  Non-binary low-density parity-check (LDPC) codes have some advantages over
their binary counterparts, but unfortunately their decoding complexity is a
significant challenge. The iterative hard- and soft-reliability based
majority-logic decoding algorithms are attractive for non-binary LDPC codes,
since they involve only finite field additions and multiplications as well as
integer operations and hence have significantly lower complexity than other
algorithms. In this paper, we propose two improvements to the majority-logic
decoding algorithms. Instead of the accumulation of reliability information in
the existing majority-logic decoding algorithms, our first improvement is a new
reliability information update. The new update not only results in better error
performance and fewer iterations on average, but also further reduces
computational complexity. Since existing majority-logic decoding algorithms
tend to have a high error floor for codes whose parity check matrices have low
column weights, our second improvement is a re-selection scheme, which leads to
much lower error floors, at the expense of more finite field operations and
integer operations, by identifying periodic points, re-selecting intermediate
hard decisions, and changing reliability information.
