Low-Density Parity-Check Lattices: Construction and Decoding Analysis

Low-Density Lattice Codes

“…We shall now compare these results with the LDPC-based coding schemes that were described in Section I: LDPC-based multi-level coding [19], [29], nonbinary alphabet LDPC coding [18] and LDPC lattice coding [15]. The comparison is summarized in Table I.…”

“…The comparison is summarized in Table I. The table shows the simulation results which are reported for each scheme in the appropriate reference, including codeword length , distance from channel capacity and error performance, measured as either bit error rate (BER), symbol error rate (SER) or normalized error probability (NEP), as defined in [15]. Note that the schemes of [29] and [18] already include an inherent shaping operation, so the simulation results for these schemes already include practical shaping performance.…”

“…Substituting in (12), we get (13) with initialization , which can be solved to get (14) We would now like to compare the amplitudes of consistent Gaussians with various values of the corresponding integer vector in order to find the lattice point whose consistent Gaussians will have largest product-of-amplitudes. From the definitions of and we then have (15) Consider two integer vectors that relate to two lattice points. Denote the corresponding product-of-amplitudes by and , respectively, and assume that for these two vectors converges to the values and , respectively.…”

“…However, there was almost no further progress from that point. Specific lattice codes that were found were based on fixed dimensional classical lattices [20] or based on algebraic error correcting codes [13], [14], and even on LDPC codes [15], but no significant effort was made in designing lattice codes directly in the Euclidean space or in finding specific capacity achieving lattice codes. Practical coding schemes for the AWGN channel were based on finite alphabet codes.…”

New results on low-density integer lattices

“…A few constructions with non-binary codes can be found in [2] (including complex constructions for E 8 and Λ 24 ). Furthermore, powerful lattices such those proposed in [5] [6] and [7] are made of nested binary codes, LDPC and Turbo codes respectively. They suffer from the small number of nested codes due to the limited complexity.…”