An iterative decoding algorithm for convolutional codes is presented. It successively processes $N$ consecutive blocks of the received word in order to decode the first block. A bound is presented showing which error configurations can be corrected. The algorithm can be efficiently used on a particular class of convolutional codes, known as doubly cyclic convolutional codes. Due to their highly algebraic structure those codes are well suited for the algorithm and the main step of the procedure can be carried out using Reed-Solomon decoding. Examples illustrate the decoding and a comparison with existing algorithms is being made.
This paper defines cyclic and minimal derivative descendants (DDs) of an extended cyclic code from the derivative of the Mattson-Solomon polynomials, respectively. First, it demonstrates that the cyclic DDs are the same extended cyclic code. It allows us to perform soft-decision decoding for extended cyclic codes based on their cyclic DDs. Then, it proves that the minimal DDs are equivalent codes. It also allows us to perform soft-decision decoding based on the minimal DDs with permutations. Simulation results show that our proposed derivative decoding can be close to the maximum likelihood de…
Recently deep neural networks have been successfully applied in channel coding to improve the decoding performance. However, the state-of-the-art neural channel decoders cannot achieve high decoding performance and low complexity simultaneously. To overcome this challenge, in this paper we propose doubly residual neural (DRN) decoder. By integrating both the residual input and residual learning to the design of neural channel decoder, DRN enables significant decoding performance improvement while maintaining low complexity. Extensive experiment results show that on different types of channel c…
We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed met…
The cyclically equivariant neural decoder was recently proposed in [Chen-Ye, International Conference on Machine Learning, 2021] to decode cyclic codes. In the same paper, a list decoding procedure was also introduced for two widely used classes of cyclic codes -- BCH codes and punctured Reed-Muller (RM) codes. While the list decoding procedure significantly improves the Frame Error Rate (FER) of the cyclically equivariant neural decoder, the Bit Error Rate (BER) of the list decoding procedure is even worse than the unique decoding algorithm when the list size is small. In this paper, we propo…
Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School（清华-伯克利深圳学院，清华大学深圳国际研究生院）
Neural decoders were introduced as a generalization of the classic Belief Propagation (BP) decoding algorithms, where the Trellis graph in the BP algorithm is viewed as a neural network, and the weights in the Trellis graph are optimized by training the neural network. In this work, we propose a novel neural decoder for cyclic codes by exploiting their cyclically invariant property. More precisely, we impose a shift invariant structure on the weights of our neural decoder so that any cyclic shift of inputs results in the same cyclic shift of outputs. Extensive simulations with BCH codes and pu…
We present novel decoding schemes for hard and soft decision decoding of block codes using the minimal weight codewords of the dual code. The decoding schemes will be described for cyclic codes where polynomials can be used, however, the modification for non-cyclic codes is possible and straight forward. The hard decision decoding calculates syndrome polynomials which are the product of the received polynomial with dual codewords. Proper cyclic shifts of these syndrome polynomials are obtained and the non-zero positions are counted componentwise for these shifts. The values of this counting ar…
As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented.
We investigate iterative low-resolution message-passing algorithms for quasi-cyclic LDPC codes with horizontal and vertical layered schedules. Coarse quantization and layered scheduling are highly relevant for hardware implementations to reduce the bit width of messages and the number of decoding iterations. As a novelty, this paper compares the two scheduling variants in combination with mutual information maximizing compression operations in variable and check nodes. We evaluate the complexity and error rate performance for various configurations. Dedicated hardware architectures for regular…
Using techniques and results from Kudekar et al. we strengthen the bounds on the weight distribution of linear codes achieving capacity on the BEC, which were shown by the first author. In particular, we show that for any doubly transitive binary linear code $C \subseteq \{0,1\}^n$ of rate $0 < R < 1$ with weight distribution $\left(a_0,...,a_n\right)$ holds $a_i \le 2^{o(n)} \cdot (1-R)^{-2 \ln 2 \cdot \min\{i, n-i\}}$. For doubly transitive codes with minimal distance at least $Ω\left(n^c\right)$, $0 < c \le 1$, the error factor of $2^{o(n)}$ in this bound can be removed at the cost of repla…
Polar codes were introduced in 2009 and proven to achieve the symmetric capacity of any binary-input discrete memoryless channel under low-complexity successive cancellation decoding. In this thesis, we construct cyclic polar codes based on a mixed-radix Cooley-Tukey decomposition of the Galois field Fourier transform. The main results are: we can, for the first time, construct, encode and decode polar codes that are cyclic, with their blocklength being arbitrary; for a given target block erasure rate, we can achieve significantly higher code rates on the erasure channel than the original pola…
In this paper, we consider how to partition the parity-check matrices (PCMs) to reduce the hardware complexity and computation delay for the row layered decoding of quasi-cyclic low-density parity-check (QC-LDPC) codes. First, we formulate the PCM partitioning as an optimization problem, which targets to minimize the maximum column weight of each layer while maintaining a block cyclic shift property among different layers. As a result, we derive all the feasible solutions for the problem and propose a tight lower bound $ω_{LB}$ on the minimum possible maximum column weight to evaluate a soluti…
