Construction D′ Lattices from Quasi-Cyclic Low-Density Parity-Check Codes

Abstract

Recently, Branco da Silva and Silva described an efficient encoding and decoding algorithm for Construction D lattices. Using their algorithm, we propose a Construction D lattice based on binary quasi-cyclic low-density parity-check (QC-LPDC) codes and single parity-check product codes. The underlying codes designed by the balanced-distances rule contribute in a balanced manner to the squared minimum distance of the constructed lattice, which results in a high lattice coding gain. The proposed lattice based on IEEE 802.16e QC-LDPC codes is shown to provide competitive error-rate performance on the power-unconstrained additive white Gaussian noise channel.

Construction D' Lattices for Power-Constrained Communications

“…Motivated by [13], we construct quasi-cyclic (QC)-LDPC codes to form Construction D' lattices (termed LDPC Construction D' lattices), because QC-LDPC codes are widely used in recent wireless communication standards. We give a design of parity-check matrices for nested QC-LDPC codes that can be easily triangularized and thus efficient encoding and indexing is allowed.…”

“…In this section, we consider two-level Construction D' lattices. One approach of lattice construction can employ QC-LDPC codes and single parity-check product codes [13]. The first level code parity-check matrix consists of a top matrix that is modified from a QC-LDPC code [28, Table I] and bottom rows which contribute to parity checks for the product code.…”

Encoding and Decoding Construction D' Lattices for Power-Constrained Communications

“…The first level component code C 0 has an M -by-N prototype parity-check matrix while the second level component code C 1 is a 2-by-N prototype paritycheck matrix. The code C 0 has a design rate 1 ´M {N , and C 1 is a high-rate code-a column weight 2, row weight N parity-check matrix is sufficient; column weight 2 was also used in [13]. The code C 0 is a subcode of C 1 thus the prototype matrix of C 1 is a matrix obtained from linear combinations of a C 0 prototype submatrix.…”

“…Motivated by [13], we also construct QC-LDPC codes to form Construction D' lattices (termed LDPC code lattices), because QC-LDPC codes are widely used in recent wireless communication standards. A design of QC-LDPC code C 0 with a parity-check matrix H 0 is presented, where the position of non-zero blocks is found by binary linear programming.…”