Multilevel Structured Low-Density Parity-Check Codes

Low-Density Parity-Check (LDPC) codes are typically characterized by a relatively high-complexity description, since a considerable amount of memory is required in order to store their code description, which can be represented either by the connections of the edges in their Tanner graph or by the non-zero entries in their parity-check matrix (PCM). This problem becomes more pronounced for pseudo-random LDPC codes, where literally each non-zero entry of their PCM has to be enumerated, and stored in a look-up table. Therefore, they become inadequate for employment in memoryconstrained transceivers. Motivated by this, we are proposing a novel family of structured LDPC codes, termed as Multilevel Structured (MLS) LDPC codes, which benefit from reduced storage requirements, hardware-friendly implementations as well as from low-complexity encoding and decoding. Our simulation results demonstrate that these advantages accrue without any compromise in their attainable Bit Error Ratio (BER) performance, when compared to their previously proposed more complex counterparts of the same code-length. In particular, we characterize a half-rate quasi-cyclic (QC) MLS LDPC code having a block length of 8064 that can be uniquely and unambiguously described by as few as 144 edges, despite exhibiting an identical BER performance over both Additive White Gaussian Noise (AWGN) and uncorrelated Rayleigh (UR) channels, when compared to a pseudorandom construction, which requires the enumeration of a significantly higher number of 24,192 edges

Repeat-Accumulate Codes for Reconciliation in Continuous Variable Quantum Key Distribution

This paper investigates the design of low-complexity error correction codes for the verification step in continuous variable quantum key distribution (CVQKD) systems. We design new coding schemes based on quasi-cyclic repeat-accumulate codes which demonstrate good performances for CVQKD reconciliation

Lattices from Codes for Harnessing Interference: An Overview and Generalizations

In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction $\pi_A$ (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction $\pi_D$ subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers

Low-Complexity LP Decoding of Nonbinary Linear Codes

Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. LP decoding has been derived for binary and nonbinary linear codes. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remains prohibitively large for codes of moderate to large block length. To address this problem, two low-complexity LP (LCLP) decoding algorithms for binary linear codes have been proposed by Vontobel and Koetter, henceforth called the basic LCLP decoding algorithm and the subgradient LCLP decoding algorithm. In this paper, we generalize these LCLP decoding algorithms to nonbinary linear codes. The computational complexity per iteration of the proposed nonbinary LCLP decoding algorithms scales linearly with the block length of the code. A modified BCJR algorithm for efficient check-node calculations in the nonbinary basic LCLP decoding algorithm is also proposed, which has complexity linear in the check node degree. Several simulation results are presented for nonbinary LDPC codes defined over Z_4, GF(4), and GF(8) using quaternary phase-shift keying and 8-phase-shift keying, respectively, over the AWGN channel. It is shown that for some group-structured LDPC codes, the error-correcting performance of the nonbinary LCLP decoding algorithms is similar to or better than that of the min-sum decoding algorithm.Comment: To appear in IEEE Transactions on Communications, 201