# LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis
## 1 Introduction  
Low-density parity-check (LDPC) codes exhibit remarkable performance within the framework of iterative decoding. Several structural properties of LDPC codes significantly influence their iterative decoding performance, including their minimum distance, row redundancy, and degree distribution of their parity-check matrices, as well as the girth and cycle distribution of their Tanner graphs [1], among others. For an LDPC code, the iterative decoder, which operates based on belief propagation, functions on its Tanner graph. The presence of cycles causes the decoder to operate locally until a solution is found or the termination condition is met. In other words, the decoding solution is derived primarily around cycles, particularly short cycles, which prevents the decoder from achieving globally optimal solutions. Consequently, iterative decoders operating on Tanner graphs with short cycles are unable to identify maximum-likelihood (ML) code words [2]. As a result, cycles, especially short ones, adversely affect the decoding performance of LDPC codes when decoded using iterative algorithms. This makes the study of cycle distribution in LDPC codes a topic of significant interest [3]. Numerous effective algorithms have been developed for the counting of cycles in LDPC codes (or their Tanner graphs). In bipartite graphs, cycles of lengths \(g,g+2\), and \(g+4\) were enumerated in [4], where \(g\) represents the girth of the graph. However, the computational complexity of these methods increases rapidly with the number of variable nodes. Several alternative approaches have been proposed based on graph properties suchas eigenvalues, degree distribution, and spectrum [5; 6; 7; 8]. Additionally, leveraging message-passing algorithms, two competitive methods were presented in [9; 10]. Nevertheless, for the methods described in [5; 6; 7; 8; 9; 10], the lengths of the counted cycles do not exceed \(2g\), where \(g\) is the girth of the LDPC code. Furthermore, necessary and sufficient conditions for the existence of cycles in quasi-cyclic LDPC (QC-LDPC) codes have been established in [11; 12; 13; 14], along with corresponding cycle-counting algorithms. Determining the number of cycles in the Tanner graphs of well-designed LDPC codes based on design rules or structural properties remains a topic of significant interest. Complementary to these, random-like constructions with carefully designed cycle-avoidance rules have also been developed [15; 16; 17]. A unifying framework emerges through partial geometries, which provides a geometric perspective for the analysis of code structures [18]. This approach has proven particularly valuable, as partial geometry-based LDPC codes demonstrate exceptional iterative decoding performance [19; 20]. Notably, balanced incomplete block designs (BIBDs) constitute an important subclass of partial geometries [21], making them particularly attractive for code construction. The design of LDPC codes with large girth and a minimal number of short cycles has been approached through various mathematical frameworks. Algebraic constructions utilizing finite fields [22], finite geometries [23], Sidon sequences [24], protographs [25], and transversal designs [26] have shown promising results. Additionally, random-like LDPC codes designed under specific rules (e.g., avoiding or eliminating short cycles) have also been proposed [15; 16; 17]. Connections among algebraically constructed LDPC codes were explored in [27]. Notably, a well-known general framework called partial geometries was introduced, under which many existing LDPC code constructions can be unified [18]. By leveraging the concept of partial geometries, key structural properties of LDPC codes and their parity-check matrices can be analyzed from a geometric perspective. Research has demonstrated that LDPC codes constructed based on partial geometries exhibit excellent performance under iterative decoding [19; 20].