Constructions Of Structured Nonbinary LDPC Codes

Low-density parity-check (LDPC) code is a class of linear block code, which provides near Shannon-Limit capacity performance. The decoding complexity increases linearly with the code length. Nonbinary LDPC codes with short and medium length gain an advantage over binary LDPC codes. Quasi-cyclic LDPC (QC-LDPC) codes are getting more attention due to their easy implementation and other graceful feature. In this thesis, three new algebraic constructions of nonbinary LDPC codes are proposed.Firstly, prime "difference codes are presented, based on the prime difference sequence over GF(q). With a prime k, we construct a base matrix over GF(q), which satisfies the row-distance constraint (RD-constraint). A new concept, called generalized q-ary location vector, instead of the special q-ary location vector is used to expand the elements in GF(q). With generalized q-ary location vector, the generalized a-multiplied circulant permutation matrix (CPM) can be obtained. Replacing the elements in the base matrix by their a-multiplied CPMs, a sparse low-density parity-check matrix can be got. The null space of the matrix gives a class of nonbinary QC-LDPC codes, called prime difference codes, which satisfiy the RC-constraint. Simulation results show that the constructed codes perform better than the PEG codes with the same parameters.Secondly, standard array codes are introduced. With the help of the standard array for a linear block code, a base matrix satisfying the RD-constraint can be obtained. Expand the base matrix by substituting each entry in the base matrix with its a-multiplied CPM. This replacement results in a low-density parity-check matrix, which satisfies the RC-constraint. With this method, low decoding complexity nonbinary LDPC codes as short as60symbols can be built. The simulation results show that the standard array codes and PEG codes have almost the same error performance. Although the EG-LDPC codes have a much steeper waterfall and outperform the standard-array codes below the BER of10-4, the standard-array codes have a smaller field size and a higher rate than the EG-LDPC codes. In point of the decoding complexity, only960FFT operations are needed for the standard-array codes in each iteration, whereas the EG-LDPC codes require8,192.Finally, cyclotomic coset codes are proposed based on cyclotomic A cylcotomic coset is composed of the exponents of all the roots for one minimal polynomial. With the aid of cyclotomic coset, a base matrix over GF(g) can be built. Replacing the entries of the base matrix with their α-multiplied CPMs, we can form a parity-check matrix. Simulation results show that the cyclotomic coset codes outperform the PEG codes with almost the same parameters by about1dB throughout the whole range of Eb/No considered. Lin’s codes still have a much steeper waterfall, but the cyclotomic coset codes exceed Lin’s codes till the BER of10-7. In addition, the cyclotomic coset code reaches Shannon Limit at the BER of10-4. Importantly, the choice of the cyclotomic coset codes is insensitive to the performance, which enhances the flexibility of construction.Furthermore, all the codes constructed by three methods mentioned above have excellent error floor "and decoding convergence characteristics.