Michael. LDPC-коды основанные на квази-циклических блочных кодах

1 Pages·0.069 MB·Russian

by  Tanner R.

ISIT2002,Lausanne,Switzerland,June30{July5,2002
A Construction for Low Density Parity Check Convolutional
Codes Based on Quasi-Cyclic Block Codes
Arvind Sridharan, Daniel J. Costello Jr., R. Michael Tanner
Deepak Sridhara, Thomas E. Fuja1 Departmentof Computer Science
Universityof Notre Dame Universityof California, Santa Cruz
IN46556, U.S.A. SantaCruz, CA 95064, U.S.A.
fasridhar, costello.2, dsridhar, [email protected] [email protected]
Abstract | A set of convolutional codes with low Performance of rate 2/5 LDPC convolutional codes with BP
100
density parity check matrices is derived from a class [1055,424] QC code, 50 iters.
R = 2/5 conv code, 15 iters. ([1055,424])
of quasi-cyclic low density parity check block codes. 10−1 R[2 1=0 25/,58 4c4o]n Qv Cco cdoed, e5,05 0it eitresr.s (.[1055,424])
Their performance when decoded using the belief R = 2/5 conv code, 15 iters. ([2105,844])
R = 2/5 conv code, 50 iters. ([2105,844])
propagation algorithm is investigated. 10−2
I. Introduction
Tbaasnenderondpeseirgmneudtataion[15m5a,6t4ri,c2e0s] [1sp].arTsehisgrcaopnhstr(uLcDtiPonC)cacnobdee Bit error rate1100−−43
generalized togiveaclass of sparse graph (LDPC)codes [2],
wellsuitedtodecodingwiththebeliefpropagation(BP)algo- 10−5
rithm. These LDPC codes are quasi-cyclic and hence admit
a convolutional representation, obtained by unwrapping the 10−6
Threshold limit for (3,5) LDPCs Cut−off rate limit
quasi-cyclic block code.
10−7
0.5 0.965 1 1.5 2 2.19 2.5 3
II. Code Construction Eb/No
The quasi-cyclic codes constructed in [2] are (j;k) regular
LDPC codes, where j and k are among the prime factors of quasi-cycliccode,butwiththeperiodofthequasi-cycliccode
m(cid:0)1, m a prime. Their parity check matrices consist of extended to in(cid:12)nity. Hence, they are equally well suited to
blocks of circularly shifted identity matrices. Each circulant decodingwiththeBPalgorithm. Further,asin[4],thecodes
matrix can equivalently be described by a polynomial. The can be decoded in a continuous fashion, so that after an ini-
corresponding convolutional code is obtained by interpreting tial delay decoding results are output continuously, an ad-
the block code’s polynomial-form parity check matrix as the vantage derived from using the convolutional representation.
analogous convolutional code construct, with the change in The above (cid:12)gure shows simulation results obtained for rate
thepolynomials’ indeterminateto’D’asistheconventionfor R = 2=5 convolutional codes and the corresponding block
convolutional codes [3]. Thus, LDPC convolutional codes de- codes, with j =3, k =5. The (3;5) block LDPC codes were
scribed byj(cid:2)k paritycheckmatrices of the form constructed by choosing primes, m = 211 and m = 421 re-
spectively, from which the convolutional codes were obtained
1 Da(cid:0)1 ::: Dak(cid:0)1(cid:0)1 asdescribed. Theconvolutionalcodesshowgoodperformance
2 Db(cid:0)1 Dab(cid:0)1 ::: Dak(cid:0)1b(cid:0)1 3 beyondthecut-o(cid:11)ratelimitwithBPdecoding. Interestingly,
H(D)=
6 ::: ::: ::: ::: 7 they outperform their block code counterparts, which is pos-
64 Dbj(cid:0)1(cid:0)1 Dabj(cid:0)1(cid:0)1 ::: Dak(cid:0)1bj(cid:0)1(cid:0)1 75 siblyduetothehigherfreedistanceoftheconvolutionalcode.
(j(cid:2)k) Moreover, the sparse graph natureof these algebraically con-
are obtained. Here a and b are non-zero elements of GF(m) structed codes makes them well suited for high speed VLSI
with multiplicative orders k and j respectively, and all implementations.
powers are taken modulo m, i.e., by Dapbq(cid:0)1, we mean References
D(apbq(cid:0)1) mod m. [1] R.M.Tanner,\A[155,64,20]sparsegraph(LDPC)code."Pre-
sented at the recent results session, IEEE Intl. Symposium on
III. Decoding and Simulation Results
InformationTheory,Sorrento,Italy,June2000.
Therate R=1(cid:0)j=k LDPCconvolutional codes constructed [2] D. Sridhara, T. Fuja, and R. M. Tanner, \Low density parity
in this fashion typically have large constraint lengths, which check matrices from permutation matrices," in Proceedings of
makes the use of trellis based decoding impractical. Sequen- 2001ConferenceonInformationSciencesandSystems,p.142,
tial decoding, although close to maximum likelihood, is com- JohnsHopkinsUniversity,Baltimore,MD,March2001.
putationally feasible only for rates below the channel cut-o(cid:11) [3] R. M. Tanner, \Convolutional codes from quasi-cyclic codes:
rate. An alternative to these methods is decoding based on Alinkbetween thetheories ofblockandconvolutional codes."
Technical Report, Computer Research Laboratory, UC Santa
graphs. The convolutional codes can be represented by con-
Cruz,November1987.
straint graphs, which are essentially the same as that of the
[4] A. J. Felstrom and K. S. Zigangirov, \Time-varying peri-
1ThisworkwassupportedinpartbyNSFGrantCCR00-75514, odicconvolutionalcodeswithlow-densityparity-checkmatrix,"
NSF Grant CCC99-96222, NASA Grant NAG5-10503, and MIT IEEE Transactions on Information Theory, vol. 45,pp.2181{
LincolnLaboratoryGrantCX-24535. 2191,September 1999.