Eliminating trapping sets in low-density parity-check codes by using Tanner graph covers

Abstract

We discuss error floor asympotics and present a method for improving the performance of low-density parity check (LDPC) codes in the high SNR (error floor) region. The method is based on Tanner graph covers that do not have trapping sets from the original code. The advantages of the method are that it is universal, as it can be applied to any LDPC code/channel/decoding algorithm and it improves performance at the expense of increasing the code length, without losing the code regularity, without changing the decoding algorithm, and, under certain conditions, without lowering the code rate. The proposed method can be modified to construct convolutional LDPC codes also. The method is illustrated by modifying Tanner, MacKay and Margulis codes to improve performance on the binary symmetric channel (BSC) under the Gallager B decoding algorithm. Decoding results on AWGN channel are also presented to illustrate that optimizing codes for one channel/decoding algorithm can lead to performance improvement on other channels.

Lowering the Error Floor of LDPC Codes Using Cyclic Liftings

“…This is achieved by removing the short cycles which form the dominant trapping sets. Our work has similarities to [10] and [20]. The similarity with both [10] and [20] is that we also use graph covers or liftings to improve the error floor performance of a base code.…”

“…One category of such literature, focusses on modification of iterative decoding algorithms, see, e.g., [9], while another category is concerned with the code construction. In the second category, some researchers use indirect measures such as girth [18] or approximate cycle extrinsic message degree (ACE) [19], while others work with direct measures of error floor performance such as the distribution of stopping sets or trapping sets [20], [10], [11]. In [20], edge swapping is proposed as a technique to increase the stopping distance of an LDPC code, and thus to improve its error floor performance over the BEC.…”

“…Random cyclic liftings are also studied in [20] and shown to improve the average performance of the ensemble in the error floor region compared to the base code. Ivkovic et al [10] apply the same technique of edge swapping between two copies of a base LDPC code to eliminate the dominant trapping sets of the base code over the BSC.…”

Deriving Good LDPC Convolutional Codes from LDPC Block Codes

“…In this section we briefly discuss some other graph-coverbased LDPC code constructions proposed in the literature, namely by Ivkovic et al [44], by Divsalar et al [43], [45], by Lentmaier et al [46], [47], and by Kudekar et al [48].…”

“…Namely, in terms of our notation, Ivkovic et al [44] start with a parity-check matrix H, choose the set L {0, 1}, a collection of zero-one matrices [43], [45] is the so-called rate-1/2 AR4JA LDPC code construction, which was also considered earlier in Example 19. A particularly attractive, from an implementation perspective, version of this code construction is obtained by an iterated graph-cover construction procedure, where each graph-cover construction is based on a cyclic cover, as in the application of GCC1 in Example 4.…”