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.
Low density parity check (LDPC) codes have stimulated a lot of interest in recent years due to their capacity approaching performance and availability of fast decoding algorithms. The existing encoding scheme for LDPC code usually incurs too high complexity and should be changed to encoding scheme with low complexity. An elementary turbo structured Tanner graph construction and encoding scheme for a family of regular LDPC codes attaining desired high girth is studied in this paper. The construction provides flexibility in choice of parameters like column weight (j > =2), row weight k and arbitrary girth g. Turbo structured low density parity check ( TS-LDPC) codes are regular LDPC codes with their Tanner graph consists of two trees connected by an interleaver. Linear complexity is attained with few alterations and restrictions of in Tanner graph of turbo structured LDPC codes. Simulation results demonstrates that not only the bit error rate (BER) performance of TSLDPC codes at low signal to noise ratio (SNR) is comparable with error performance of corresponding random LDPC codes, with better error performance at high SNR but linear encoding considerably reduces simulation time.
The main subjects of the thesis are LDPC codes constructed from combinatorial structures. Combinatorial structures that have been used for the construction are:
μ-geodetic graphs obtained from block designs, strongly regular graphs with parameters
(v,k,0,1), and block designs with parameters
(45,5,1). In the thesis we have constructed a family of LDPC codes using
μ-geodetic graphs obtained from block
(v,k,λ) designs. We have shown that the Tanner graphs of LDPC codes constructed from
μ-geodetic graphs obtained from block designs with
k=3 are free of cycles of length four, and we have analyzed properties of the constructed codes. We have examined a connection between the parameters of the constructed codes and the parameters of the initial design. The existence and the structures of absorbing sets in the Tanner graphs of the constructed codes have been analyzed. Analysis of structures of the absorbing sets enabled us to determine the exact number of the absorbing sets of certain size using parameters of the initial design. For construction of LDPC codes we have used strongly regular graphs with parameters
(v,k,0,1) since the Tanner graph of the corresponding LDPC code is free of cycles of length four. We have examined the existence and the structures of absorbing sets in the Tanner graphs of the constructed codes. LDPC codes from some block designs with parameters
(45,5,1) have been constructed. Simulation results for the constructed LDPC codes have been presented.