Novel Code-Construction for (3, k) Regular Low Density Parity Check Codes

Communication system links that do not have the ability to retransmit generally rely
on forward error correction (FEC) techniques that make use of error correcting codes
(ECC) to detect and correct errors caused by the noise in the channel. There are
several ECC’s in the literature that are used for the purpose. Among them, the low
density parity check (LDPC) codes have become quite popular owing to the fact that
they exhibit performance that is closest to the Shannon’s limit.
This thesis proposes a novel code-construction method for constructing not only (3, k)
regular but also irregular LDPC codes. The choice of designing (3, k) regular LDPC
codes is made because it has low decoding complexity and has a Hamming distance,
at least, 4. In this work, the proposed code-construction consists of information submatrix
(Hinf) and an almost lower triangular parity sub-matrix (Hpar). The core design
of the proposed code-construction utilizes expanded deterministic base matrices in
three stages. Deterministic base matrix of parity part starts with triple diagonal matrix
while deterministic base matrix of information part utilizes matrix having all elements
of ones. The proposed matrix H is designed to generate various code rates (R) by
maintaining the number of rows in matrix H while only changing the number of
columns in matrix Hinf.
All the codes designed and presented in this thesis are having no rank-deficiency, no
pre-processing step of encoding, no singular nature in parity part (Hpar), no girth of
4-cycles and low encoding complexity of the order of (N + g2) where g2«N. The
proposed (3, k) regular codes are shown to achieve code performance below 1.44 dB
from Shannon limit at bit error rate (BER) of 10
−6
when the code rate greater than
R = 0.875. They have comparable BER and block error rate (BLER) performance
with other techniques such as (3, k) regular quasi-cyclic (QC) and (3, k) regular
random LDPC codes when code rates are at least R = 0.7. In addition, it is also shown
that the proposed (3, 42) regular LDPC code performs as close as 0.97 dB from
Shannon limit at BER 10
−6
with encoding complexity (1.0225 N), for R = 0.928 and
N = 14364 – a result that no other published techniques can reach.