  The cyclically equivariant neural decoder was recently proposed in [Chen-Ye,
International Conference on Machine Learning, 2021] to decode cyclic codes. In
the same paper, a list decoding procedure was also introduced for two widely
used classes of cyclic codes -- BCH codes and punctured Reed-Muller (RM) codes.
While the list decoding procedure significantly improves the Frame Error Rate
(FER) of the cyclically equivariant neural decoder, the Bit Error Rate (BER) of
the list decoding procedure is even worse than the unique decoding algorithm
when the list size is small. In this paper, we propose an improved version of
the list decoding algorithm for BCH codes and punctured RM codes. Our new
proposal significantly reduces the BER while maintaining the same (in some
cases even smaller) FER. More specifically, our new decoder provides up to
$2$dB gain over the previous list decoder when measured by BER, and the running
time of our new decoder is $15\%$ smaller. Code available at
https://github.com/improvedlistdecoder/code
