In one embodiment of the present invention, the efficiency of an encoder can
be
maintained while reducing the complexity of a decoder. In one embodiment, the invention
includes a communication system in which a transmitter has a pipelined encoder
that includes a controller that does not access a look-up table to encode a block
of data bits into a binary codeword, and in which a receiver has a decoder to receive
the binary codeword after transmission through a communications medium, and to
correct an error in the binary codeword introduced during transmission without
using a look-up table.