and - 上海交通大学数学系math.sjtu.edu.cn/conference/8shcc/ppt/blake.pdf · 8 2 = exp (+ 1)...

LDPC Codes and Finite Geometries

Ian F. Blake (UBC)(joint with Qiuju Dao (LSI Logi ) and Shu Lin (UC Davis))May 20148JCC, Shanghai, May 2014

Purpose of talk

Comments on the uses of �nite geometries inthe onstru tion of LDPC ode to a hieve lowprobability of error �oors

8JCC, Shanghai, May 2014 1

Outline

I. The elements of LDPC oding for low error �oorsII. Classi al �nite geometries and odingIII. Partial geometries and generalized quadrangles


I. The elements of LDPC oding for low error �oors


Preliminaries

I using hard de isions at the re eiver in urs a ostof 2dB of Eb=N0 - annot a�ord this - ) soft de isionsI using soft de isions ) annot use algebrai de oding algsI must use soft de ision algorithms and try for ML performan eI ) message passing algorithms - an approximationI su h "real number" algorithms a dynami al system) onvergen e questionsI trapping sets


Model of a digital ommuni ation system

+n(t) psd N0=2fsi (t); i = 1; : : : ;Mg î

I Eb=N0 = SNR (power ratio) = (Eb=T )=(N0W )Eb = energy in transmitted signal per bit over (0; T )I Eb=N0dB = 10(log10(Eb=N0))I 3dB = doubling of power/energy per bitI many parameters of interest, power, bandwidth baud rate,modulation, signal shaping, : : :


Hard vs soft de isions

Re eived sequen e: r = (r1; r2; : : : ; rn)

` I Hard de ision: r 2 Fn2, Soft de ision: r 2 RnI If ri independent, hard de ision okayIf ri orrelated (as in oding), need soft de ision


Shannon oding theorem - informalTheorem (Shannon): For a dis rete memoryless hannel, there existsa quantity, apa ity, C, su h that ommuni ation with arbitrarily lowerror probability is possible for any rate R less than apa ity and thisis not possible at rates above apa ity. In parti ular, there exists afun tion Er(R) su h thatPe < exp(NEr(R); R < C; Er(R) � 0; 0 � R < C:Er(R)

RC1


Capa ity - AWGN hannel

AWGN: CAWGN = W log(1 + PN0W ) bps

BIAWGN: CBIAWGN = � ∫ ��(x) log2 ��(x)dx � 12 log2(2��2)where ��(x) � p8��2 = exp(�(x + 1)2=2�2+exp(�(x � 1)2=2�2:� a fun tion of Eb=N0.


Hard vs softEb=N0

C0

1soft hard2dB

PeEb=Nosoft hard2dB

ShannonI for a given rate ode - Shannon limit Eb=N0I oding to a hieve apa ity8JCC, Shanghai, May 2014 9

Error orre ting odes - binary

I (n; k) ode - k � n generator matrix, (n � k)� n parity he k matrixI 2k odewords, rate = k=n,I De oding: for a given re eived word r �nd losest odeword xI Hard de ision - algebrai de oding algorithms -in Fn2I Soft de ision - approximations to maximum likelihood (ML) - in Rn- minimizes Pe for equally likely inputs- ML algorithms too omputationally omplexI algebrai soft de ision algorithms known but not e�e tive?ML de oding: ^x = argmaxx2C P (x j r)


Maximum likelihood de oding - the problem

re eived r = (r1; r2; : : : ; rn) 2 Rn �nd argmaxx2C P (x j r)I Sear hing all 2k possibilities in Rn very omplexI We annot a�ord to give up 2-dB ) soft de isionI For good performan e we have to ome lose to MLI Real number algorithms - dynami systemsI Gallager understood this problem in the early 1960's !!!The answer: message passing algorithms on graphs


Gallager A de oding algorithm - hard de ision

I alphabet f�1; 1gI initially, re eived symbols VN's transferred to he k nodesI he k node j sends to variable node vi (if onne ted)the parity (produ t) of all re eived symbols ex ept from viI a VN sends to neighbor CN the original symbol unless allin oming symbols were the same - unanimous agreementI exa t analysis available - assuming independen e of in omingsymbols - indi ates gap to apa ity for biregular odesI Gallager B - variable thresholds


Belief propagation (BP)/message passing algorithms

I for soft de ision (BIAWGN) hannelsI an "approximation" to ML - although often far from itI omplexity of algorithm number of edges in graph= number of ones in parity he k matrixI onvergen e of algorithms, number of iterations, performan ewhen to stop?


Belief propagation algorithm equations

Important prin iple: message sent from VM v to CN must not take intoa ount message sent from to v in previous round - and for CN's to VN'sThe equations: log likelihood ratios:L(x j y) = P r(x = 0 j y)=P r(x = 1 j y); ì = lnL(xi j yi)ì =1) xi = 0m(`)v = { mv ` = 0mv + ∑ 02Cvn m`�1 0v ` � 1m` v = ln 1 + ∏v 0inV nv tanh(m`v 0 =2)1�∏v 0inV nv tanh(m`v 0 =2)


Belief propagation algorithm diagram

v ... ......

... m(`�1) vm(`)v I iterations build up belief - +1) x = 0


The error �oor problem

I a dynami al system over the realsI omplexity proportional to no. edges in graph

PeEb=N0

error �oor


Trapping sets

1. A (�; �) trapping set for the ode de�ned by the Tanner graph G is asubset � of VN's su h that the subgraph of G indu ed by the set �,denoted by G[�℄, has exa tly � odd degree CN's (and an arbitrary numberof even degree CN's).2. The trapping set is said to be elementary if all the asso iated he k nodesin G[�℄ have degree 1 or 2.3. The trapping set is alled small if � � pn (n is the ode length) and�=� � 4.8JCC, Shanghai, May 2014 17

Trapping sets

In essen e, if there is a small trapping set, there is a relativelyhigher probability the BP algorithm will fail to onverge asit iterates. The updates " ontradi t" or os illate as thealgorithm iterates.


LDPC odes - summing up

A good ode should have:I good minimum distan e for its rateI be e� ient to en odeI no small trapping sets for good error �oorI have sparse parity he k matrix for algorithm e� ien yI girth � 6?I performan e lose to the Shannon limit - about 1dB at 10�6?


II. Classi al �nite geometries and oding


The geometri odes

I a variety of related odes studied extensivelyI these in lude Reed-Muller (RM) odes, Generalized RM, proje tive RMEu lidean geometry odes, proje tive geometry odesI several ways of de�ning these odes:generator polynomial, geometri , multivariable (evaluation)I the lass of polynomial odes (Kasami, Lin and Peterson, 1968) in ludesall these lasses of odesI the book of Assmus and Key


Generalized Reed-Muller (GRM) odes - y li over FqI a natural generalization of the binary RM odesI Constru t (m + 1)� (qm � 1) array over Fqzeroth row all ones - rows i asso with xi ; i = 1; : : : ; m olumns = all nonzero m-tuples over Fq; u0; u1; : : : ; uqm�1"pun tured" GRM ode GRM� - drop the zero ol.I Pr(m; q) = all polys degree � r in m variables over FqI the "evaluation" ode: f = (f (u1) : : : ; f (uqn�1)); ui 2 FqGRM�q(r;m) = f f j f 2 Pr(m; q)g


Properties of GRM odes

For r < q n = qm � 1; k = (r +mm ); d = (q � r)q(m�1) � 1;where wq(j) = ∑i ji ; j = ∑i jiqi ; 0 � ji < q:Theorem: � 2 Fqm primitive. GRM�(r;m) y li with �j a root i�fj j 0 < j < qm � 1; 0 < wq(j) � m(q � 1)� r � 1g:(relates single variable to multivariable approa hes)


Proje tive geometry odes - PG(m; q)

De�nition 1: A q-ary r -th order proje tive geometry ode is the largest linear ode PG(q;m) with all r -�ats in its dual spa e.De�nition 2: A q-ary r -th order y li proje tive geometry ode of length�m;q = (qm+1 � 1)=(q � 1) has de�ning set of roots for the dual ode�h(q�1); � primitive 2 Fqm+10 < min0�`<mwq(p`h(q � 1)) � �(q � 1)

Theorem (Hamada): The two de�nitions are equivalent.


Parameters of the proje tive geometry odes - PG(m; q)

n = (m+1�1)=(q � 1); k = n � Rr(m; q); d � (qm�r�1 � 1)=(q � 1) + 1

where Rr(m; q) is the rank of r -�ats in PG(m:q). For the dual ode

d = (qr+1 � 1)=(q � 1)


Eu lidean geometry odes

De�nition 1: A q-ary r -th order Eu lidean geometry ode is the largest linear ode EG(q;m) with all r -�ats not going through the origin in its dual spa e.De�nition 2: A q-ary r -th order Eu lidean geometry ode of length qm is anextended y li ode with a de�ning set of roots for the dual ode�h; � 2 Fqm 0 < min0�`<mwq(p`h) < �(q � 1)Theorem (Hamada): The two de�nitions are equivalent.


Properties of EG odes

An r -th order q-ary EG ode of r -�ats in EG(m; q) in the dual ode is y li with parameters:n = qm � 1; k = n � fRr(m; q)� Rr(m � 1; q)g; d � qm�r + pqm�r�1

and the dual ode has minimum distan e d = qr � 1.I It is noted the GRM odes over prime �elds are also geometri odes.I expressions for Rr(m; q)? Hamada


Hamada's theorem

Theorem (Hamada): The q-rank of the in iden e matrix of pointsand r -�ats in PG(m; q) is equal to :

Rr(m; q) = ∑fs0;:::;smgm�1∏j=0 L(sj+1;sj)

∑i=0 (�1)i(m + 1i )(m + sj+1p � sj � ipm )

where q = ps and summation is taken over all ordered sets(s0; s1; : : : ; sm) denoted by Sm;r(pm) of (m + 1) integers s`; ` =0; 1; : : : ;, su h thatsm = s0; 0 � sj � m�r; and 0 � sj+1p�sj � (m+1)(p�1); j = 0; 1; : : : ; s�1:and L(sj+1; sj) = bsj+1p�sjp :8JCC, Shanghai, May 2014 28

Spe ial ases of Hamada's theorem

I Hamada's formula di� ult to gain insight fromI some parti ular ases of points and �ats wherethe formula is "simple" are summarizedI in many ases the formulae are of use in odingI let d(r;m; q) = Rr(m; q) be the p-rank, q = ph of points andr -�ats in PG(m; q) and d�(r;m; q) the rank of the dual ode8JCC, Shanghai, May 2014 29

Spe ial ases of Hamada's theorem - ontd.1d�(1; m; p) = (m+p�1p�1 )d�(m � 1; m; q) = qm+1�1q�1 � (m+p�1m )h � 1d�(2; m; p) = (m+2p�2m )� (p � 2)(m+p�2m�1 )d�(3; m; p) = (m+3p�3m )� (2p � 3)(m+2p�3m�1 ) + (p�22 )(m+p�3m�2 )

1Ce herini and Hirs hfeld, 1992, Hirs hfeld and Shaw, 19948JCC, Shanghai, May 2014 30

Spe ial ases of Hamada's theorem - ontd.2d�(r;m; 2) = ∑ri=1 (m+1i )d�(r;m; p) = pm+1�1p�1 �∑m�rs=0 ∑[s(p�1)=p℄i=0 (�1)i(m+1i )(m+s(p�1)�ipm )(S)d�(r;m; p) = ∑r�1j=0(�1)j((r�j)(p�1)�1j )(m+(r�j)p�rm�j ) (HS)

Note impli ations for d�(1; m; p)Hamada: rank of all r -�ats in EG(m; q) is = Rr(m; q)� Rr(m � 1; q)2Ce herini and Hirs hfeld, 1992, Hirs hfeld and Shaw, 19948JCC, Shanghai, May 2014 31

Spe ial ases of Hamada's theorem - Hirs hfeld, Shaw,van Lint

pm+1 � 1p � 1 �m�1

∑s=0 [s(p�1)=p℄∑i=0 (�1)i(m + 1i )(m + s(p � 1)� ipm ) = (m + p � 1p � 1 )


Two results in Eu lidean geometries

1) Key and Ma kenzie, 1991: The rank of m-�ats in EG(2m; p):m�1∑i=0 (�1)i(2mi )(m + (m � i)p2m )

2) Sin, DCC 2004: V a ve tor spa e, dimension n + 1 over Fpt In iden ematrix of e �ats and d-�ats in V where "in iden e" means nonzerointerse tion formula given for the p-rank of the in iden e matrix generalizesformula of Hamada Other "simple" formulae?8JCC, Shanghai, May 2014 33

III. Partial geometries and generalized quadrangles


Partial geometries

I in iden e stru tures of points and lines (in l. EG, PG)I a set N of n points and a set M of mI in iden e relations - parallel lines et I the geometri aspe t is helpful to analyze trapping sets


Partial geometries - ontd

A partial geometry PaG( ; �; Æ) if for � � 2; � 2 and Æ � 1:1. any two points are on at most one line;2. ea h point is on lines;3. Ea h line passes through � points;4. If a point v is not on a line L, then there areexa tly Æ lines, ea h passing through v and a point on L.5. When Æ = � 1, a net


Partial geometries - ontd

I n = (�(�� 1)( � 1) + Æ)=Æ pointsI m = ((�� 1)( � 1) + Æ)=Æ linesProfessor Lin will talk on these odes


Example! " # $ % & '( '' ') '* '!

'(!'"

'(!'!

'(!')

'(!'(

'(!%

'(!#

'(!!

'(!)

'((

+'#')&,'"*$)-./!012/34567869:;<5

=>?,@A

+'#')&,'"*$)-,AB?,CD6"(

+'#')&,'"*$)-,A0B?,CD6"(

+'#')&,'"*$)-,AB?,CD6'(

+'#')&,'"*$)-,A0B?,CD6'(

+'#')&,'"*$)-,AB?,CD6"

+'#')&,'"*$)-,A0B?,CD6"

=E:;;8;30C9CD

F;<8@5@3A2=G

Figure 1: Performan e urves for the (16129,15372)quasi- y li PaG8JCC, Shanghai, May 2014 38

Trapping sets for PaG odes

Theorem:3 If G is the Tanner graph of an LDPC ode with girth at least 6,with V N 0s of degree , and G ontains a (�; �) trapping set with � < then

� � ( + 1� �)�:

(Thus, if + 1� � > 4, not a small trapping set)3Diao et al, IT Trans, 20138JCC, Shanghai, May 2014 39

Trapping sets for PaG odes

I geometri stru ture allows interpretation of trapping setsI VN's are points, CN's linesI � � V N 0s a (�; �) trapping set )� lines passing through odd number points in �I mi = number of lines (CN's) passing through ipoints of � (number of CN 0s of degree i)


Trapping sets for PaG odes - ontd.

Theorem : Let GPaG be the Tanner graph of a PaG( ; �; Æ).If � is a (�; �) trapping set and � � then:

� � ( + 1� �)�+ ∑i odd(i � 1)2mi + ∑i even i(i � 2)mi

and equality holds if Æ = � and the sums go to 2b(�+1)=2 �1and 2b�=2 , respe tively.(Note that the �rst term on the right hand side of thisexpression is the general bound noted earlier.)


Trapping sets for PaG odes - ontd.

In the parti ular ase of a net the bound an be improved somewhat. LetL1; L2; : : : ; L� be a set of parallel lines in PaG( ; �; � 1) and for a set � ofV N 0s let �i =j � \ Li j. Then:Theorem: If the set � of V N 0s of a net is a (�; �) trapping set, then

� � ( � 1)�� 2 + �∑i=1 �2i + j f` : 1 � ` � �; �` oddg j :

(Agrees with previous bound if whenever �i � 2; 8i - better in other ases)


Codes from generalized quadrangles

I spe ial ase of partial geometries Æ = 1 i.e. for v =2 L thereis a unique line point v 0 2 L and line through v ; v 0I graph has girth 8- better performan e?I a deeply studied subje t - books by Thas (2), Payne and van MaldeghemI GQ odes - the dual spa e of the in iden e matrix of a GQI a subset of generalized polygonsI results on trapping sets for PaG's appli able hereI studied for odes for over 30 years



I good survey on GQ odes by Liu and Pados (IT Trans, 2005)I stronger stru ture but more limited sets of parametersI girth 8 - bounds or exa t expressions fro dimension/ distan eI in iden e matrix an be made quasi- y li - blo ks of ir ulants- useful for en odingI di�erent size ir ulants for di�erent ordersI limited simulation - so far poor results - short odes



I GQ of order (s; t) if ea h line is in ident with(s + 1) points and ea h ea h point is in ident with (t + 1) linesI only orders (q; q); (q; q2); (q2; q); (q2; q3); (q3; q2)and (q � 1; q + 1) and (q + 1; q � 1) (duals) knownI point set N and line set M then:j N j= (s + 1)(st)2 � 1st � 1 ; jM j= (t + 1)(st)2 � 1st � 1


Trapping sets in Codes from GQ's

Case i): An ovoid of the generalized quadrangle is a set of points O su h thatevery line in the quadrangle is in ident with a unique point of O. Su h ovoids,if they exist for a parti ular onstru tion, must have 1 + st points. Suppose aset of � � 1 + st VN's has the property that ea h line is in ident with atmost one point of S. The CN's orresponding to these VN's then is just theset of �(1 + t) lines that interse t the � VN's. Ea h su h CN has degree 1.Sin e it is assumed t will be quite large and, in parti ular t > 3, the ratio�=� = 1 + t > 4 and su h a set of VN's annot be a small trapping set.


Trapping sets in Codes from GQ's - ontd

Case ii) A spread S is a set of lines of the GQ su h that ea h point of thequadrangle is in ident with a unique line in S i.e. the lines of S are parallel.By duality if a spread exists it must have 1 + st lines. Suppose the � VN'sare ollinear. The orresponding CN's in the subgraph generated by su h a on�guration will have � = t� CN's and for large t does not orrespond to asmall trapping set. Again ea h CN has degree 1.


Trapping sets in Codes from GQ's

I the above suggests the geometri stru ture of GQ's usefuland an be exploited further for trapping set sizeI Question: For a given � of VN's, what is thesmallest � of CN's possible?I eviden e of usefulness of girth 8?


Summary of talk

I some understanding of stru ture of LDPC odes forlow error �oorI �nite geometries seem to give odes - good minimum distan evs rate, girth � 6, useful hara terization of trapping sets,e� ient en odings/quasi- y li I an analyti relationship between trapping set size anderror �oor would be very useful


Thank you for your attention8JCC, Shanghai, May 2014 50

Date post:	20-Aug-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

and - 上海交通大学数学系math.sjtu.edu.cn/conference/8shcc/ppt/blake.pdf · 8 2 = exp (+ 1)...

Documents