# LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis
## 1 Introduction
Notably, BIBDs constitute an important subclass of partial geometries [21], making them particularly attractive for code construction. Consequently, the use of BIBDs to construct high-performance LDPC codes has become a topic of significant interest in coding theory [28; 29; 30; 31; 32; 33]. It is worth noting that the construction methods for some BIBDs were detailed in [34]. In this paper, we investigate LDPC codes constructed based on balanced incomplete block designs (BIBDs). By analyzing the cycle structure, we aim to understand how the design parameters of BIBDs, such as block size and the number of points, affect the presence and distribution of short cycles, which are critical factors in determining the performance of LDPC codes under belief propagation decoding. Unlike previous works reported in [28; 29; 30; 31; 32; 33], we utilize the complete structure of a BIBD to design LDPC codes. Specifically, given a BIBD, the constructed LDPC code exhibits well-defined parameters, such as code length, row weight, and column weight the parity-check matrix. Moreover, we establish the incidence relationship between points and blocks in a BIBD and analyze the cycle structure within the Tanner graphs of the resulting LDPC codes. Based on this analysis, we propose a systematic method for counting cycles and determine the exact number of cycles of lengths 6 and 8 in the Tanner graphs. Finally, we provide numerical simulation results to demonstrate the performance of the constructed LDPC codes based on BIBDs. The main contributions of this paper are outlined as follows. * 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.