Related papers: Sparsifying Parity-Check Matrices
Large Language Models (LLMs) are increasingly applied to complex tasks that require extended reasoning. In such settings, models often benefit from diverse chains-of-thought to arrive at multiple candidate solutions. This requires two…
In this paper we study reaction and timing attacks against cryptosystems based on sparse parity-check codes, which encompass low-density parity-check (LDPC) codes and moderate-density parity-check (MDPC) codes. We show that the feasibility…
The error floor phenomenon observed with LDPC codes and their graph-based, iterative, message-passing (MP) decoders is commonly attributed to the existence of error-prone substructures -- variously referred to as near codewords, trapping…
We consider spatially coupled low-density parity-check codes with finite smoothing parameters. A finite smoothing parameter is important for designing practical codes that are decoded using low-complexity windowed decoders. By optimizing…
Channel coding aims to minimize errors that occur during the transmission of digital information from one place to another. Low-density parity-check (LDPC) codes can detect and correct transmission errors if one encodes the original…
A Bounded-Degree Low-Rank Parity-Check (BD-LRPC) code is a rank-metric code that admits a parity-check matrix whose support is generated by a set of powers of an element. This specific structure of the parity-check matrix was employed to…
We present an analytic method of assessing the typical performance of low-density parity-check codes on finite-state Markov channels. We show that this problem is similar to a spin-glass model on a `small-world' lattice. We apply our…
We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank…
As machine learning is increasingly used in essential systems, it is important to reduce or eliminate the incidence of serious bugs. A growing body of research has developed machine learning algorithms with formal guarantees about…
Channel coding is vital for reliable sixth-generation (6G) data transmission, employing diverse error correction codes for various application scenarios. Traditional decoders require dedicated hardware for each code, leading to high…
A lower bound on the maximum likelihood (ML) decoding error exponent of linear block code ensembles, on the erasure channel, is developed. The lower bound turns to be positive, over an ensemble specific interval of erasure probabilities,…
We study pattern matching problems on two major representations of uncertain sequences used in molecular biology: weighted sequences (also known as position weight matrices, PWM) and profiles (i.e., scoring matrices). In the simple version,…
In this paper, we present a generalized version of the matrix chain algorithm to generate efficient code for linear algebra problems, a task for which human experts often invest days or even weeks of works. The standard matrix chain problem…
The discovery of the family of balanced product codes was pivotal in the subsequent development of 'good' low density quantum error correcting codes that have optimal scaling of the key parameters of distance and storage density. We review…
Minimum-weight perfect matching (MWPM) has been been the primary classical algorithm for error correction in the surface code, since it is of low runtime complexity and achieves relatively low logical error rates [Phys. Rev. Lett. 108,…
An efficient decoding algorithm for horizontally u-interleaved LRPC codes is proposed and analyzed. Upper bounds on the decoding failure rate and the computational complexity of the algorithm are derived. It is shown that interleaving…
Moderate Density Parity Check (MDPC) codes are defined here as codes which have a parity-check matrix whose row weight is $O(\sqrt{n})$ where $n$ is the length $n$ of the code. They can be decoded like LDPC codes but they decode much less…
This paper proposes two approaches for reducing the impact of the error floor phenomenon when decoding quantum low-density parity-check codes with belief propagation based algorithms. First, a low-complexity syndrome-based linear…
Quantum error correction (QEC) is essential for fault-tolerant quantum computation. Often in QEC errors are assumed to be independent and identically distributed and can be discretised to a random Pauli error during the execution of a…
In this paper we present an extended variant of low rank parity check matrix (LRPC) codes that have received significant interests in recent years. It is shown that the extension indeed yields a superfamily of LRPC codes, which are termed…