Generalized Product Codes Research Articles

Density Evolution for Deterministic Generalized Product Codes on the Binary Erasure Channel at High Rates

Generalized product codes (GPCs) are extensions of product codes (PCs), where code symbols are protected by two component codes but not necessarily arranged in a rectangular array. We consider a deterministic construction of GPCs (as opposed to randomized code ensembles) and analyze the asymptotic performance over the binary erasure channel under iterative decoding. Our code construction encompasses several classes of GPCs previously proposed in the literature, such as irregular PCs, blockwise braided codes, and staircase codes. It is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity. We show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph. Using a convergence result for these graphs, we derive the density evolution equations that characterize the asymptotic decoding performance. As an application, we discuss the design of irregular GPCs, employing a mixture of component codes with different erasure-correcting capabilities.

Approaching Capacity at High Rates with Iterative Hard-Decision Decoding

A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with message-passing decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. In this paper, we show that one can approach capacity at high rates using iterative hard-decision decoding (HDD) of generalized product codes. Specifically, a class of spatially-coupled GLDPC codes with BCH component codes is considered, and it is observed that, in the high-rate regime, they can approach capacity under the proposed iterative HDD. These codes can be seen as generalized product codes and are closely related to braided block codes. An iterative HDD algorithm is proposed that enables one to analyze the performance of these codes via density evolution (DE).

Generalized product codes and their performance

AbstractProduct codes show improved correcting capability by a comparatively simple method of combining codes and are also effective in correcting bursty errors. In this study, product codes using several kinds of single‐error correcting and double‐error detecting (SEC‐DED) codes, different by rows, are proposed. A method of probabilistically correcting bursty errors by using SEC‐DED codes different by rows and utilizing the differences in these error‐correcting characteristics is proposed. In addition, a decoding method that can correct random errors of the type that could not be corrected by the conventional product codes for this code is proposed.