It is well-established "folk knowledge" that in order to be iteratively decodable, a code should have a sparse parity-check matrix, as it has some qualitative and quantitative properties for finite-length codes.
The sum-product iterative decoder, conventionally used for low-density parity-check (LDPC) codes, hold promise as a decoder for general linear block code decoding. However, the promise is only partly fulfilled because, as we show experimentally, the decoder performance degrades rapidly as a function of parity check matrix weight. Even in the case of decoder failure, however, we demonstrate that there is information present in the decoder output probabilities that can still help with the decoding problem.
A new method of finding binary cyclic codes from the finite-field transform domain is presented. These cyclic codes have sparse parity-check matrix and thus are suitable for iterative decoding. Some interesting properties of the proposed construction method include the knowledge of the minimum distance and the ability to trade the increase in code dimension with a reduction in the parity-check matrix sparsity. By means of simulations, we show that the error correcting performance of the codes under iterative decoding is very close to the sphere-packing-bound constrained for binary transmission.
In this paper, we present a novel approach to improve belief propagation (BP) decoding performance by sparsifying the Tanner graph. Our proposed method first constructs a low-weight parity-check (LWPC) matrix with increased rows based on the original parity-check matrix. Then, all 4-cycles in the LWPC matrix are eliminated by adding auxiliary variable nodes and corresponding parity-check rows, resulting in the generalized LWPC (G-LWPC) matrix. Compared to the original matrix, the sparsity is greatly improved without altering the encoding constraints. Simulation results show that BP decoding based on the G-LWPC matrix is particularly effective for short code lengths. For a (128,20) polar code with 24-bit cyclic redundancy check, our proposed G-LWPC matrix reduces the density from 17.4% to 1.3%, and with 10 iterations, BP decoding can achieve the performance of successive cancellation list decoding with a list size of 32.
Belief propagation (BP) is a soft‐decision (SD) decoder that is commonly used to obtain near‐optimal decoding performance for linear codes defined over sparse parity‐check matrix. Nevertheless, the BP yields poor performance when used in the decoding of algebraic block codes, usually described by a dense parity‐check matrix. Hence, enhanced BP decoders transform the parity check matrix of such codes at each iteration for efficient decoding. In this article, the performance of the transformed parity‐check matrix of the iterative SD decoder is analysed for the class of binary cyclic codes using a perfect knowledge model (PKM). The PKM computes a list of candidate matrices and selects a baseline parity‐check matrix according to a distance metric. The selected matrix is optimal since it minimizes the probability of error over various choices in the list. Results show that, for a given channel condition, the conventional transformed matrix obtained by Gaussian elimination is sub‐optimal and does not necessarily contain unitary weighted columns at corresponding columns of the unreliable bits. Moreover, PKM can be used to verify the performances of newly developed iterative SD decoders for binary cyclic codes based on parity‐check equations instead of maximum‐likelihood decoding.