Constellations design for multiple transmissions maximizing the minimum squared Euclidean distance

During the so-called ARQ the same date is retransmitted many times to reduce the probability of error. In this paper, we consider the case when each transmission carries exactly the same information content (bits) as the first one and we focus on the constellation design aimed at improvement of the receiver performance. The performance for high SNR depends on the minimum Euclidean distance between symbols which grows as fast as the number of transmissions if the same modulation and bit-to-symbol mapping are used. It was shown in the literature that by modifying the bit-to-symbol mapping the Euclidean distance may be made to increase faster than the number of transmissions. Despite possible performance gains, no method was proposed to optimally design the mapping constellation for each of the transmissions, mainly, because the exhaustive search is out of reach of modern computers for constellations with more than 8 symbols. In this paper, we propose an algorithm which efficiently finds optimal (unlike the known heuristics which yield sub-optimal solutions) mappings for constellation sizes 16 and 32.