Plausibility Analysis of Shift-Sum Decoding for Cyclic Codes

摘要

Using the minimum weight dual codewords (MWDCs) of a cyclic code, the shift-sum decoding can correct errors beyond half of the code's minimum Hamming distance. It utilizes the frequency of the syndrome polynomials' coefficients to identify the erroneous positions and correct the errors. This paper analyzes the plausibility of the shift-sum decoding for both binary and non-binary cyclic codes. It first determines the probability distributions of the frequency of the syndrome polynomials' coefficients as well as their expected values for the erroneous and non-erroneous positions. Based on these characterizations, this work further provides an analysis for the iterative shift-sum decoding, unveiling the statistical rationale on the shift-sum decoding's capability of correcting errors beyond the half distance bound.