LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis
Abstract
:1. Introduction
- Novel structured construction: We present a systematic framework for constructing LDPC codes using complete BIBDs. The proposed method generates parity-check matrices with guaranteed structural properties that satisfy the row–column (RC) constraints, extending the partial geometry approach established in [18]. The construction method provides a mathematically rigorous yet practically viable coding solution that bridges the gap between theoretical design and communication requirements. This makes it particularly attractive for applications demanding both excellent error correction capability and implementation feasibility.
- Girth guarantees and performance analysis: We provide a rigorous proof that the BIBD-LDPC codes achieve a minimum girth of 6, effectively eliminating detrimental short cycles that impair iterative decoding. The inherent properties of BIBDs, including balanced connectivity and optimized cycle structure, naturally prevent small trapping sets and low-weight code words, thereby enhancing both waterfall-region performance and error-floor characteristics.
- Comprehensive cycle analysis: We develop a systematic methodology for analyzing the cycle structure of the constructed codes. The proposed technique enables precise enumeration of cycles (particularly lengths 6 and 8) in the Tanner graphs, with a generalizable framework that can be extended to count longer cycles (e.g., 10, 12, etc.). This analysis provides valuable insights into the code’s graphical properties and decoding behavior.
2. LDPC Codes Constructed from BIBDs and Their Tanner Graphs
2.1. LDPC Codes and Tanner Graphs
2.2. Balanced Incomplete Block Designs (BIBDs)
- (1)
- Each block in contains exactly points for .
- (2)
- Every pair of distinct points in is contained in exactly blocks.
2.3. LDPC Codes Constructed Based on BIBDs
3. Girth and Cycle Structure Analysis of BIBD-LDPC Codes
3.1. Cycle Structure of BIBD-LDPC Codes
3.2. Girth of BIBD-LDPC Codes
3.3. A Method for Counting Short Cycles of BIBD-LDPC Codes
- Selection of i points from : The number of ways to choose i distinct points from the points in is given by the combination formula expressed as
- Exclusion of invalid cases: Among the chosen i points, some cases must be excluded where any t adjacent points (for ) lie in the same block. These cases require a detailed combinatorial analysis, and the number of such invalid cases is denoted by .
- Counting valid cycles of length : For the remaining valid selections of i points, the number of cycles of length that can be formed is denoted by . This involves determining how many distinct cycles of length exist for each valid set of i points. This is analogous to the circle permutation problem but without considering the order of permutation. A simple counting argument shows that
3.3.1. Number of Cycles of Length 6
3.3.2. Number of Cycles of Length 8
- (1)
- For each block, the number of ways to choose four points is expressed as
- (2)
- The total number of blocks in the BIBD is expressed as
- (3)
- Thus, the total number of cases where four points lie in the same block is expressed as
- (1)
- For each block, the number of ways to choose three points is expressed as
- (2)
- The fourth point must be chosen from the remaining points (since it cannot be in the same block as the first three points).
- (3)
- The total number of blocks in the BIBD is
- (4)
- Thus, the total number of cases where three points lie in the same block and the fourth point lies in a different block is expressed as
4. Short Cycles of Specific BIBD-LDPC Codes
- (1)
- If or , there exists a BIBD.
- (2)
- If or , there exists a BIBD.
- (3)
- If or , there exists a BIBD.
- (4)
- If or for 16, 21, 36, 46, 51, 61, 81, 166, 226, 231, 256, 261, 286, 316, 321, 346, 351, 376, 406, 411, 436, 441, 471, 501, 561, 591, 616, 646, 651, 676, 771, 796, and 801, there exists a BIBD.
- (5)
- If for 1, 2, 3, 5, 6, 12, 14, 17, 19, 22, 27, 33, 37, 39, 42, 47, 59, and 62 or for 3, 19, 34, and 39, there exists a BIBD.
5. Simulation Results
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
LDPC | Low-density parity check |
BIBD | Balanced incomplete block design |
BIBD-LDPC codes | LDPC codes constructed from BIBDs |
ML | Maximum likelihood |
RC | Row–column |
QC-LDPC | Quasi-cyclic LDPC |
Number of cycles of length 6 | |
Number of cycles of length 8 | |
AWGN | Additive white Gaussian noise |
BPSK | Binary phase-shift keying |
SPA | Sum-product algorithm |
PEG | Progressive edge growth |
PEG-LDPC | LDPC code based on the progressive edge-growth algorithm |
BER | Bit-error rate |
WER | Word-error rate |
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BIBDs | BIBD-LDPC Codes | |||
---|---|---|---|---|
Code Length | Row Weight | Column Weight | Code Rate | |
3 | ||||
4 | ||||
5 | ||||
6 | ||||
7 |
BIBDs | Cycle Lengths and Numbers of BIBD-LDPC Codes | ||
---|---|---|---|
Length 4 | Length 6 | Length 8 | |
0 | |||
0 | |||
0 | |||
0 | |||
0 |
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Xu, H.; Zhang, X.; Xu, M.; Yu, H.; Zhu, H. LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis. Entropy 2025, 27, 476. https://doi.org/10.3390/e27050476
Xu H, Zhang X, Xu M, Yu H, Zhu H. LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis. Entropy. 2025; 27(5):476. https://doi.org/10.3390/e27050476
Chicago/Turabian StyleXu, Hengzhou, Xiaodong Zhang, Mengmeng Xu, Haipeng Yu, and Hai Zhu. 2025. "LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis" Entropy 27, no. 5: 476. https://doi.org/10.3390/e27050476
APA StyleXu, H., Zhang, X., Xu, M., Yu, H., & Zhu, H. (2025). LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis. Entropy, 27(5), 476. https://doi.org/10.3390/e27050476