Low density parity check lattices

Low density parity check codes can have a remarkable performance under iterative decoding algorithms. This idea is used to construct a class of lattices with relatively high coding gain and low decoding complexity.; The lattice construction is based on the so-called Construction D'. This lattice construction converts a set of parity checks defining a family of nested low density parity check codes into congruences for a lattice called a low density parity check lattice (LDPC lattice, for short). Bounds on the minimum distance and coding gain of the corresponding lattice in terms of the minimum distance of the underlying codes are provided. Given a parity check matrix of a lattice, a practical way of finding the lattice parameters such as cross sections and label group sizes is given.; A new approach based on iterative decoding of lattices is taken. For the decoding of a low density parity check lattice, the Min-Sum algorithm is generalized to group codes over different alphabet sizes and is applied to the Tanner graph of the lattice. An upper bound on the decoding complexity using the generalized Min-Sum algorithm is given. Also for the decoding complexity of LDPC lattices the exact number of computations among lower and upper bounds is derived. The analysis of decoding complexity confirms the use of LDPC codes for the lattice construction. It is shown that the decoding complexity grows linearly with the dimension, exponentially with row weights and is polynomial time in terms of coding gain of the lattice.; The progressive edge growth algorithm is extended to what we call the E-PEG algorithm. This algorithm is used to construct a class of nested regular binary codes to generate the corresponding lattice. A class of 2-level lattices is constructed. The performance of this class is compared with the corresponding 1-level construction and other well known constructions.