Constructions of LDPC Codes Based on Finite Fields

Chao Chen

Huaan Li, Hengzhou Xu, Baoming Bai +2 more

There are many algebraic tools, e.g., finite fields and combinatorial designs, for constructing good quasi-cyclic LDPC (QC-LDPC) codes. In particular, QC-LDPC codes designed based on two arbitrary subsets of a given finite field perform very well and some well-known algebraic constructions of QC-LDPC codes have been verified as special cases of this method. How to choose these two subsets to further improve the performance of finite field based QC-LDPC codes is of interest. In this paper, we study such codes and analyze their structural properties from isomorphism perspective, and generalize some rules to efficiently determine the non-isomorphic finite field based codes. By comparing the cycle distributions of non-isomorphic codes, we can easily construct the finite field based QC-LDPC codes with larger girth and fewer short cycles. These codes generally perform very well and can be used as the ingredients of some further processings, e.g., masking. Numerical results show that the constructed codes have better performance with the iterative decoding algorithms.

Valentina Pepe

Huang Shen

This paper presents a construction method of quasi-cyclic non-binary low density parity-check(NB-LDPC)code based on the finite field three-dimensional construction.After construction of base matrix based on the finite field,a cyclic matrix is constructed with the method of three-dimensional extension.Finally,non-binary parity check matrix with quasi-cyclic feature is constructed.Compared with the GF(29)-based RS(511,305)and NB-LDPC code constructed by the three-dimensional construction can increase 3.3dB net code gain(NCG)at the bit error rate(BER)of 10-7.Compared with the binary QC-LDPC codes constructed by the traditional way,the binary LDPC codes randomly constructed,the32-ary QC-LDPC codes constructed based on finite fields and the 64-ary cyclic LDPC codes constructed based on Euclidean geometry,the NB-LDPC code constructed by the three-dimensional construction can increase 0.56dB,0.56dB,0.03dB and 0.83dB NCGrespectively at the BERof 10-7.Through performance simulation of the NB-LDPC code constructed based on this three-dimensional construction,it shows that such an NB-LDPC code not only has quasi-cyclic feature,conducive to hardware implementation,but also shows better error correction performance in the short code length.

B. N. Sindhu Tejaswini, Rajendra Prasad Lal, Vadlamudi Ch. Venkaiah

Mona Nasseri, Xin Wei Xiao, Bane V. Vasic +1 more

This paper presents two types of concatenated LDPC coding schemes which are viewed asgeneralized globally coupled(GC)LDPCcoding schemes in which outer codes serve as thelocal codesfor correcting local errors and inner codes serve asglobal coupling codesto correct global errors. The first type of concatenated LDPC coding scheme globally couples a finite geometry (FG) LDPC code as the local code and a finite field (FF) LDPC code as the global coupling code. This type of global coupling, calledGC-FG/FF-LDPC coupling, combines the distinct features of both FG- and FF-LDPC codes to achieve low error rates at a rapid decoding convergence and an error performance close to the Shannon limit. Decoding of a GC-FG/FF-LDPC code is carried out intwo iterative phases, global/local or local/global. In the second type of concatenated LDPC coding scheme, both local and global coupling codes are FF-LDPC codes. If both local and global coupling codes are constructed from the same finite field and have the same graphical structures, a GC-FF/FF-LDPC code can be decoded in one phase or two phases iteratively, otherwise, it can be decoded in two phases. Construction of GC-FF/FF-LDPC codes is very flexible in lengths and rates. The proposed two-phase iterative decoding is practically implementable.