Low-Density Lattice Codes Based on Golomb Rulers

Low-density lattice codes (LDLC) are a new type of lattice codes that can be decoded effectively and achieve performance levels near the capacity of the additive white Gaussian noise (AWGN) channel. Earlier designs of LDLC, such as those utilizing Latin squares, relied on high-complexity computer searches to eliminate short cycles. The proposed code construction offers a systematic representation and inherently avoids these issues. This study presents a design for LDLC that is based on Golomb rulers. This approach is deterministic and results in a parity-check matrix that is free of short cycles. In all scenarios examined, the LDLC built on Golomb rulers demonstrated superior frame error rate performance compared to those based on Latin squares. For instance, there is a 1.5 dB improvement observed for a dimension of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{n}=73$</tex>.