Quasi-Cyclic LDPC Code Construction for Very Long Code Lengths
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Solution Overview
Problem
There is no known method to construct quasi-cyclic check matrices for long geometric LDPC codes exceeding 20000 bits, limiting the efficiency and scalability of digital communication coding systems.
Innovation Solution
A single equation derived from a geometry is used to construct quasi-cyclic parity check matrices for LDPC codes of exceedingly long length, up to and exceeding 400,000 bits, ensuring good error correcting properties and low complexity decoding.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Length of stationary object
If column splitting operations are used to construct longer codes from finite projective and affine spaces, then code length increases up to approximately 500,000 bits, but the codes lose their geometric properties
Solution Approach 1:
The code construction is segmented into two parts: using finite geometry (projective, affine, or polar spaces) to define the parity-check matrix structure, and applying column splitting operations only to specific columns while preserving the geometric incidence structure. This segmentation allows long codes to be constructed while maintaining geometric properties that ensure good error-correcting performance.
2Productivity
If iterative decoding algorithms are used for very long LDPC codes, then decoding efficiency is maintained, but computational cost increases significantly
Solution Approach 1:
The LDPC codes constructed from finite geometries exhibit local quality variations in their Tanner graphs, with certain structural properties (such as girth and degree distribution) that are locally optimized. This allows the use of simplified decoding algorithms that exploit these local structural characteristics, reducing computational cost while maintaining decoding efficiency for very long codes.
3Device complexity
If quasi-cyclic structure is imposed on LDPC codes, then encoding complexity is reduced and fast encoding is achieved, but code design flexibility is limited
Solution Approach 1:
The invention uses a universal construction method based on finite geometries (projective spaces, affine spaces, and polar spaces) that can generate quasi-cyclic LDPC codes with different lengths and rates by varying the geometric parameters. This universal approach maintains the quasi-cyclic structure for efficient encoding while providing flexibility through different geometric configurations and parameter selections.
Data Source
AI summary
In an aspect, digital communication coding methods (or simply coding methods) are provided, said methods comprising providing families of quasi-cyclic low-density parity check, LDPC, codes each represented by a single equation. Said single equation is derived from a geometry whose nature implies good error correcting properties for each of the LDPC codes and allows quasi-cyclic parity check matrices to be constructed for geometric LDPC codes of exceedingly long length, up to and exceeding four hundred thousand bits. The construction of such check matrices is not possible by any other known method. The LDPC codes have different lengths and different rates and the families of LDPC codes are provided with a guarantee of minimum distance. The geometric and quasi-cyclic properties of the LDPC codes allow low complexity decoding with very low frame error rates. Systems, computing systems and computer programs suitable to perform such coding methods are also provided.