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Novel bitwise retransmission schemes are devised which retransmit only the bits received with small reliability. The retransmissions are used to accumulate the reliabilities of individual bits. Unlike the conventional automatic repeat request (ARQ) schemes, the proposed scheme does not require a checksum for the error detection. The bits to be retransmitted are reported as a combination number, or two synchronized random number generators (RNGs) at the transmitter and receiver are used to greatly compress the feedback message. The bitwise retransmission decisions and/or combining can be performed after the demodulation or after the channel decoding at the receiver. The bit-error rate (BER) expressions are derived for the case of one and two retransmissions, and verified by computer simulations. Assuming three specific retransmission strategies, the scheme parameters are optimized to minimize the overall BER. For the same number of retransmissions and packet length, the proposed schemes always outperform the frequently used stop-and-wait ARQ. The impact of feedback errors is also considered. Finally, practical designs of the bitwise retransmissions for data fusion from sensor nodes in Zigbee, Wifi and Bluetooth networks are presented.