Shift-Sum Decoding of Non-Binary Cyclic Codes

This paper proposes a novel shift-sum decoding method for non-binary cyclic codes, which only requires finite field operations but yields advanced decoding performance. Using the cyclically different minimum-weight dual codewords (MWDCs) and their proper shifts, a frequency matrix can be obtained as a reliability metric for identifying the error positions and magnitudes. By analyzing the statistical distributions of the matrix entries, the rationale for the shift-sum decoding’s advanced error-correction capability is revealed. Based on this decoding method, a hard-decision iterative shift-sum (HISS) decoding algorithm is first proposed. It can correct errors beyond half of the code’s minimum Hamming distance. By further utilizing the reliability information obtained from the channel, a soft-decision iterative shift-sum (SISS) decoding algorithm is then proposed to improve the decoding performance. Both the HISS and the SISS algorithms are realized only with polynomial multiplications and numerical comparisons, which are hardware-friendly. To further improve the error-correction performance, the HISS and SISS algorithms can be integrated in a Chase decoding mechanism for handling the test-vectors. Simulation results on Reed-Solomon (RS) and non-binary BCH (NB-BCH) codes show that the proposed algorithms yield a competent decoding and complexity performances in comparison with the existing decoding algorithms.