DOI: 10.5061/dryad.8gtht76zs
title: Quantum LDPC code orthogonal parity-check matrix pairs H_Gamma, H_Delta

description: Quantum low-density parity-check (QLDPC) codes require orthogonal
 parity-check matrices to satisfy the quantum orthogonality condition,
 ensuring their validity for quantum error correction. This dataset
 provides orthogonal parity-check matrix pairs $H_\Gamma, H_\Delta$, which
 are essential for constructing QLDPC codes. The matrices were generated
 using predefined parameters such as code size, finite field order, and
 cycle length constraints. The dataset includes a variety of matrix
 instances stored in structured text format. The dataset was developed as
 part of a study addressing two fundamental challenges in quantum error
 correction: approaching the hashing bound—a fundamental performance
 benchmark and lower bound of quantum capacity—and avoiding the error floor
 phenomenon, where the error rate stagnates under low noise levels. These
 challenges have hindered the realization of practical quantum error
 correction schemes. In our research, we propose a novel construction and
 decoding method for QLDPC codes that significantly improves upon existing
 quantum error-correcting codes. Our approach is the first to demonstrate
 performance approaching the hashing bound while also showing no observable
 error floor phenomenon in the tested parameter range. This dataset
 contains the parity-check matrices used in our study, allowing for
 reproducibility and further exploration of QLDPC code performance. Each
 parity-check matrix is represented by three files: a structure file
 containing nonzero element positions, a size descriptor file specifying
 matrix dimensions and finite field order, and a value file containing
 corresponding finite field elements. File names follow a systematic
 convention encoding key parameters such as the number of rows and columns,
 the field order, and random seed values used for matrix generation. This
 dataset is a valuable resource for researchers in quantum error
 correction, enabling exploration of QLDPC codes under different parameter
 settings. It facilitates testing of decoding algorithms, performance
 evaluation, and further development of quantum coding theory. The dataset
 is publicly available on Dryad and can be freely accessed for research and
 development purposes. There are no legal or ethical constraints associated
 with the use of this dataset. The matrices were generated entirely within
 our research and do not contain any personally identifiable or sensitive
 information. This dataset supports the reproducibility of QLDPC code
 studies and contributes to advancements in quantum error correction
 techniques.
description: This dataset represents $H<em>{\Gamma}$ and
 $H</em>{\Delta}$, the error-correcting codes evaluated in the
 manuscript.
description: # Quantum LDPC Code Orthogonal Parity-Check Matrix Pairs $H\_\Gamma,
 H\_\Delta$ [Access this dataset on
 Dryad](https://doi.org/10.5061/dryad.8gtht76zs) This dataset contains
 orthogonal parity-check matrix pairs $H_\Gamma, H_\Delta$ used for quantum
 low-density parity-check (QLDPC) codes. These matrices are designed to
 satisfy the quantum orthogonality condition, ensuring the validity of the
 quantum code. The dataset includes various instances of $H_\Gamma,
 H_\Delta$ generated under different conditions, along with metadata
 describing their structural properties. These matrices play a crucial role
 in the construction and evaluation of quantum error-correcting codes. ##
 Description of the data and file structure The dataset consists of
 orthogonal parity-check matrix pairs $H_\Gamma, H_\Delta$ used in quantum
 LDPC codes. These matrices are stored in plain text (.txt) format and
 follow a structured naming convention that encodes important parameters.
 ### **File Naming Convention** Each file name follows the format: ```
 {QC-LDPC="QC", APM-LDPC="APM"}_{H_Δ="Delta",
 H_Γ="Gamma"}_n{cols×p}_m{rows×p}_N{cols}_M{cols}_J{J}_L{L}_P{P}_alpha{QC-LDPC="1", APM-LDPC="2"}_GF{2^p}_GIRTH{girth}_SEED{seed} ``` where: * `QC` or `APM`: Specifies whether the matrix is Quasi-Cyclic LDPC (QC-LDPC) or Array-Based LDPC (APM-LDPC). * `Delta` or `Gamma`: Specifies whether the file contains $H_\Delta$ or $H_\Gamma$. * `n`: Number of columns in the parity-check matrix multiplied by $p$. * `m`: Number of rows in the parity-check matrix multiplied by $p$. * `N`: Number of columns in the parity-check matrix. * `M`: Number of columns in the parity-check matrix. * `J, L, P`: Parameters defining the matrix structure. * `alpha`: Value `1` for QC-LDPC, `2` for APM-LDPC. * `GF`: The size of the finite field $GF(2^p)$. * `GIRTH`: The minimum cycle length in the parity-check matrix. * `SEED`: The random seed used for generating the matrix. ### **File Types and Contents** Each matrix is represented by three different files: 1. **Matrix structure file** (`{prefix}.txt`): * Contains the nonzero entry column indices for each row. 2. **Size descriptor file** (`{prefix}_size.txt`): * Contains three values: 1. Number of rows in the parity-check matrix. 2. Number of columns in the parity-check matrix. 3. The field size $2^p$. 3. **Finite field value file** (`{prefix}_value.txt`): * Stores the finite field nonzero values corresponding to each nonzero position in the parity-check matrix. * Values are listed in the same order as in the structure file. This structured format allows users to reconstruct the full parity-check matrix while keeping the storage efficient. ## Sharing/Access information The dataset is publicly available on Dryad and can be accessed via the following link:\ [Access this dataset on Dryad](https://doi.org/10.5061/dryad.8gtht76zs) There are no other publicly accessible locations where this dataset is hosted. ### **Data Origin** The data in this repository was entirely generated as part of our research. No external datasets were used or modified to create this dataset.