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Constructing Grith Eight GC-LDPC Codes Based on the GCD FLRM Matrix with a New Lower Bound.

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Researchers developed a new method for constructing globally coupled (GC) low-density parity-check (LDPC) codes. This approach reduces complexity and improves performance by optimizing the circulant permutation matrix (CPM) size, enabling more flexible code design.

Keywords:
full-length row multiplier matrixglobally coupled LDPCgreatest common divisorlarge girth

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Area of Science:

  • Coding Theory
  • Information Theory
  • Digital Communications

Background:

  • Globally coupled (GC) low-density parity-check (LDPC) codes offer high performance and low complexity by combining local and global parity checks.
  • Quasi-cyclic (QC) LDPC codes are commonly used as local codes, with their design linked to the size of the circulant permutation matrix (CPM).
  • Existing methods for constructing GC-LDPC codes, particularly those using greatest common divisor (GCD)-based full-length row multiplier (FLRM) matrices, have constraints on CPM size to avoid cycles, impacting efficiency.

Purpose of the Study:

  • To identify a new, tighter lower bound for the circulant permutation matrix (CPM) size in GC-LDPC code construction.
  • To propose an efficient algorithm for determining the minimum CPM size and the corresponding FLRM matrix.
  • To develop novel methods for constructing GC-QC-LDPC codes with a girth of 8, leveraging the new lower bound and algorithm.

Main Methods:

  • Derivation of a new lower bound for the CPM size, improving upon existing constraints for avoiding short cycles.
  • Development of an algorithm to compute the minimum required CPM size and identify the associated FLRM matrix.
  • Construction of GC-QC-LDPC codes with girth 8 using two distinct methods based on the proposed algorithm and GCD-based FLRM matrices.

Main Results:

  • The proposed algorithm achieves a significant reduction in CPM size, up to 45% smaller than previously established sufficient conditions for girth 8.
  • The newly constructed GC-QC-LDPC codes demonstrate improved performance compared to conventional GC-LDPC constructions.
  • The method offers enhanced flexibility in designing code length and code rate for GC-LDPC codes.

Conclusions:

  • The novel lower bound and construction algorithm provide a more efficient way to design GC-LDPC codes.
  • The proposed GC-QC-LDPC codes offer superior performance and greater design flexibility, making them suitable for various communication applications.
  • This research contributes to the advancement of efficient and high-performance error-correcting codes.