Contents
7.1 Summary of Contributions . . . . 136 7.2 Possible Future Work . . . . 137
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This thesis addressed some issues of the binary and the NB LDPC codes. This chapter summarizes the main contributions of the thesis and then proposes some directions of research that may be useful for future researchers.
7.1 Summary of Contributions
The contributions of the thesis can be divided into four parts. The first two parts deal with the TSs and the short cycles which are responsible for the error floor in the iterative decoding of the binary LDPC codes. The last two parts consider the design of optimum puncturing patterns for the NB LDPC codes in order to achieve rate-compatibility. The contributions in these four parts are summarized below:
• Finding of a list of dominant trapping sets of an LDPC code is an important task. The list aids in the construction and the decoder design of an LDPC code. The literature contains several approaches to find a list of dominant trapping sets. We considered the hierarchical approach where a smaller trapping set is gradually expanded by adding one or more VNs to obtain larger trapping sets. Based on the hierarchical approach, we proposed a technique to find the dominant trapping sets of the irregular LDPC codes. We considered only six types of sources to find the trapping sets of a particular class. The proposed technique has been able to find the dominant trapping sets of several commonly considered irregular LDPC codes. Numerical results have shown that it can find more trapping sets of a class than those by other methods available in the literature.
• A dominant trapping set of an LDPC code usually contains one or more harmful short cycles. In addition to the length, the connectivity of a cycle is also crucial parameter. A poorly connected cycle affects the performance of the iterative decoder significantly. The connectivity of a cycle is measured by the EMD. Because of simplicity, the ACE is usually used as a substitute for the EMD. However, in many cases, the ACE is not equal to the EMD. A cycle with unequal ACE and EMD contains some sub-cycles. For an ACE spectrum constrained code, these sub-cycles must satisfy the girth and the ACE constraints specified by the particular ACE spectrum. Considering these constraints, we derived three sufficient conditions for the equality of the ACE and the EMD.
The first condition is based on the girth constraint and the remaining two are derived by taking
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7.2 Possible Future Work
the ACE constraints into account. Investigations on two ACE spectrum constrained codes were carried out to examine the efficiency of these sufficient conditions.
• We proposed an EXIT chart model to analyze bitwise and symbolwise puncturing patterns for the NB LDPC codes. We observed that the performance of a puncturing pattern is determined by the degrees of the VNs. The grouping algorithm [32] is a standard tool to identify the VNs which can be recovered after puncturing. It normally selects the low-degree VNs for puncturing.
By keeping the constraints from the grouping algorithm in mind, an EXIT chart based strategy is devised to obtain the optimum recoverable puncturing pattern for the NB LDPC codes. The novelty of this puncturing scheme arises from the joint use of both asymptotic (EXIT charts) and finite-length (grouping algorithm) techniques. The optimized recoverable puncturing patterns performed better than the other puncturing schemes available in the literature at different rates.
• We have studied the effects of the cycles in the context of the iterative decoding of the NB LDPC codes. The cycles which satisfy a certain condition based on the edge labels are harmful for the iterative decoder. These cycles are known as NB cycles. We find an expression for the probability of a cycle becoming an NB cycle for a code defined over GF(q). This probability depends only on q. The Tanner graph of an NB LDPC code defined over a very high order field, contains very few short NB cycles. However, some NB LDPC codes of higher mean column weights and defined over a smaller field may contain a sizable number of short NB cycles. We developed an effective RC puncturing technique for such NB LDPC codes. For that, we formulated a cycle- based rule to select the punctured VNs. The proposed rule selects those VNs which are involved in a lower number of short NB cycles with low EMD values. Simulation results in different contexts confirmed the usefulness and the efficiency of the cycle-based selection rule.
7.2 Possible Future Work
In continuation of the works presented in the thesis, there are several problems that can be the subject of future research. Some possible directions are outlined below.
• Using the list of dominant trapping sets, a decoder-based strategy may be devised to improve the error floor performance. The conventional decoding steps can be modified or some new steps can be appended in order to reduce the effects of the known dominant trapping sets.
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• The ACE is an approximation for the EMD. An algorithm may be developed to construct a parity-check matrix based on the EMD spectrum instead of ACE spectrum.
• There is a scope of deriving more sufficient conditions for the equality of the ACE and the EMD of a cycle of the ACE spectrum constrained LDCPC codes. In addition to the sufficient conditions, the necessary conditions may be derived. The necessary and the sufficient conditions together will result in the complete characterization of the equality of the ACE and the EMD of a cycle.
• Shortening reduces the rate of a code. An EXIT chart model can be formulated in the context of shortening for the NB LDPC codes. With the help of this EXIT chart, the optimum shortening pattern can be identified.
• The improved RC puncturing scheme for the NB LDPC codes discussed in Chapter 6 takes the EMD values of the short NB cycles into account. The performance of the proposed cycle-based scheme is investigated for a regular code. An equivalent cycle-based scheme involving the ACE values of the short NB cycles can be investigated in the case of the irregular codes. By comparing this scheme with the proposed scheme, the benefit of considering the EMD instead of the ACE can be analyzed.
• The cycles influence the binary LDPC codes more significantly than the NB LDPC codes. The cycle-based puncturing scheme proposed in Chapter 6 for the NB LDPC codes can be examined in the case of the binary LDPC codes.
• For the RC coding schemes in a fast changing environment, the selection of the proper trans- mission rate is very crucial. There have to be some feedback mechanisms from the receiver to the transmitter so that the coding system can be adapted to the suitable rate depending on the current SNR value. This means that the receiver must send some information back to the transmitter regarding the current SNR value. The transmitter can thus judiciously select the convenient rate. The proposed RC puncturing schemes for the NB LDPC codes may be studied in the context of a more practical environment with the inclusion a feedback system.