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LOW-DENSITY PARITY-CHECK CODESBASEDONSPARSE BINARYSEQUENCES

Shanbao He, Chunming Zhao

Published Jan 1, 2005• 0 citations

Abstract

theconstruction ofLDPCcodes andtheconstruction isbasedonthelines andpoints ofafinite geometry. Inthispaper, wepropose anovel andsimple con- Itcangetquasi-cyclic LDPCcodes withlowencoding struction ofregular LDPCcodes using sparse binary complexity. sequences. Thedistribution matrix ofonesinaparity- Inthis paper wepresent aneffective algorithm which check matrix isintroduced into this newconstruction. cangenerate LDPC codeswithperformance slightly Thecyclic cross correlation function andthecyclic better thanthatofthePEGcodes, while maintaining autocorrelation function areemployed toensure that fairly lowencoding complexity. Inthenextsection, we thegirth ofTanner graphisatleast eight. Thenew describe LDPC codesfromsparse binary sequences, codes perform wellusing thesum-product algorithmtheir structures andtheir girth distribution. Section foriterative decoding. Lowencoding complexity can 3lists thedetailed construction ofLDPCcodes based also beachieved duetotheinherent quasi-cyclic struc- onsparse sequences. Performance comparisons ofdiftureoftheLDPCcodes. ferent classes ofLDPC codes aregiveninSection 4. Section 5concludes this paper.

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theconstruction ofLDPCcodes andtheconstruction isbasedonthelines andpoints ofafinite geometry. Inthispaper, wepropose anovel andsimple con- Itcangetquasi-cyclic LDPCcodes withlowencoding struction ofregular LDPCcodes using sparse binary complexity. sequences. Thedistribution matrix ofonesinaparity- Inthis paper wepresent aneffective algorithm which check matrix isintroduced into this newconstruction. cangenerate LDPC codeswithperformance slightly Thecyclic cross correlation function andthecyclic better thanthatofthePEGcodes, while maintaining autocorrelation function areemployed toensure that fairly lowencoding complexity. Inthenextsection, we thegirth ofTanner graphisatleast eight. Thenew describe LDPC codesfromsparse binary sequences, codes perform wellusing thesum-product algorithmtheir structures andtheir girth distribution. Section foriterative decoding. Lowencoding complexity can 3lists thedetailed construction ofLDPCcodes based also beachieved duetotheinherent quasi-cyclic struc- onsparse sequences. Performance comparisons ofdiftureoftheLDPCcodes. ferent classes ofLDPC codes aregiveninSection 4. Section 5concludes this paper.

In this paper, we propose a novel and simple construction of regular LDPC codes using sparse binary sequences. The distribution matrix of ones in a parity-check matrix is introduced into this new construction. The cyclic cross correlation function and the cyclic auto correlation function are employed to ensure that the girth of Tanner graph is at least eight. The new codes perform well using the sum-product algorithm for iterative decoding. Low encoding complexity can also be achieved due to the inherent quasi-cyclic structure of the LDPC codes.

A novel and simple construction of regular LDPC codes was proposed, which used sparse binary sequences, through studying construction methods of LDPC codes. The distribution matrix of ones in parity-check matrix was introduced into the new construction. The cyclic cross correlation function and the cyclic auto correlation function were utilized to ensure that the girth of Tanner graph was eight. The new codes perform well using the sum-product algorithm for iterative decoding. Low encoding complexity can also be achieved due to the inherent quasi-cyclic structure of the LDPC codes.

Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.

Wepropose abit-serial LDPCdecoding scheme toreduceA.LDPCcodes andmin-sum decoding interconnect complexity infully-parallel low-density parity-check de- A binary (NN -M) LDPCcode, C,isthenull space ofasparse coders. Bit-serial decoding alsofacilitates efficient implementation of M N wordlength-programmable LDPCdecoding whichisessential forgear M xN parity-check matrix, H.Itcanalso bedescribed byabipartite shift decoding. Tosimplify theimplementation ofbit-serial decodinggraph, orTannergraph. Tannergraph ofanLDPC codeconsists of wepropose anewapproximation tothecheck update function inthe twosetofnodes. Checknodes{Cl, C2,...,CM}represent therows min-sum decoding algorithm. Thenewcheck update rule computes onlyofH andvariable nodes{VI V2,*.* VN}represent thecolumns. An theabsolute minimumandapplies acorrection tooutgoing messages if ' ' t required. Wepresent a650-Mbps bit-serial (480, 355)RS-based LDPC edgeconnects thechecknodecmtothevariable nodevnifandonly decoder implemented onasingle Altera Stratix EP1S80 FPGAdevice. To ifHmn isnonzero. Foracodewithafull-rank M xN parity-check ourknowledge, this isthefastest FPGA-based LDPCdecoder reported matrix, H,thecoderate isR = - M/N.We denote thesetof intheliterature. variables thatparticipate incheckCm asN(m) = {n:Hmn= 1}andthesetofchecks inwhichthevariable vnparticipates as