# LDPC Codes on Balanced Incomplete Block Designs: Construction, Girth, and Cycle Structure Analysis

## TL;DR

This paper explores how to build LDPC codes using BIBD designs and analyzes their cycle structures to improve performance.

## Contribution

A novel method for constructing LDPC codes from BIBDs with guaranteed girth and detailed cycle enumeration.

## Key findings

- LDPC codes from BIBDs have a girth of 6.
- Exact counts of cycles of lengths 6 and 8 are determined.
- Simulation results show excellent performance of the constructed codes.

## Abstract

In this paper, we investigate the cycle structure inherent in the Tanner graphs of low-density parity-check (LDPC) codes constructed from balanced incomplete block designs (BIBDs). We begin by delineating the incidence structure of BIBDs and propose a methodology for constructing LDPC codes based on these designs. By analyzing the incidence relations between points and blocks within a BIBD, we prove that the resulting LDPC codes possess a girth of 6. Subsequently, we provide a detailed analysis of the cycle structure of the constructed LDPC codes and introduce a systematic approach for enumerating their short cycles. Using this method, we determine the exact numbers of cycles of lengths 6 and 8. Simulation results demonstrate that the constructed LDPC codes exhibit excellent performance.