Gradient-descent decoding of one-step majority-logic decodable codes

In this paper, a new low-complexity gradient-descent based iterative majority-logic decoder (GD-MLGD) is proposed for decoding One-Step Majority-Logic Decodable (OSMLD) codes. We give a formulation of the decoding problem of binary OSMLD codes, as a maximization problem of a derivable objective function. The optimization problem is solved using a pseudo gradient-descent algorithm, which performs iteratively an update towards the optimal estimated codeword been transmitted, based on the first-order partial derivatives of each variable calculated in the previous iteration. The proposed decoding scheme achieves a fast convergence to an optimum codeword compared to other decoding techniques reviewed in this paper, at the cost of lower computational complexity. The quantized version (QGD-MLGD) is also proposed in order to further reduce the computational complexity. Simulation results show that the proposed decoding algorithms outperform all the existing majority-logic decoding schemes, and also various gradient-descent based bit-flipping algorithms, and performs nearly close to the belief propagation sum–product (BP-SP) decoding algorithm of LDPC codes, especially for high code lengths, providing an efficient trade-off between performance and decoding complexity. Moreover, the proposed quantized algorithm has shown to perform better than all the existing decoding techniques. The proposed decoding algorithms have shown to be suitable for ultra reliable, low latency and energy-constrained communication systems where both high performances and low-complexity are required.