Decoding Reed-Muller and Polar Codes by Successive Factor Graph Permutations

Abstract

Reed-Muller (RM) and polar codes are a class of capacity-achieving channel coding schemes with the same factor graph representation. Low-complexity decoding algorithms fall short in providing a good error-correction performance for RM and polar codes. Using the symmetric group of RM and polar codes, the specific decoding algorithm can be carried out on multiple permutations of the factor graph to boost the errorcorrection performance. However, this approach results in high decoding complexity. In this paper, we first derive the total number of factor graph permutations on which the decoding can be performed. We further propose a successive permutation (SP) scheme which finds the permutations on the fly, thus the decoding always progresses on a single factor graph permutation. We show that SP can be used to improve the error-correction performance of RM and polar codes under successive-cancellation (SC) and SC list (SCL) decoding, while keeping the memory requirements of the decoders unaltered. Our results for RM and polar codes of length 128 and rate 0.5 show that when SP is used and at a target frame error rate of 10 −4 , up to 0.5 dB and 0.1 dB improvement can be achieved for RM and polar codes respectively.

Hardware Implementation of Iterative Projection-Aggregation Decoding of Reed-Muller Codes

“…Successive-cancellation (SC) decoding [7] and SC list (SCL) decoding [8], make use of the decomposable structure of RM codes to provide recursive decoding methods with reasonable complexity. The work of [9] improved the performance of SC and SCL decoding methods by exploring several carefully selected permutations of the factor graph of RM codes. The work of [10] exploits the symmetric structure of RM codes and applies an iterative decoding method to provide near maximum likelihood (ML) performance.…”

Decoding Reed–Muller Codes Using Redundant Code Constraints

“…Due to the intractability of ML decoding, an important question is how to approach ML performance with reasonable decoding complexity in practice. Many recent approaches exploit the symmetry of RM codes by making use of their large automorphism group 1 [4], [5], [7], [8]. This typically results in code representations with many redundant code constraints, e.g., overcomplete parity-check matrices where the number of rows is much larger than the rank [7], [9].…”

Belief Propagation With Permutated Graphs of Polar Codes

“…Polar codes, with their low encoding and decoding complexity (successive cancellation, SC) [1], have been selected as the channel coding scheme for the control channel of the 5G communications [2] (enhanced mobile broadband scenarios, eMBB). The performance of the original SC decoding is further improved from various decoding techniques, such as the SC list (SCL) decoding [3] [4], the belief propagation (BP) decoding [5]- [8], and the SC and SCL decoding helped by a successive permutation scheme in [9].…”

“…empirical construction of permutated graphs with better performance than that of [5] and [8]. The authors in [9] allow the subcodes in each layer permutating freely, and thus producing more permutations of the factor graph than n!. A successive permutation is performed in [9] to find a permutation in each layer.…”

“…The authors in [9] allow the subcodes in each layer permutating freely, and thus producing more permutations of the factor graph than n!. A successive permutation is performed in [9] to find a permutation in each layer. Note that the permutations considered in this paper follow that of [5], [8] where subcodes in each layer permutate in the same way.…”