  We present novel decoding schemes for hard and soft decision decoding of
block codes using the minimal weight codewords of the dual code. The decoding
schemes will be described for cyclic codes where polynomials can be used,
however, the modification for non-cyclic codes is possible and straight
forward. The hard decision decoding calculates syndrome polynomials which are
the product of the received polynomial with dual codewords. Proper cyclic
shifts of these syndrome polynomials are obtained and the non-zero positions
are counted componentwise for these shifts. The values of this counting are a
reliability measure and can be used for locating the error and also the
non-error positions. This reliability measure is the basis for various variants
of hard decision decoding algorithms. Decoding schemes with iterative error
reduction are possible as well as information set decoding using the inherent
reliability information of the measure even if there is no reliability
information from the channel. Further, we will show how reliability information
from the channel can be included in order to obtain soft decision decoding
schemes. We derive the relation between bit flipping, believe propagation, and
majority logic decoding to the novel schemes. As examples to illustrate the
functioning we use BCH and Reed-Muller codes as examples for binary codes, and
RS codes for non-binary codes. Besides the Plotkin construction we recall a
known result that Reed-Muller codes punctured by one position are cyclic and
thus, are equivalent to special cases of BCH codes. Simulation results for hard
and soft decision decoding will be given for several examples and compared with
results from literature. Finally, we analyze the soft decision decoding of the
Plotkin construction and derive that one of the two codes uses a $3$ dB better
channel (also known as channel polarization).
