Splitting Sequences for Coding and Hybrid Incremental ARQ with Fragment Retransmission

This paper proposes a code defined on a finite ring ℤpM, where pM = 2m−1 is a Mersenne prime, and m is a binary size of ring elements. The code is based on a splitting sequence (splitting set) S, defined for the given multiplier set E=±20, ±21,…, ±2m−1. The elements of E correspond to the weights of binary error patterns that can be corrected, with the bidirectional single-bit error being the representative that occurs the most. The splitting set splits the code-word into sub-words, which inspired the name splitting code. Each sub-word, provided with auxiliary control symbols that are a byproduct of the coding procedure, corrects a single symbol error. The code can be defined, with some constraints, for general Mersenne numbers as well, while the multiplier set can be adjusted for adjacent binary errors correction. The application proposed for this code is a hybrid three-stage incremental ARQ procedure that transmits the code-word in the first stage, auxiliary control symbols in the second stage, and retransmits the sub-words detected as incorrect in the third stage. At each stage, error correction can be turned on or off, keeping both the retransmission rate and residual error rate at a low level.