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Recent works showed how low-density parity-check (LDPC) erasure correcting codes, under maximum likelihood (ML) decoding, are capable of tightly approaching the performance of an ideal maximum-distance-separable code on the binary erasure…
We prove that approximating the size of stopping and trapping sets in Tanner graphs of linear block codes, and more restrictively, the class of low-density parity-check (LDPC) codes, is NP-hard. The ramifications of our findings are that…
This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum…
We consider a special family of SC-LDPC codes, that is, time-invariant LDPCC codes, which are known in the literature for a long time. Codes of this kind are usually designed by starting from QC block codes, and applying suitable unwrapping…
The low-density parity-check (LDPC) lattices perform very well in high dimensions under generalized min-sum iterative decoding algorithm. In this work we focus on 1-level LDPC lattices. We show that these lattices are the same as lattices…
We present a new family of low-density parity-check (LDPC) convolutional codes that can be designed using ordered sets of progressive differences. We study their properties and define a subset of codes in this class that have some desirable…
An algorithm for constructing parity-check matrices of non-binary quasi-cyclic low-density parity-check (NB QC-LDPC) codes is proposed. The algorithm finds short cycles in the base matrix and tries to eliminate them by selecting the…
It is widely accepted that quantum error correction is essential for realizing large-scale fault-tolerant quantum computing. Recent experiments have demonstrated error correction codes operating below threshold, primarily using local planar…
Quantum low-density parity-check (QLDPC) codes offer a promising route to scalable fault-tolerant quantum computation, but their performance under iterative decoding is strongly influenced by short-cycle structure and other harmful…
Low-rank parity-check (LRPC) codes are the rank-metric analogue of low-density parity-check codes and they found important applications in code-based cryptography. In this paper we investigate a sub-family of LRPC codes, which have a…
Low-density parity-check (LDPC) codes together with belief propagation (BP) decoding yield exceptional error correction capabilities in the large block length regime. Yet, there remains a gap between BP decoding and maximum likelihood…
The non-binary low-density parity-check (NB-LDPC) codes can offer promising performance advantages but suffer from high decoding complexity. To tackle this challenge, in this paper, we consider NB-LDPC codes over finite fields as codes over…
One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in…
Decoders for Low Density Parity Check (LDPC) codes are usually tailored to an application and optimized once the specific content and structure of the parity matrix are known. In this work we consider the parity matrix as an argument of the…
This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields.…
The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block…
Low-density parity check (LDPC) codes are a well known class of Pauli stabiliser Hamiltonians that furnish fixed-point realisations of nontrivial gapped phases such as symmetry breaking and topologically ordered (including fracton) phases.…
We present a mathematical connection between channel coding and compressed sensing. In particular, we link, on the one hand, \emph{channel coding linear programming decoding (CC-LPD)}, which is a well-known relaxation o maximum-likelihood…
In the literatures, it is well-known that Fossorier code has the girth among LDPC codes. In this paper, we introduce a new class of low-density parity-check (LDPC) codes, with higher girth than other previous constructed codes. Especially…
A generalized low-density parity-check (GLDPC) code is a class of codes, where single parity check nodes in a conventional low-density parity-check (LDPC) code are replaced by linear codes with higher parity check constraints. In this…