The stopping distance and stopping redundancy of a linear code are important concepts in the analysis of the performance and complexity of the code under iterative decoding on a binary erasure channel. In this paper, we studied the stopping distance and stopping redundancy of Finite Geometry LDPC (FG-LDPC) codes, and derived an upper bound of the stopping redundancy of FG-LDPC codes. It is shown from the bound that the stopping redundancy of the codes is less than the code length. Therefore, FG-LDPC codes give a good trade-off between the performance and complexity and hence are a very good choice for practical applications.
The stopping distance and stopping redundancy of a linear code are important concepts in the analysis of the performance and complexity of the code under iterative decoding on a binary erasure channel. In this paper, we studied the stopping distance and stopping redundancy of Finite Geometry LDPC (FG-LDPC) codes, and derived an upper bound of the stopping redundancy of FG-LDPC codes. It is shown from the bound that the stopping redundancy of the codes is less than the code length. Therefore, FG-LDPC codes give a good trade-off between the performance and complexity and hence are a very good choice for practical applications.
In this letter, the stopping sets and stopping distance of finite geometry LDPC (FG-LDPC) codes are studied. It is known that FG-LDPC codes are majority-logic decodable and a lower bound on the minimum distance can be thus obtained. It is shown in this letter that this lower bound on the minimum distance of FG-LDPC codes is also a lower bound on the stopping distance of FG-LDPC codes, which implies that FG-LDPC codes have considerably large stopping distance. This may explain in one respect why some FG-LDPC codes perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs.
On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.
Stopping set properties determine the iterative decoding performance of any linear block code on the Binary Erasure Channel (BEC). Furthermore, there is a common conception that they are crucial factors on other channels, like the Binary Symmetric Channel (BSC), as well. In order to investigate whether this is true indeed, we study stopping set properties and iterative decoding performance on the BSC of Finite-Geometry Low-Density Parity-Check (FG-LDPC) codes, which have been chosen due to their eectiveness and the good structure of their parity-check matrices (PCMs). Also modied versions of these PCMs are considered. As a rather surprising result, we show that a PCM with better stopping set properties than another PCM of the same code does not necessarily lead a better decoding performance on the BSC. Also, PCMs of the same code with the same stopping set properties, do not always have the same decoding performance on the BSC.
It is now well known that the performance of a linear code C under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for C. Several recent papers refer to this parameter as the \emph{stopping distance} s of C. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for C depends on the corresponding choice of a parity-check matrix. It is easy to see that s≤d, where d is the minimum Hamming distance of C, and we show that it is always possible to choose a parity-check matrix for C (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, termed the \emph{stopping redundancy} of C, defined as the minimum number of rows in a parity-check matrix H for C such that the corresponding stopping distance s(H) attains its largest possible value, namely s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.
It is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, the stopping redundancy of Copf, defined as the minimum number of rows in a parity- check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely, s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes
Low-density parity-check (LDPC) codes in conjunction with iterative decoding based on message-passing algorithms have been shown to achieve excellent performance over a variety of communication channels. In this dissertation, we investigate LDPC codes under two scenarios which present different challenges for providing robust communications. First, LDPC codes are analyzed and designed for time-selective, frequency-nonselective complex-fading channels where both the amplitude and the phase of the transmitted signal are altered by the channel. To combat the fading, we use a pilot symbol-assisted scheme where known pilot symbols are added to the LDPC code and used to help estimate the channel at the receiver. Using asymptotic, infinite-length analysis, we investigate several iterative, message-passing, joint-estimation-and-decoding strategies; optimal energy distribution between pilot and data symbols; and LDPC code design. Several interesting results are obtained regarding unification of analysis and code design. Next, we address the issue of achieving very low error rates with practical, finite-length LDPC codes for binary erasure channels. Finite-length LDPC codes suffer from error floors which limit the achievable error rate, and this error-floor performance is determined by stopping sets. Asymptotic analysis of weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with lower error floors but does not reflect the behavior of sublinearly-sized stopping sets, which can dominate the iterative-decoding, error-floor performance. Thus, we provide a perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, advantages of protograph structure, and effects of precoding. We show for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly-growing stopping sets, and this polynomial relationship can be approximately captured by a single parameter. Further, we begin to address the question, Given finite stopping-set sizes and finite codelengths, do the stopping sets follow the behavior predicted by linear or sublinear analysis?