Encoding-system includes a check-bit-generation-unit configured to generate N-ary-parity-bits by processing information composed of N-ary-symbols (where N is a power of 2) in modulo-N with a LDPC-matrix composed of binary elements, an encoded-sequence-generation-unit configured to generate an encoded sequence including the information composed of the N-ary-symbols and the N-ary-parity-bits, a modulation unit configured to modulate the encoded sequence in a modulation scheme having N-ary-modulation-symbols to produce a modulated signal, a demodulation-unit configured to demodulate the modulated signal to produce a demodulated signal, a metric-generation-unit configured to generate a metric for each of N-modulation-signal-points from the demodulated signal to obtain a plurality of metrics, and a decoding-unit configured to decode the encoded sequence by obtaining posteriori probabilities of the symbols in accordance with a state transition having an N-state defined by a binary LDPC-matrix corresponding to the LDPC-encoder encoding the encoded sequence, on the basis of the metrics.