Low-density parity-check (LDPC) codes are a good liner block parity codes. In this paper, the fundamental of LDPC codes are introduced briefly first. Then an algorithm called the minimum-weight codeword searching is discussed, which can be used to compute the minimum distance of LDPC codes. The effectiveness and the practicability of the algorithm are demonstrated by some examples. It is significant for us to analyze and optimize the design of LDPC codes.
Low-density parity-check (LDPC) codes are a good liner block parity codes. In this paper, the fundamental of LDPC codes are introduced briefly first. Then an algorithm called the minimum-weight codeword searching is discussed, which can be used to compute the minimum distance of LDPC codes. The effectiveness and the practicability of the algorithm are demonstrated by some examples. It is significant for us to analyze and optimize the design of LDPC codes.
Low-density parity-check (LDPC) codes in their broader-sense definition are linear codes whose parity-check matrices have fewer 1s than 0s. Finding their minimum distance is therefore in general an NP-hard problem. We propose a randomized algorithm called nearest nonzero codeword search (NNCS) approach to tackle this problem for iteratively decodable LDPC codes. The principle of the NNCS approach is to search codewords locally around the all-zero codeword perturbed by minimal noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum-Hamming- weight codeword whose Hamming weight is equal to the minimum distance of the linear code. This approach has its roots in Berrou et al.'s error-impulse method and a form of Fossorier's list decoding for LDPC codes.
Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.
In this letter, we obtain lower and upper bounds on the minimum distance dminof low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value amax, exhaustively. We also derive a lower bound Lneon the weight of non-elementary code words. By performing the search with amax= Lne, we either obtain an elementary code word with the smallest weight dmin, or establish the lower bound of Lneon dmin. For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on dmin. We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, dmin. Finding dmin, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.
In this paper, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m q. In the literature, the minimum/stopping distance of these codes (denoted by d(q, m) and h(q, m), respectively) has been thoroughly studied for m 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m = 6, the best known minimum distance upper bound, derived by Mittelholzer, is d(q, 6) 32. In this paper, we derive an improved upper bound of d(q, 6) 20 and a new upper bound d(q, 7) 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m 7 and low-to-moderate values of q ≤79.
In this work, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m <= q. In the literature, the minimum/stopping distance of these codes (denoted by d(q,m) and h(q,m), respectively) has been thoroughly studied for m <= 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m=6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q,6) <= 32. In this work, we derive an improved upper bound of d(q,6) <= 20 and a new upper bound d(q,7) <= 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m <= 7 and low-to-moderate values of q <= 79.