  Polar codes are of great interests because they provably achieve the capacity
of both discrete and continuous memoryless channels while having an explicit
construction. Most existing decoding algorithms of polar codes are based on
bit-wise hard or soft decisions. In this paper, we propose symbol-decision
successive cancellation (SC) and successive cancellation list (SCL) decoders
for polar codes, which use symbol-wise hard or soft decisions for higher
throughput or better error performance. First, we propose to use a recursive
channel combination to calculate symbol-wise channel transition probabilities,
which lead to symbol decisions. Our proposed recursive channel combination also
has a lower complexity than simply combining bit-wise channel transition
probabilities. The similarity between our proposed method and Arikan's channel
transformations also helps to share hardware resources between calculating bit-
and symbol-wise channel transition probabilities. Second, a two-stage list
pruning network is proposed to provide a trade-off between the error
performance and the complexity of the symbol-decision SCL decoder. Third, since
memory is a significant part of SCL decoders, we propose a pre-computation
memory-saving technique to reduce memory requirement of an SCL decoder.
Finally, to evaluate the throughput advantage of our symbol-decision decoders,
we design an architecture based on a semi-parallel successive cancellation list
decoder. In this architecture, different symbol sizes, sorting implementations,
and message scheduling schemes are considered. Our synthesis results show that
in terms of area efficiency, our symbol-decision SCL decoders outperform both
bit- and symbol-decision SCL decoders.
