are used as examples and illustrations of the general results. 2.1 Collections of differences, sp ecial case Collections of differences are usually defined with resp ect to a set a non-negative integers, see for example [2]. Collections of group differences are defined in Section 2.3 and the collections of (integer) differences are sp ecial cases of these when the group is a cyclic group. The integer definition is recapp ed here and used to giv e examples of the general definition. Let S = { i 1 , i 2 , . . . , i r } b e a set of non-negative unequal in tegers and n an integer with n > i j for all j = 1 , 2 , . . . , r . Then the c ol le ction of cyclic differ enc es of S mo d n is defined by DS ( n ) = { i j − i k mo d n | 1 ≤ j, k ≤ r , j 6 = k } . This collection has p ossibly repeated elemen ts. 3 F or example if S = { 1 , 3 , 7 , 8 } and n = 12 then DS (12) =                2 6 7 4 5 1 10 6 5 8 7 11                = { 2 , 6 , 7 , 4 , 5 , 1 , 10 , 6 , 5 , 8 , 7 , 11 } . In this case 6 , 7 , 5 o ccur twice. If | S | = r then counting rep eats | D S ( n ) | = r ( r − 1). 2.2 Cyclic group ring differences Consider the group ring R C n where C n is the cyclic group of order n generated by g . Supp ose u = α i 1 g i 1 + α i 2 g i 2 + . . . + α i r g i r ∈ RC n with α i j 6 = 0 (and 0 ≤ i j < n ). F or each g i , g j in u with non-zero co efficien ts form g i g − j , g j g − i and define D S ( u ) to be the collection of all such g i g − j , g j g − i . Set S = { i 1 , i 2 , . . . , i r } and define the collection of cyclic differences DS ( n ) as ab o v e. It is clear that D S ( n ) and D S ( u ) are equiv alent, the only difference b eing in the notation used. The pro of of the following theorem is a direct corollary of the more general Theorem 2.2 b elow. Theorem 2.1 U has no 4-cycles in its gr aph if and only if DS ( u ) has no r ep e ate d elements. 2.2.1 Example Set u = 1 + g + g 3 + g 7 in Z 2 C 15 . The collection of differences is formed from { 0 , 1 , 3 , 7 } and is th us D S ( u ) = { 1 , 3 , 7 , 2 , 6 , 4 , 14 , 12 , 8 , 13 , 9 , 11 } whic h has no rep eats. Hence the matrix formed from u , whic h is circulant in this case, has no short cycles. Set u = 1 + g + g 3 + g 7 in Z 2 C 13 . The collection of differences formed from { 0 , 1 , 3 , 7 } is { 1 , 3 , 7 , 2 , 6 , 4 , 12 , 10 , 6 , 11 , 7 , 9 } and has repeats 6 , 7. Th us the matrix formed from u has short cycles – but w e can identify where they o ccur. 2.3 Collection of differences in a general group ring Let RG denote the group ring of the group G ov er the ring R . Let G b e listed by G = { g 1 , g 2 , . . . , g n } . Let u = n X i =1 α i g i in RG . F or eac h (distinct) pair g i , g j o ccurring in u with non-zero coefficients, form the (group) differ enc es g i g − 1 j , g j g − 1 i . Then the c ol le ction of differ enc e of u , D S ( u ), consists of all suc h differences. Th us: D S ( u ) =  g i g − 1 j , g j g − 1 i | g i ∈ G, g j ∈ G, i 6 = j, α i 6 = 0 , α j 6 = 0  . Note that the collection of differences of u consists of group elements and for eac h g , h , g 6 = h , o ccurring with non-zero co efficien ts in u b oth gh − 1 and its inv erse hg − 1 are formed as part of the collection of differences. Theorem 2.2 The matrix U has no short cycles in its gr aph if and only if D S ( u ) has no r ep e ate d (gr oup) elements. Pro of: The rows of U corresp ond in order to ug i , i = 1 , . . . , n , see [5]. 4 Then U has a 4-cycle ⇐ ⇒ for some i 6 = j and some k 6 = l , the co efficien ts of g m , g l , in ug i and ug j are nonzero. ⇐ ⇒ ug i = ... + α g k + β g l + . . . and ug j = ... + α 1 g k + β 1 g l + . . . ⇐ ⇒ u = ... + αg k g − 1 i + β g l g − 1 i + . . . and u = ... + α 1 g k g − 1 j + β 1 g l g − 1 j + . . . . ⇐ ⇒ D S ( u ) contains b oth g k g − 1 i g i g − 1 l = g k g − 1 l and g k g − 1 j g − 1 l = g k g − 1 l . This happ ens if and only if D S ( u ) has a rep eated element.  2.4 Rep eated elements Supp ose no w u is such that D S ( u ) has rep eated elements. Hence u = ... + α m g m + α r g r + α p g p + α q g q + ... , where the display ed α i are not zero, so that g m g − 1 r = g p g − 1 q . The elements causing a short cycle are displa y ed and note that the elements g m , g r , g p , g q are not necessarily in the order of the listing of G . Since we are interested in the graph of the elemen t and th us in the non-zero co efficien ts, replace a non-zero co efficien t b y the co efficien t 1. Th us write u = ... + g m + g r + g p + g q + ... so that g m g − 1 r = g p g − 1 q . Include the case where one p, q could b e one of m, r in which case it should not b e listed in the expression for u . Then ug − 1 m g p = .. + g p + g r g − 1 m g p . + ... = ... + g p + g q + .. and ug − 1 p g m = .... + g m + g q g − 1 p g m = ... + g m + g r + ... . (Note that ug − 1 m g p = ug − 1 r g q and ug − 1 p g m = ug − 1 q g r ) Th us to av oid short cycles, do not use the row determined by g − 1 m g p or the row determined by g − 1 p g m in U if using the first row or in general if g i ro w o ccurs then g i g − 1 m g p , and g i g − 1 p g m ro ws must not o ccur. Similarly when DS ( u ) has rep eated elements by av oiding certain columns in U , it is p ossible to finish up with a matrix without short cycles. 2.5 Sp ecial group cases The sp ecial case when G = C n w as dealt with in Section 2.2. Let G = C n × C m b e the direct pro duct of cyclic groups C n , C m generated by g , h resp ectiv ely . These groups are particularly useful in practice. List the elements of G by { 1 , g , g 2 , . . . , g n − 1 , h, hg , hg 2 , . . . , hg n − 1 , . . . , h m − 1 , h m − 1 g , . . . , h m − 1 g n − 1 } . Then ev ery element in RG is of the form u = a 0 + ha 1 + . . . + h m − 1 a m − 1 with eac h a i ∈ C n . The collection of differences of u is easy to determine and elements with no rep eats in their collection of differences are thus easy to construct. Relativ e to this listing the matrix of an element in RG is a circulant-b y-circulan t matrix of size mn × mn , 5 [3]. Another particularly useful group which is relativ ely easy to w ork with is the dihedral group D 2 n giv en b y h a, b | a 2 = 1 , b n = 1 , ab = b − 1 a i , [11]. This group is non-ab elian for n ≥ 3. Every element u in RD 2 n ma y b e written as u = f ( b ) + ag ( b ) with f ( b ) , g ( b ) ∈ RC n where C n is generated b y b . The collection of differences of u is easy to determine. The corresp onding matrix U of u has the form  A B B A  where A is circulan t and B is reverse circulan t, [3]. This gives non-comm utativ e matrices and non-comm utativ e co des. 3 Examples, sim ulations and comparisons In this section examples and simulations of the metho d are giv en and some comparisons are made with kno wn co des. The sizes of the examples are chosen in order to compare with kno wn examples. Ho w ever there is no theoretical limit on size, the constructions are easy to p erform and there is complete freedom as to choice of ro ws or columns to delete from a particular unit in order obtain LDPC co des with no short cycles. The simulations compare v ery fav ourably with kno wn examples and in some cases outstrip these. This algebraic metho d for construction has other adv antages such as for applications where low storage and low p o w er are requiremen ts. The co de may b e stored by an algebraic form ula with few parameteres and the chec k matrix restored as required line-by-line without the need to store the whole structure in memory . 3.1 The examples generally In general the examples are taken from unit-deriv ed co des within Z 2 ( C n × C 4 ), where Z 2 = F 2 is the field of t wo elements. The matrices derived are then submatrices of circulan t-b y-circulant matrices and are easy to program. Assume that C n is generated by g and C 4 is generated by h . Ev ery elemen t in the group ring is then of the form: n − 1 X i =0 ( α i g i + hβ i g i + h 2 γ i g i + h 3 δ i g i ), with α i , β i , γ i , δ i ∈ Z 2 . 3.2 (96,48) examples These are derived from Z 2 ( C 24 × C 4 ). The chec k element v = g 24 − 9 + g 24 − 15 + g 24 − 19 + hg 24 − 3 + hg 24 − 20 + h 2 g 24 − 22 + h 3 g 24 − 22 + h 3 g 24 − 12 is used to define the LDPC co de TFI-96-59-8. It is easy to c hec k from Theorem 2.2 that v has no short cycles in its matrix V . A pattern to delete half the columns from the matrix V of v is chosen to pro duce the rate 1 / 2 co de TFI-96-59-8 . TFI-96-59-8 is compared to pseudo-random co de MK-96-33-964 (size=96, rate=1/2) of MacKa y [7]. 6 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 S/N (dB) BER (AWGN BPSK) BER TFI-96-59-8 BER MK-96-33-964 3.3 Random selection 7 F or the abov e, 10 random LDPC (96 , 49) co des were taken from the unit v in Section 3.2 and simulated. The sim ulation of TFI-96-59-8, used in the previous graph where it is compared to MK-96-33-964, is included for comparison. 3.4 (504,252) example The next example is deriv ed from Z 2 ( C 126 × C 4 ). v = g 126 − 10 + g 126 − 99 + hg 126 − 47 + h 2 ( g 126 − 15 + g 126 − 25 + g 126 − 81 ) + h 3 ( g 126 − 6 + g 126 − 23 + g 126 − 64 ). Sp ecific column deletions are chosen from V to give the LDPC rate 1 / 2 co de TFI-504-91-0. The per- formance of TFI-504-91-18 is compared to that of PEGReg252x504 Progressiv e Edge Growth, Xiao-Y u Hu, IBM Zurich Research Labs. 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 S/N (dB) BER (AWGN BPSK) BER TFI-504-91-18 BER PEGReg252x504 3.5 Size: 816; rate =1/2 and 3/4 Here Z 2 ( C 204 × C 4 ) is used. Set v = g 204 − 75 + h ( g 204 − 13 + g 204 − 111 + g 204 − 168 ) + h 2 ( g 204 − 29 + g 204 − 34 + g 204 − 170 ) + h 3 ( g 204 − 27 + g 204 − 180 ). Half the columns of V are deleted in a sp ecific manner to get the TFI-816-0p5-29-4 rate 1/2 code. The same v is taken and sp ecific three quarters of the columns of V are deleted to get the 3 / 4 rate (816 , 612) LDPC code TFI-816-0p75-29-4. In the first graph the p erformances of TFI-816-0p5-29-4 and TFI-816-0p75-29-4 are compared. In the second graph the p erformances of TFI-816-0p5-29-4 and MK-816-55-156, a pseudo-random rate 1 / 2 co de due to MacKay , [7], are compared. 8 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 S/N (dB) BER (AWGN BPSK) BER TFI-816-0p5-29-4 BER TFI-816-0p75-29-4 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 S/N (dB) BER (AWGN BPSK) BER TFI-816-0p5-29-4 BER MK-816-55-156 9 3.6 Industry Standards Here comparisons of Bit Error Rate (BER) and Blo c k Error Rate (BLER) p erformance of LDPC co des defined in the 802.11n & 802.16e standard with equiv alent co des generated by presen t metho d are given. 3.6.1 Case 1, 802.11n: Co d e s i ze = 6 4 8 ; Co d e r a t e = 2 / 3 Ga u ssi a n Ch a n n e l w i t h QA M 6 4 m o d u l a t i o n ( Ba n d w id t h = 4 0 M H z, B i t R a t e = 3 0 0 M b i t / s ) 1 .E - 0 7 1 .E - 0 6 1 .E - 0 5 1 .E - 0 4 1 .E - 0 3 1 .E - 0 2 1 .E - 0 1 1 .E + 0 0 0 2 4 6 8 1 0 1 2 Eb / N 0 ( d B) BER / BLER - BE R 2 1 6 x6 48 IEE E BLE R 2 1 6 x 6 4 8 IEE E BE R 2 1 6 x6 48 TFI BLE R 2 1 6 x 6 4 8 T F I Co d e s iz e = 6 4 8 ; Co d e r a t e = 2 / 3 G a u ssi a n Ch a n n e l w i t h QA M 6 4 m o d u l a t i o n ( Ba n d w i d t h = 4 0 M H z, Bi t Ra t e = 3 0 0 M b i t / s) 1 1 0 1 0 0 0 2 4 6 8 1 0 1 2 Eb / N 0 ( d B) Av e r a g e n o . o f i t e r a t i on s - I T E R 2 1 6 x 6 4 8 I E EE I T E R 2 1 6 x 6 4 8 T F I Matrix size: 216 by 648; Co de size = 648; Co de rate = 2/3. Matrix structure: The last 189 columns con tain a ‘staircase’ structure whic h is identical as in the IEEE matrix. The remaining part was generated using the group ring algebraic algorithm which tak es 15 initial parameters as input. 10 3.6.2 Case 2, (802.11n): Co d e s i ze = 1 2 9 6 ; Co d e r a t e = 3 / 4 G a u s si a n Ch a n n e l w i t h QA M 6 4 m o d u l a t i o n ( Ba n d w id t h = 4 0 M H z , Bi t Ra t e = 3 0 0 M b i t / s) 1 .E - 0 7 1 .E - 0 6 1 .E - 0 5 1 .E - 0 4 1 .E - 0 3 1 .E - 0 2 1 .E - 0 1 1 .E + 0 0 0 2 4 6 8 1 0 1 2 Eb / N 0 ( d B) BER / BLER - BE R 3 2 4 x 1 2 9 6 I E EE BLE R 3 2 4 x1 2 9 6 IEE E BE R 3 2 4 x 1 2 9 6 T F I BLE R 3 2 4 x1 2 9 6 TFI Co d e s iz e = 1 2 9 6 ; Co d e r a t e = 3 / 4 Ga u ssi a n Ch a n n e l w i t h QA M 6 4 m o d u l a t i o n ( Ba n d w i d t h = 4 0 M H z, Bi t Ra t e = 3 0 0 M b i t /s) 1 1 0 1 0 0 0 2 4 6 8 1 0 1 2 Eb / N 0 ( d B) Av e r a g e n o. of i t e r a t i on s - I T E R 3 2 4 x 1 2 9 6 I E EE I T E R 3 2 4 x 1 2 9 6 T FI Matrix size: 324 by 1296; Co de size = 1296; Code rate = 3/4. Matrix structure: The last 270 columns con tain a ‘staircase’ structure whic h is identical as in the IEEE matrix. The remaining part was generated using the algebraic group ring algorithm which tak es 17 initial parameters as input. 11 3.6.3 Case 3, (802.16e): Co d e s i ze = 1 1 5 2 ; Co d e r a t e = 2 / 3 G a u ss i a n C h a n n e l w i t h QA M 6 4 m o d u l a t i o n ( Ba n d w i d t h = 4 0 M H z, Bi t R a t e = 3 0 0 M b i t / s) 1 .E - 0 7 1 .E - 0 6 1 .E - 0 5 1 .E - 0 4 1 .E - 0 3 1 .E - 0 2 1 .E - 0 1 1 .E + 0 0 0 2 4 6 8 1 0 1 2 Eb / N 0 ( d B) BER / BLER - BE R 3 8 4 x1 1 5 2 I E E E BLE R 3 8 4 x1 1 5 2 IEE E BE R 3 8 4 x1 1 5 2 T F I BLE R 3 8 4 x1 1 5 2 TFI Co d e s i ze = 1 1 5 2 ; Co d e r a t e = 2 / 3 Ga u s si a n Ch a n n e l w i t h QA M 6 4 m o d u l a t i o n ( Ba n d w i d t h = 4 0 M H z , Bi t Ra t e = 3 0 0 M b i t / s ) 1 1 0 1 0 0 0 2 4 6 8 1 0 1 2 Eb / N 0 ( d B) Av e r a g e n o . o f i t e r a t i o n s - I T E R 3 8 4 x1 1 5 2 I E E E I T E R 3 8 4 x1 1 5 2 T F I Matrix size: 384 by 1152; Co de size = 1152; Code rate = 2/3. Matrix structure: The last 336 columns con tain a ‘staircase’ structure whic h is identical as in the IEEE matrix. The remaining part was generated using the algebraic group ring algorithm which tak es 17 initial parameters as input. References [1] R.G. Gallager, L ow-density p arity-che ck c o des , MIT Monograph, 1963. [2] v an Lint, J.H. and Wilson, R.M., A c ourse in Combinatorics , Cambridge Univ ersit y Press, 2001. [3] T ed Hurley , “Group rings and rings of matrices”, In ter. J. Pure & Appl. Math., 31, no.3, 2006, 319-335. [4] Paul Hurley & T ed Hurley , “Mo dule co des in group rings”; ISIT2007, Nice, 1981-1985, 2007. [5] T ed Hurley , “Co des from zero-divisors and units in Group Rings”, arXiv: 0710:5873. [6] David MacKay , Information The ory , CUP , 2005. [7] http://www.inference.ph y .cam.ac.uk/mac k ay/codes/data.html [8] D.J.C. MacKay & R.M. 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