Construction Of Structured Low Density Lattice Codes Based On Finite Fields

Low-density lattice codes(LDLC), inspired by low-density parity-check(LDPC)and in the quest of finding practical capacity achieving lattice codes, were proposed by N. Sommer etc. in 2008. These lattices codes are generated directly in the Euclidean space, by restricting the inverse of its generating matrix to sparse. These codes have high coding gain and can be decoded with linear complexity iterative decoding.It is important to construct LDLC with linear complexity which have large girth and low decoding complexity, lower triangular structure, good performance and parameters flexibility. Finite fields were successfully used to construct algebraic LDPC codes, especially Quasi-Cyclic LCPC codes. These LDPC codes with large minimum distances have lower error floor, linear complexity of encoding and are more practical for hard-decision algebraic decoding. In this paper, we show that finite fields can also be successfully used to construct algebraic LDLC, denoted by structured LDLC. A general framework to construct algebraic LDLC is presented in this paper. Firstly, we employ matrices constructed from Berlekamp-Justesen(B-J) codes and array code, as binary base matrix of paritycheck matrix of LDLC. Then we generate corresponding Latin square matrices with degree d, as masking matrix, which are square matrices where every row and every column has the same d distinct symbols. Based on these binary base matrices and masking matrices, we obtain many classes of practical and highly structured LDLC with various size. These LDLC have comparable performance to the computer-generated random-like codes over addition white Gaussian noise(AWGN) channel with iterative soft-decision decoding in terms of symbol-error probability. Furthermore, structured LDLC with large girth can be constructed more easily than random codes. Note that iterative decoding of LDLC with big girth can be more easily converged to the exact solution.Complex low-density lattice codes(CLDLC) yields better SER for small degrees or small dimensions and better suits MIMO communication systems. Therefore, the general framework is extended to complex low-density lattice codes(CLDLC) and results in algebraic CLDLC which can perform just as well as random-like CLDLC.