Reliability-Based Decoding of Low-Density Lattice Codes Using Gaussian and Eisenstein Integers

Abstract

This paper proposes reliability-based decoding for complex low-density lattice codes (CLDLC) which can be applied to both Gaussian and Eisenstein integers. Two major contributions are: first, a decoding algorithm for CLDLC using a likelihood-based reliability function is used to determine the number of complex Gaussian functions at the variable node. This allows each message to be approximated by a variable number of Gaussian functions depending upon its reliability. An upper bound on the Kullback-Leibler (KL) divergence of the approximation is formed to find selection thresholds via linear regression. Second, a construction of CLDLC using Eisenstein integers is given. Compared to Gaussian integers, this reduces the complexity of CLDLC decoding by exploiting the structure of the Eisenstein integers. The proposed CLDLC decoding algorithm has higher performance and lower complexity compared to existing algorithms. When the reliability-based algorithm is applied to Eisenstein integer CLDLC decoding, the complexity is reduced to $\mathcal {O}(n\cdot t \cdot 1.35^{d-1})$ at the volume-to-noise ratio of 6 dB, for lattice dimension n, with degree d inverse generator matrix and t decoding iterations. Decoding CLDLC using Eisenstein integers has lower complexity than CLDLC using Gaussian integers when $n \geq 49$ .

Keywords

• Complex low-density lattice codes
• low-density lattice codes
• parametric decoder
• lattice decoder
• Eisenstein integers
• Gaussian integers