Patent Publication Number: US-11038597-B2

Title: Communication system and communication apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2019-164929, filed on Sep. 10, 2019, the entire contents of which are incorporated herein by reference. 
     FIELD 
     The embodiments discussed herein are related to an encoding circuit, a decoding circuit, an encoding method, a decoding method, a transmission device, and an optical transmission system. 
     BACKGROUND 
     With the increase in the transmission capacity of the optical transmission device, for example, multi-level modulation schemes are used such as quadrature phase shift keying (QPSK), 16 quadrature amplitude modulation (16QAM), and 64QAM. In the multi-level modulation schemes, among symbols arranged in a constellation, a symbol corresponding to a value of each bit string in a frame signal to be modulated is assigned to a bit string, whereby an optical signal is generated having a phase and intensity in accordance with the symbol. 
     Probabilistic shaping (PS) technology (hereinafter referred to as “PS”) forms a probability distribution of symbol assignment by converting the value of the bit string so that a symbol closer to the center of the constellation is assigned more. The noise tolerance is therefore improved of the signal light generated from the frame. 
     For the PS, for example, distribution matching (DM) processing is used for increasing the mark rate of the bit string to a rate greater than 50(%) (for example, 80(%)). The probability of symbol assignment is therefore biased toward the center of the constellation only in a specific quadrant among the first to fourth quadrants that divide the constellation. Thereafter, a quadrant in which the symbol to be assigned is located is determined from the first to fourth quadrants. 
     In determination of the quadrant, a parity bit of an error correction code such as forward error correction (FEC) may be used (for example, see Non-Patent Document that is F. Buchali, et al., “Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration”,  JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL., 34, NO. 7, Apr. 1, 2016). Since the mark rate of the error correction code is maintained close to 50(%), the first to fourth quadrants are selected with substantially the same probability, and a probability distribution biased toward the center of the constellation is formed over all quadrants. 
     Examples of the error correction code encoding method include, for example, bit-interleaved coded modulation (BICM) and multilevel coding (MLC) (for example, see Patent Document and Non-Patent Documents, the patent document is Japanese Laid-open Patent Publication No. 2008-187706, the Non-patent Documents are U. Wachsmann, et al., “Multilevel Codes: Theoretical Concepts and Practical Design Rules”,  IEEE TRANSACTIONS ON INFORMA TION THEORY , VOL 45, NO. 5, July 1999, A. Bisplinghoff, et al., “Low-Power, Phase-Slip Tolerant, Multilevel Coding for M-QAM”,  JOURNAL OF LIGHTWAVE TECHNOLOGY , VOL., 35, NO. 4, Feb. 15, 2017, and Y. Koganei, et al., “ Multilevel Coding with Spatially - Coupled Codes for beyond  400  Gbps Optical Transmission ”, OK, 2018, Tu3C.2). The BICM is an encoding method that collectively encodes each bit string without discrimination by level (most significant bit (MSB) or least significant bit (LSB)). The MLC is an encoding method that divides bit strings for respective levels and individually generates error correction codes. 
     SUMMARY 
     According to an aspect of the embodiments, an optical communication system comprising a first communication device configured to transmit optical signals and a second communication device configured to receive the optical signals, wherein the first transmission device includes an encoding circuit configured to assign, to a plurality of bit strings, symbols each corresponding to a value of every one of the plurality of bit strings, the symbols being among a plurality of symbols in a constellation of a multi-level modulation scheme, convert values of bit strings that are among the plurality of bit strings and other than a first bit string to cause a symbol that is among the plurality of symbols and closer to a center of the constellation to be assigned more, generate a first error correction code from the plurality of bit strings for each of frames, generate a second error correction code from a second bit string among the plurality of bit strings in every one of a plurality of periods that divide a period of a frame, delay the first error correction code and inserts the first error correction code into the first bit string in a frame subsequent to the frame, and delay the second error correction code and inserts the second error correction code into the first bit string in a period that is later than a period of the second bit string used to generate the second error correction code, the periods being among the plurality of periods, wherein the encoding circuit uses the delayed first error correction code and the delayed second error correction code to convert a value of the second bit string. 
     The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a configuration diagram illustrating an example of an optical transmission system; 
         FIG. 2  is a configuration diagram illustrating an example of a transponder; 
         FIG. 3  is a configuration diagram illustrating an example of an encoding circuit using BICM; 
         FIG. 4  is a configuration diagram illustrating an example of a decoding circuit using BICM; 
         FIG. 5  is a diagram illustrating an example of processing of PS; 
         FIG. 6  is a diagram illustrating an example of symbol mapping; 
         FIG. 7  is a diagram illustrating an example of a probability distribution of symbol assignment before and after XOR; 
         FIG. 8  is a configuration diagram illustrating an example of an encoding circuit using MLC; 
         FIG. 9  is a configuration diagram illustrating an example of a decoding circuit using MLC; 
         FIG. 10A  is a diagram illustrating another example of the symbol mapping; 
         FIG. 10B  is a diagram illustrating an example of the probability distribution of symbol assignment before and after XOR; 
         FIG. 11  is a configuration diagram illustrating another example of the encoding circuit using MLC; 
         FIG. 12  is a diagram illustrating a frame format of n output signal output from an encoding circuit of a first embodiment; 
         FIG. 13  is a configuration diagram illustrating the encoding circuit of the first embodiment; 
         FIG. 14  is a configuration diagram illustrating a decoding circuit of the first embodiment; 
         FIG. 15  is a diagram illustrating a frame format of an output signal output from an encoding circuit of a second embodiment; 
         FIG. 16  is a configuration diagram illustrating the encoding circuit of the second embodiment; 
         FIG. 17  is a configuration diagram illustrating a decoding circuit of the second embodiment; 
         FIG. 18  is a diagram illustrating a frame format of an output signal output from an encoding circuit of a third embodiment; 
         FIG. 19  is a configuration diagram illustrating the encoding circuit of the third embodiment; and 
         FIG. 20  is a configuration diagram illustrating a decoding circuit of the third embodiment. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
       FIG. 1  is a configuration diagram illustrating an example of an optical transmission system. The optical transmission system includes a pair of wavelength multiplexing optical transmission devices  7   a  and  7   b  connected to each other via transmission lines Da and Db such as optical fibers. The wavelength multiplexing optical transmission devices  7   a  and  7   b  mutually transmit and receive wavelength multiplexed optical signals S in which a plurality of optical signals having respective different wavelengths is subjected to wavelength multiplexing. 
     The wavelength multiplexing optical transmission device  7   a  includes a plurality of transponders  1   a , an optical multiplexing unit  30   a , an optical demultiplexing unit  31   a , optical amplifiers  32   a  and  33   a , and a management unit  6   a . Furthermore, the wavelength multiplexing optical transmission device  7   b  includes a plurality of transponders  1   b , an optical multiplexing unit  30   b , an optical demultiplexing unit  31   b , optical amplifiers  32   b  and  33   b , and a management unit  6   b.    
     The transponders  1   a  and  1   b  are examples of first and second transmission devices, respectively, and transmit and receive optical signals. The optical signals have, for example, an OTUCn frame format defined in ITU-T Recommendation G.709. 
     The transponders  1   a  and  1   b  are connected to network (NW) devices  9  such as routers on the client network side. The transponders  1   a  and  1   b  transmit and receive a plurality of client signals to and from the network devices  9 . The transponders  1   a  and  1   b  store the plurality of client signals from the network devices  9  in a common frame and output the signals to the optical multiplexing units  30   a  and  30   b , and extract the plurality of client signals from the frame from the optical demultiplexing units  31   a  and  31   b  to transmit the signals to the network devices  9 . 
     The optical multiplexing units  30   a  and  30   b  are, for example, optical selection switches or optical filters, and perform wavelength multiplexing on optical signals input from the plurality of transponders  1   a  and  1   b  to obtain the wavelength multiplexed optical signals S, and output the wavelength multiplexed optical signals S to the optical amplifiers  32   a  and  32   b . The optical amplifiers  32   a  and  32   b  amplify the wavelength multiplexed optical signals S and output the amplified wavelength multiplexed optical signals S to the transmission lines Da and Db. 
     The wavelength multiplexed optical signals S are input from the transmission lines Da and Db to the optical amplifiers  33   a  and  33   b . The optical amplifiers  33   a  and  33   b  amplify the wavelength multiplexed optical signals S and output the amplified wavelength multiplexed optical signals S to the optical demultiplexing units  31   a  and  31   b.    
     The optical demultiplexing units  31   a  and  31   b  are, for example, optical selection switches or optical filters, and separate the wavelength multiplexed optical signals S into optical signals of respective wavelengths. The optical signals are input from the optical demultiplexing units  31   a  and  31   b  to the plurality of transponders  1   a  and  1   b.    
     The management units  6   a  and  6   b  are, for example, circuits including a processor such as a central processing unit (CPU), and control the wavelength multiplexing optical transmission devices  7   a  and  7   b . The management units  6   a  and  6   b , for example, set gains for the optical amplifiers  32   a  and  32   b , and set wavelength multiplexing target frames for the optical multiplexing units  30   a  and  30   b . Furthermore, the management units  6   a  and  6   b , for example, set optical signals to be separated by the optical demultiplexing units  31   a  and  31   b , and perform settings related to storing of the client signals in the frame by the transponders  1   a  and  1   b.    
       FIG. 2  is a configuration diagram illustrating an example of the transponders  1   a  and  1   b . The transponders  1   a  and  1   b  include a plurality of transmission/reception modules  10 , a framer chip  11 , a digital signal processor (DSP)  12 , an analog-digital conversion unit (DA/AD)  13 , an analog coherent optics (ACO)  14 , and a setting processing unit  15 . 
     The transmission/reception modules  10  are optical modules detachable from a circuit board on which the framer chip  11  is mounted via an electrical connector, for example. The transmission/reception modules  10  transmit and receive client signals to and from the network devices  9 . Examples of the frame format of the client signals include a synchronous optical network (SONET) frame and a GigabitEthernet (registered trademark) (GbE) frame, but are not limited to this. 
     First, description will be given of processing in the uplink direction toward the ACO  14  from the transmission/reception modules  10 . 
     The transmission/reception modules  10  perform light-electricity conversion on client signals received from the network devices  9 , and output the converted client signals to the framer chip  11 . The framer chip  11  stores the client signals input from the respective transmission/reception modules  10  in a frame. In the present embodiment, an example of the frame is an OTUCn frame, but is not limited to this, and another frame may be used. 
     The framer chip  11  outputs the frame to the DSP  12 . The DSP  12  generates an error correction code for the frame, modulates the frame with a multi-level modulation scheme, and outputs the modulated frame to the analog-digital conversion unit  13 . The analog-digital conversion unit  13  converts the frame from a digital signal to an analog signal and outputs the converted frame to the ACO  14 . The ACO  14  converts the frame from an electric signal to an optical signal and outputs the optical signal to the optical multiplexing units  30   a  and  30   b.    
     Next, description will be given of processing in the downlink direction toward the transmission/reception modules  10  from the ACO  14 . 
     The ACO  14  receives an optical signal, converts the optical signal into an electric signal, and outputs the electric signal to the analog-digital conversion unit  13 . Note that the electric signal has the frame structure described above. The analog-digital conversion unit  13  converts the electric signal from an analog signal to a digital signal, and output the digital signal to the DSP  12 . The DSP  12  performs demodulation processing on the electric signal to reproduce the frame, performs error correction, and then outputs the frame to the framer chip  11 . Note that the ACO  14  is an example of first and second conversion circuits. 
     The framer chip  11  extracts client signals from the frame and outputs the client signals to the transmission/reception modules  10 . The transmission/reception modules  10  convert the client signals from electric signals to optical signals and transmit the optical signals to the network devices  9 . 
     Furthermore, the setting processing unit  15  performs various settings for the framer chip  11 , the DSP  12 , and the ACO  14  in accordance with instructions of the management units  6   a  and  6   b.    
     Furthermore, the DSP  12  includes an encoding circuit  120  for encoding a plurality of bit strings in an uplink direction frame, and a decoding circuit  121  for decoding a plurality of bit strings in a downlink direction frame. Each of the bit strings is a series of bit values obtained by performing parallel conversion on serial data of the frame. 
     (Encoding and Decoding by BICM) 
       FIG. 3  is a configuration diagram illustrating an example of the encoding circuit  120  using BICM. The encoding circuit  120  includes a PS conversion unit  29 , an HD-FEC generation unit  24 , an SD-FEC generation unit  25 , and a symbol mapping unit  27 . The PS conversion unit  29  includes DM processing units  21   a  and  21   b  and an exclusive OR (XOR) operator  23 . Note that, in the present embodiment, 64 QAM is given as an example of the multi-level modulation scheme, but the multi-level modulation scheme is not limited to this. 
     A frame signal Sin input from the framer chip  11  is divided into three bit strings of level- 0  to level- 2  by serial-parallel conversion, for example. Here, the bit string of level- 2  is the MSB, and level- 0  is the LSB. Each of the bit strings of level- 0  to level- 2  is transmitted onto an individual lane. 
     The PS conversion unit  29  forms a probability distribution of symbol assignment for each of the bit strings of level- 0  to level- 2  by PS. The DM processing unit  21   a  performs DM processing on the bit string of level- 1  and the DM processing unit  21   b  performs DM processing on the bit string of level- 0 . The mark rate of each of the bit strings of level- 0  and level- 1  therefore increases to a rate greater than 50(%) (for example, 80(%)), and the number of values of “1” becomes greater than that of “0” in each of the bit strings of level- 0  and level- 1 . 
     The XOR operator  23  performs XOR the value of the bit string of level- 0  with the value of the bit string of level- 1 . The value of the bit string of level- 0  therefore becomes a value obtained by XORing the original value of the bit string of level- 0  with the value of the bit string of level- 1 . Each of the bit strings is output from the PS conversion unit  29  to the HD-FEC generation unit  24 . 
     The HD-FEC generation unit  24  generates an HD-FEC parity that is a hard decision code, from the bit strings of level- 0  to level- 2 . The HD-FEC generation unit  24  inserts the HD-FEC parity into the bit string of level- 2 . Each of the bit strings is output from the HD-FEC generation unit  24  to the SD-FEC generation unit  25 . Note that the HD-FEC parity is an example of a first error correction code used for correction of an error in a result of hard decision on each bit string. 
     The SD-FEC generation unit  25  generates an SD-FEC parity that is a soft decision code, from the bit strings of level- 0  to level- 2 . The SD-FEC generation unit  25  inserts the SD-FEC parity into the bit string of level- 2 . Each of the bit strings is output from the SD-FEC generation unit  25  to the symbol mapping unit  27 . Note that the SD-FEC parity is an example of a second error correction code used for correction of an error in a result of soft decision on each bit string. 
     The symbol mapping unit  27  assigns, to the bit strings, symbols corresponding to values of the bit strings of level- 0  to level- 2 , the symbols being among a plurality of symbols in a constellation of 64QAM. The symbol mapping unit  27  outputs an output signal Sout corresponding to the assigned symbol to the analog-digital conversion unit  13 . 
     Reference numeral  90  indicates contents of the bit strings in the frame input to the symbol mapping unit  27 . The bit strings of level- 0  and level- 1  include data # 0  and data # 1  on which DM processing has been performed, respectively. 
     Furthermore, the bit string of level- 2  includes data # 2  that has not been subjected to DM processing, the HD-FEC parity, and the SD-FEC parity. The HD-FEC parity and the SD-FEC parity are inserted into a period Ta within a period T of the frame, and the data # 2  is inserted into a period Tb within the period T of the frame. The periods Ta and Tb are set so that the HD-FEC parity and the SD-FEC parity are, for example, about 20(%) of the amount of data of the entire frame. 
       FIG. 4  is a configuration diagram illustrating an example of the decoding circuit  121  using BICM. The decoding circuit  121  includes a soft decision unit  41 , an SD-FEC decoding unit  42 , an HD-FEC decoding unit  45 , and a PS inverse conversion unit  49 . The PS inverse conversion unit  49  includes an XOR operator  47  and inverse-DM (IDM) processing units  48   a  and  48   b.    
     The soft decision unit  41  performs restoration by performing soft decision on a value of each of the bit strings of level- 0  to level- 2  from an input signal Sin′ input from the analog-digital conversion unit  13 . The soft decision unit  41  determines probability of values “0” and “1” of the bit strings from symbols indicated by the input signal Sin′. Each of the bit strings of level- 0  to level- 2  is transmitted to an individual lane. The soft decision unit  41  outputs the value of each of the bit strings of level- 0  to level- 2  to the SD-FEC decoding unit  42 . 
     The SD-FEC decoding unit  42  corrects the values of the bit strings of level- 0  to level- 2  on the basis of the SD-FEC parity. For example, the SD-FEC decoding unit  42  performs decoding by using the SD-FEC parity. The SD-FEC decoding unit  42  outputs each of the bit strings of level- 0  to level- 2  to the HD-FEC decoding unit  45 . 
     The HD-FEC decoding unit  45  corrects the value of each of the bit strings of level- 0  to level- 2  on the basis of the HD-FEC parity. For example, the HD-FEC decoding unit  45  performs decoding by using the HD-FEC parity. The HD-FEC decoding unit  45  outputs each of the bit strings of level- 0  to level- 2  to the PS inverse conversion unit  49 . 
     The PS inverse conversion unit  49  performs conversion reverse to the conversion by the PS conversion unit  29  for each of the bit strings of level- 0  to level- 2 . The XOR operator  47  performs XOR the value of the bit string of level- 0  with the value of the bit string of level- 1 . The value of the bit string of level- 0  therefore becomes the original value of the bit string of level- 0  before being performed XOR by the XOR operator  23  of the encoding circuit  120 . 
     The bit string of level- 1  is input to the IDM processing unit  48   a , and the bit string of level- 0  is input from the XOR operator  47  to the IDM processing unit  48   b.    
     The IDM processing units  48   a  and  48   b  perform inverse-DM processing that is inverse conversion processing of DM processing of the DM processing units  21   a  and  21   b  on the bit strings of level- 1  and level- 0 , respectively. The bit strings of level- 1  and level- 0  therefore become values before being converted by the DM processing units  21   a  and  21   b , respectively, of the PS conversion unit  29  in the encoding circuit  120 . The bit strings of level- 0  to level- 2  are output to the framer chip  11  as an output signal Sout′. 
     The PS conversion unit  29  of the encoding circuit  120  converts the value of each of the bit strings of level- 0  and level- 1  so that a symbol closer to the center of the constellation of 64QAM is assigned more. A probability distribution is therefore formed in which a symbol closer to the center of the constellation has a higher probability of symbol assignment. 
       FIG. 5  is a diagram illustrating an example of processing of PS. In the present embodiment, a constellation of 16QAM is given as an example for convenience of description. In the constellation, symbols P 11  to P 14 , P 21  to P 24 , P 31  to P 34 , and P 41  to P 44 , which are signal points, are equally divided and arranged in the first to fourth quadrants. 
     The size of a circle indicating each of the symbols P 11  to P 14 , P 21  to P 24 , P 31  to P 34 , and P 41  to P 44  indicates a value of the probability of symbol assignment. The probabilities of symbol assignment before PS are equal to each other among the symbols P 11  to P 14 , P 21  to P 24 , P 31  to P 34 , and P 41  to P 44 . 
     The probabilities of symbol assignment after PS become higher as the symbols P 11  to P 14 , P 21  to P 24 , P 31  to P 34 , and P 41  to P 44  are closer to a center point O. For example, the symbols P 22 , P 23 , P 32 , and P 33  having the shortest distance from the center point O have the maximum probability of symbol assignment, and the symbols P 11 , P 14 , P 41 , and P 44  having the longest distance from the center point O have the minimum probability of symbol assignment. 
     In the formation of the probability distribution of symbol assignment, the value of each of the bit strings of level- 0  and level- 1  is converted so that the symbols P 22 , P 23 , P 32 , and P 33  closer to the center point O have higher probabilities of symbol assignment, and quadrants of the symbols P 11  to P 14 , P 21  to P 24 , P 31  to P 34 , and P 41  to P 44  are determined by the value of the bit string of level- 2 . 
       FIG. 6  is a diagram illustrating an example of symbol mapping. The symbol mapping unit  27  maps each of the bit strings of level- 0  to level- 2  to a symbol by Gray code mapping. 
     The symbol mapping unit  27  assigns the value of each of the bit strings of level- 0  to level- 2  to an I value and a Q value. For example, the symbol mapping unit  27  may assign the same value of each of the bit strings of level- 0  to level- 2  to both the I value and the Q value. For example, in a case where the value of the bit string of level- 0  is “1”, the I value and the Q value are each “1”. 
     Furthermore, the symbol mapping unit  27  may alternately assign the value of each of the bit strings of level- 0  to level- 2  to the I value and the Q value. For example, in a case where the values of two consecutive bits in the bit string of level- 0  are “1” and “0”, the I value is “1” and the Q value is “1”. 
     The I value and Q value of the bit string of level- 2  determine a quadrant of a symbol to be assigned. In a case where the I value=“0” and the Q value=“0”, a symbol in the first quadrant is assigned, and in a case where the I value=“1” and the Q value=“0”, a symbol in the second quadrant is assigned. Furthermore, in a case where the I value=“1” and the Q value=“1”, a symbol in the third quadrant is assigned, and in a case where the I value=“0” and the Q value=“0”, a symbol in the fourth quadrant is assigned. 
     The PS conversion unit  29  XORs the value of the bit string of level- 1  with the value of the bit string of level- 0  by the XOR operator  23  so that a symbol closer to the center point O has a higher symbol assignment probability. 
       FIG. 7  is a diagram illustrating an example of the probability distribution of symbol assignment before and after performing XOR. Note that, the values of the bit strings of level- 0  to level- 2  in  FIG. 7  may be any of the I value and the Q value. 
     The DM processing units  21   a  and  21   b  respectively convert the values of the bit strings of level- 1  and level- 0  so that the number of values of “1.” becomes greater than that of “0”, For this reason, a probability increases that the values of the bit strings of level- 0  and level- 1  are both “1” (see reference sign m 2 ), and a symbol P 1  closest to the center point O has a lower probability than that of a symbol P 2  that is on the outer side of the symbol P 1  in the probability distribution of symbol assignment before performing XOR. 
     However, in the arrangement of the Gray code, a probability increases that the value of the bit string of level- 0  becomes “0” by being XORed with the value of the bit string of level- 1 . For this reason, a probability increases that the values of the bit strings of level- 0  and level- 1  becomes “0” and “1”, respectively (see reference sign m 1 ), and the symbol P 1  closest to the center point O has a higher probability than that of the symbol P 2  that is on the outer side of the symbol P 1  in the probability distribution of symbol assignment after XOR. 
     Furthermore, the value of the bit string of level- 2  is the HD-FEC parity and the SD-FEC parity generated respectively by the HD-FEC generation unit  24  and the SD-FEC generation unit  25 . Since the mark rates of the HD-FEC parity and the SD-FEC parity are maintained close to 50(%), the first to fourth quadrants are selected with substantially the same probability, and a probability distribution is formed biased toward the center point O of the constellation over all quadrants. For this reason, the noise immunity of the output signal is increased. 
     However, as illustrated in  FIG. 3 , the SD-FEC generation unit  25  sets a whole of the bit strings of level- 0  to level- 2  (see the dotted frame) as an encoding target area (operation area for the SD-FEC parity). For example, the SD-FEC generation unit  25  generates an SD-FEC parity from the bit strings of level- 0  to level- 2 . In comparison with the hard decision code, the soft decision code has a higher correction capability, but has a larger power consumption in encoding and decoding. 
     (Encoding and Decoding by MLC) 
       FIG. 8  is a configuration diagram illustrating an example of the encoding circuit  120  using MLC. In  FIG. 8 , the same components as those in  FIG. 3  are denoted by the same reference numerals, and description thereof will be omitted. 
     The encoding circuit  120  includes a PS conversion unit  29   x , an HD-FEC generation unit  24   x , an SD-FEC generation unit  25   x , and a symbol mapping unit  27   x . The PS conversion unit  29   x  includes the DM processing units  21   a  and  21   b  and an XOR operator  23   x . Note that, in the present embodiment, 64QAM is given as an example of the multi-level modulation scheme, but the multi-level modulation scheme is not limited to this. 
     The PS conversion unit  29   x  converts the value of each of the bit strings of level- 0  and level- 1  so that a symbol closer to the center of the constellation of 64QAM is assigned more. The XOR operator  23   x  performs XOR the value of the bit string of level- 0  with the value of the bit string of level- 2 . The value of the bit string of level- 0  therefore becomes a value obtained by XORing the original value of the bit string of level- 0  with the value of the bit string of level- 2 . Each of the bit strings is output from the PS conversion unit  29   x  to the HD-FEC generation unit  24   x.    
     The HD-FEC generation unit  24   x  individually generates an HD-FEC parity that is a hard decision code, from the bit strings of level- 0  to level- 2 . The HD-FEC generation unit  24   x  inserts the HD-FEC parity of the bit string of level- 1  into the bit string of level- 1 , and inserts the HD-FEC parity of the bit string of level- 2  into the bit string of level- 2 . Each of the bit strings of level- 1  and level- 2  is output from the HD-FEC generation unit  24   x  to the symbol mapping unit  27   x.    
     Furthermore, the HD-FEC generation unit  24   x  inserts the HD-FEC parity of the bit string of level- 0  into the bit string of level- 0 . The bit string of level- 0  is output from the HD-FEC generation unit  24   x  to the SD-FEC generation unit  25   x.    
     The SD-FEC generation unit  25   x  generates an SD-FEC parity that is a soft decision code, from the bit string of level- 0 . The SD-FEC generation unit  25   x  deletes the HD-FEC parity from the bit string of level- 0 , and inserts the SD-FEC parity into the bit string of level- 0 . The bit string of level- 0  is output from the SD-FEC generation unit  25   x  to the symbol mapping unit  27   x.    
     The symbol mapping unit  27   x  assigns, to the bit strings, symbols corresponding to the values of the bit strings of level- 0  to level- 2 , the symbols being among the plurality of symbols in the constellation of 64QAM. The symbol mapping unit  27   x  outputs the output signal Sout corresponding to the assigned symbol to the analog-digital conversion unit  13 . 
     Reference numeral  91  indicates contents of the bit strings in the frame input to the symbol mapping unit  27   x . The bit string of level- 0  includes data # 0  on which DM processing has been performed, and the SD-FEC parity. The bit string of level- 1  includes data # 1  on which DM processing has been performed, and the HD-FEC parity, and the bit string of level- 2  includes data # 2  on which DM processing have not been performed, and the HD-FEC parity. 
       FIG. 9  is a configuration diagram illustrating an example of the decoding circuit  121  using MLC. In  FIG. 9 , the same components as those in  FIG. 4  are denoted by the same reference numerals, and description thereof will be omitted. 
     The decoding circuit  121  includes a soft decision unit  41   x , an SD-FEC decoding unit  42   x , a hard decision unit  43 , an HD-FEC decoding unit  45   x , and a PS inverse conversion unit  49   x . The PS inverse conversion unit  49   x  includes an XOR operator  47   x  and the IDM processing units  48   a  and  48   b . The input signal Sin′ is input to each of the soft decision unit  41   x  and the hard decision unit  43 . 
     The soft decision unit  41   x  performs restoration by performing soft decision on a value of the bit string of level- 0  from the input signal Sin′. The soft decision unit  41   x  determines probability of values “0” and “1” of the bit strings from symbols indicated by the input signal Sin′. The soft decision unit  41   x  outputs the value of the bit string of level- 0  to the SD-FEC decoding unit  42   x.    
     The SD-FEC decoding unit  42   x  corrects the value of the bit string of level- 0  on the basis of the SD-FEC parity. For example, the SD-FEC decoding unit  42   x  performs decoding by using the SD-FEC parity. The SD-FEC decoding unit  42   x  outputs the bit string of level- 0  to the HD-FEC decoding unit  45   x.    
     Furthermore, the hard decision unit  43  performs restoration by performing hard decision on a value of each of the bit strings of level- 1  and level- 2  from the input signal Sin′. The hard decision unit  43  determines values “0” and “1” of the bit strings from symbols indicated by the input signal Sin′. The hard decision unit  43  outputs the value of each of the bit strings of level- 1  and level- 2  to the HD-FEC decoding unit  45   x.    
     The HD-FEC decoding unit  45   x  corrects the value of each of the bit strings of level- 0  to level- 2  on the basis of the HD-FEC parity. For example, the HD-FEC decoding unit  45   x  performs decoding by using the HD-FEC parity. The HD-FEC decoding unit  45   x  outputs each of the bit strings of level- 0  to level- 2  to the PS inverse conversion unit  49   x.    
     The PS inverse conversion unit  49   x  performs conversion reverse to the conversion by the PS conversion unit  29   x  for each of the bit strings of level- 0  to level- 2 . The XOR operator  47   x  performs XOR the value of the bit string of level- 0  with the value of the bit string of level- 2 . The value of the bit string of level- 0  therefore becomes the original value of the bit string of level- 0  before being been XOR by the XOR operator  23   x  of the encoding circuit  120 . 
     The bit string of level- 1  is input to the IDM processing unit  48   a , and the bit string of level- 0  is input from the XOR operator  47   x  to the IDM processing unit  48   b . The bit strings of level- 0  to level- 2  are output to the framer chip  11  as an output signal Sout′. 
     The PS conversion unit  29   x  of the encoding circuit  120  converts the value of each of the bit strings of level- 0  and level- 1  so that a symbol closer to the center of the constellation of 64QAM is assigned more. A probability distribution is therefore formed in which a symbol closer to the center constellation has a higher probability of symbol assignment. 
     Furthermore, the symbol mapping unit  27   x  performs symbol mapping different from the symbol mapping by the symbol mapping unit  27  of BICM. 
       FIG. 10A  is a diagram illustrating another example of the symbol mapping. In  FIG. 10A , description of contents common to  FIG. 6  will be omitted. 
     The symbol mapping unit  27   x  maps each of the bit strings of level- 0  to level- 2  to a symbol by set-partitioning. The arrangement of the values of the bit strings of level- 0  and level- 1  in the set-partitioning is different from the arrangement in the Gray code. 
     According to this arrangement, the HD-FEC decoding unit  45   x  of the decoding circuit  121  performs multi-stage decoding (MSD), whereby the Euclidean distance between the symbols in the constellation may be made longer than that in the case of the Gray code. For example, in a case where the bit string of level- 0  that is the LSB is correctly decoded with the I value=1 and the Q value=0, only solid line circle symbols in the constellation are used limitedly as decoding target symbols. 
     For this reason, even though the SD-FEC is applied only to the bit string of level- 0 , it is possible to reduce errors in the bit strings of level- 1  and level- 2  that are more significant levels, and suppress a reduction in error correction capability. 
     Furthermore, the PS conversion unit  29   x  performs XOR the value of the bit string of level- 2  with the value of the bit string of level- 0  by the XOR operator  23   x  so that a symbol closer to the center point O has a higher symbol assignment probability. 
       FIG. 10B  is a diagram illustrating an example of the probability distribution of symbol assignment before and after XOR. Note that, in  FIG. 10B , description of contents common to  FIG. 7  will be omitted. 
     Since the DM processing units  21   a  and  21   b  respectively convert the values of the bit strings of level- 1  and level- 0  so that the number of values of “1” becomes greater than that of “0”, the probability increases that the values of the bit strings of level- 0  and level- 1  are both “1” (see reference sign m 3 ). Here, in the set-partitioning, the arrangement of the I value and the Q value of the bit string of level- 0  is asymmetric with respect to the center point O. For this reason, on one side with respect to the center point O, a symbol P 3  closest to the center point O has a lower probability than that of a symbol P 4  that is on the outer side of the symbol P 3  in the probability distribution of symbol assignment before being XOR. 
     However, in the arrangement of the set-partitioning, a probability increases that the value of the bit string of level- 0  becomes “0” by being been XOR with the value of the bit string of level- 2  symmetric with respect to the center point O. For this reason, a probability increases that the values of the bit strings of level- 0  and level- 1  become “0” and “1”, respectively (see reference sign m 4 ), and the symbol P 3  closest to the center point O has a higher probability than that of the symbol P 4  that is on the outer side of the symbol P 3  in the probability distribution of symbol assignment after being XOR. 
     Furthermore, the value of the bit string of level- 2  is the HD-FEC parity generated by the HD-FEC generation unit  24   x . Since the mark rate of the HD-FEC parity is maintained close to 50(%), the first to fourth quadrants are selected with substantially the same probability. 
     As illustrated in  FIGS. 8 and 9 , in the frame using MLC, since the SD-FEC parity is generated only from the bit string of level-U, power consumption is reduced as compared with the case where BICM is used. 
     However, in a case where MLC is used, since the SD-FEC parity (see reference numeral  911 ) that may not be subjected to DM processing is inserted only into the bit string of level- 0 , there is a possibility that the effect is reduced of noise tolerance improvement by PS as compared to the case where BICM is used. Furthermore, since the HD-FEC parity (see reference numeral  910 ) that is not subjected to DM processing is inserted into the bit string of level- 1 , there also is a possibility that the effect is reduced of noise tolerance improvement by PS as compared to the case where BICM is used. 
     On the other hand, in a case where the SD-FEC parity and the HD-FEC parity are inserted into the bit string of level- 2 , which is not subject to DM processing, as in the example described below, a reduction in the effect of noise tolerance improvement is suppressed. 
       FIG. 11  is a configuration diagram illustrating another example of the encoding circuit  120  using MLC. In  FIG. 11 , the same components as those in  FIG. 8  are denoted by the same reference numerals, and description thereof will be omitted. 
     The encoding circuit  120  includes the PS conversion unit  29   x , an HD-FEC generation unit  24   x ′, an SD-FEC generation unit  25   x ′, and the symbol mapping unit  27   x . The PS conversion unit  29   x  includes the DM processing units  21   a  and  21   b  and the XOR operator  23   x . Note that, in the present embodiment, 64QAM is given as an example of the multi-level modulation scheme, but the multi-level modulation scheme is not limited to this. 
     Reference numeral  92  indicates contents of the bit strings in the frame input to the symbol mapping unit  27   x . The bit string of level- 0  includes data # 0  on which DM processing has been performed, the SD-FEC parity, and the HD-FEC parity. The bit string of level- 1  includes data # 1  on which DM processing has been performed, and the bit string of level- 2  includes data # 2  on which DM processing have not been performed. Note that, in the bit string of level- 2 , a period including data # 2  is expressed as a period Tp, and a period including SD-FEC parity and HD-FEC parity is expressed as a period Tq. 
     The HD-FEC generation unit  24   x ′ generates an HD-FEC parity similarly to the HD-FEC generation unit  24   x  so that the frame format described above may be formed, but unlike the HD-FEC generation unit  24   x , the HD-FEC parity is inserted into the bit string of level- 2 . Furthermore, the SD-FEC generation unit  25   x ′ generates an SD-FEC parity similarly to the SD-FEC generation unit  25   x , but unlike the SD-FEC generation unit  25   x , the SD-FEC parity is generated from data # 2  in the bit string of level- 2  and inserted into the bit string of level- 2 . 
     The SD-FEC parity in the period Tq has a value that is a bit string determined by operation processing on the entire data # 0  indicated by reference numeral  912 . For this reason, the value of the SD-FEC parity has not been determined at the start time of the period Tq. 
     Furthermore, the HD-FEC parity in the period Tq has a value that is a bit string determined by operation processing on a whole of data # 0  and data # 1  indicated by reference numeral  913 . For this reason, the value of the HD-FEC parity has not been determined at the start time of the period Tq. 
     Thus, an XOR operation is not correctly executed in the period Tq. The purpose of the XOR operation is to change the probability distribution of symbol assignment before and after XOR as illustrated in  FIG. 7 . For this reason, in a case where the XOR operation is not performed normally, there is a possibility that a normal probability distribution is not obtained, and the effect of PS for noise immunity improvement is reduced. 
     EMBODIMENTS 
     The encoding circuit  120  of the present embodiment therefore delays the SD-FEC parity and the HD-FEC parity, and uses the SD-FEC parity and the HD-FEC parity for the XOR operation. The HD-FEC parity is generated from data # 0  to data # 2  of level- 0  to level- 2  in a frame and inserted into the bit string of level- 2  in a subsequent frame. The SD-FEC parity is generated from data # 0  of level- 0  in each period that divides a period of the frame and is inserted into the bit string of level- 2  in a subsequent period. 
     Thus, the XOR operation may be performed using the SD-FEC parity and the HD-FEC parity that are determined bit strings, so that the probability distribution of symbol assignment is formed normally. 
     First Embodiment 
       FIG. 12  is a diagram illustrating a frame format of an output signal Sout output from an encoding circuit  120  of a first embodiment. In the present embodiment, a case where 64QAM is used as a multi-level modulation scheme will be described. In this case, a frame includes bit strings of level- 0  to level- 2 . Note that the bit string of level- 2  is an example of a first bit string, and the bit string of level- 0  is an example of a second bit string. 
     A period T of frames #n and #(n+1), (n is a positive integer) is divided into a plurality of periods Ta and Tb and a period Tc at the end. In each of the frames #n and #(n+1), the periods Tb and Ta repeatedly arrive, and the period Tc arrives last. The frames #n and #(n+1) are examples of two consecutive frames in time series. Since the frame #(n+1) subsequent to the frame #n has the same configuration as the frame #n, description thereof will be omitted. 
     The bit string of level- 0  in the frame #n includes data # 1  to data # 5  subject to DM processing. The data # 1  to data # 5  are each included in a bit string over a pair of periods Ta and Tb or a pair of periods Tb and Tc. 
     The bit string of level- 1  in the frame #n includes data # 10  subject to DM processing in any of the periods Ta to Tc. 
     The bit string of level- 2  in the frame #n includes data # 30  to data # 33  not subject to DM processing, SD-FEC parities (SD-PTY) # 0  to # 4 , and an HD-FEC parity (HD-PTY) # 0 . The SD-FEC parities # 0  to # 4  are each inserted into the bit string in a period Tb, and the data # 30  to data # 33  are each included in the bit string in a period Ta. Furthermore, the HD-PTY # 0  is inserted into the bit string in the period Tc at the end. 
     The SD-FEC parities # 1  to # 4  are respectively generated from the data # 1  to data # 4 , each in an immediately preceding pair of periods Ta and Tb. For this reason, the SD-FEC parities # 1  to # 4  are respectively inserted into bit strings with a delay of a pair of periods Ta and Tb from the data # 1  to data # 4  used to generate the SD-FEC parities # 1  to # 4 . 
     Furthermore, the SD-FEC parity # 0  is generated from data at the end of frame #(n−1) (not illustrated) immediately preceding the frame #n. Furthermore, the data # 5  within the periods Tb and Tc is used to generate an SD-FEC parity # 5  at the beginning of the frame #(n+1) next to the frame #n. In this way, the SD-FEC parities # 0  to # 5  are each inserted into periods Ta and Tb (or Tb and Tc) that are later than periods Ta and Tb (or Tb and Tc) of data used to generate the SD-FEC parity. Note that the pair of periods Ta and Tb and the pair of periods Tb and Tc are examples of a plurality of periods that divide a period T of a frame. 
     The HD-FEC parity # 0  is generated from data included in each bit string in the immediately preceding frame #(n−1). Furthermore, the data # 1  to # 5 , # 10 , and # 30  to # 33 , and the SD-FEC parities # 0  to # 4  in the frame #n are used to generate an HD-FEC parity # 1  in the frame #(n+1) next to the frame #n. For example, an area X in the frame #n excluding the HD-FEC parity # 0  is used to generate the HD-FEC parity # 1  in the frame #(n+1) next to the frame #n. In this way, the MD-FEC parities # 0  and # 1  are respectively inserted into the periods Tc in the frames #n and #(n+1) subsequent to the frames #(n−1) and #n of data used to generate the HD-FEC parities # 0  and # 1 . 
     According to the frame format of the present embodiment, only the bit string of level- 0  is used to generate an SD-FEC parity, so an operation area for the SD-FEC parity is limited to the bit string of level- 0  and the bit string of level- 2  in the period Tc where the SD-FEC parity is inserted. For this reason, the operation area for the SD-FEC parity becomes narrower than that of the BICM frame format illustrated in  FIG. 3 , and power consumption is reduced. 
     Furthermore, the frame #n includes the plurality of SD-FEC parities # 0  to # 4 , and the operation area for the SD-FEC parities # 0  to # 4  is divided into a plurality of areas. For this reason, power consumption is reduced as compared with a case where an SD-FEC parity is generated from a series of consecutive operation areas. 
     Furthermore, since the SD-FEC parities and the HD-FEC parities are inserted into the bit string of level- 2  that is not subject to DM processing, unlike the case of MLC illustrated in  FIG. 9 , a reduction of the amount of data of level- 0  and level- 1  subject to DM processing is suppressed. 
     For this reason, the encoding circuit  120  and a decoding circuit  121  respectively perform encoding processing and the decoding processing on the basis of the frame format described above, thereby reducing power consumption without reducing noise tolerance. Each of the configurations of the encoding circuit  120  and the decoding circuit  121  will be described below. 
       FIG. 13  is a configuration diagram illustrating the encoding circuit  120  of the first embodiment. In  FIG. 13 , the same components as those in  FIG. 3  are denoted by the same reference numerals, and description thereof will be omitted. Furthermore, the encoding method of the embodiment is encoding processing of the encoding circuit  120  described below. This will be described below with reference to  FIGS. 12 and 13 . 
     The encoding circuit  120  includes an operation control unit  20 , a PS conversion unit  29   y , an HD-FEC generation unit  24   y , an SD-FEC generation unit  25   y , a symbol mapping unit  27   y , selectors (SELs)  26   a  and  26   b , an HD delay generation unit  28   a , and an SD delay generation unit  28   b . The PS conversion unit  29   y  includes DM processing units  21   a  and  21   b , a selector (SEL)  22 , and an XOR operator  23   y.    
     The operation control unit  20  switches among the periods Ta to Tc within the period T of the frame. For example, the operation control unit  20  controls the selectors  22 ,  26   a , and  26   b  in accordance with the periods Ta to Tc. 
     In  FIG. 13 , “H” surrounded by a box indicates a connection relationship between an output and an input of an HD-FEC parity (HD parity), and “S” surrounded by a box indicates a connection relationship between an output and an input of an SD-FEC parity (SD parity), Note that the notation described above is also used in  FIG. 13  and subsequent drawings in a similar manner. Furthermore, “L 2 ” surrounded by a box indicates a connection relationship between an output and an input of data in the bit string of level- 2 . 
     Furthermore, signals input to the selectors  22 ,  26   a , and  26   b  are denoted by reference signs (Ta, Tb, and Tc) indicating the periods Ta, Tb, and Tc during which the corresponding input signals are selected. For example, the selector  26   a  selects an HD-FEC parity as an input signal during the period Ta, and an SD-FEC parity during the period Tb. Note that the notation described above is also used in  FIG. 13  and subsequent drawings in a similar manner. 
     Each of the bit strings of level- 0  to level- 2  is inputted to the PS conversion unit  29   y . The PS conversion unit  29   y  is an example of a conversion unit, and converts the value of each of the bit strings of level- 0  and level- 1  other than level- 2  so that a symbol closer to the center of a constellation is assigned more. Data # 30  to data # 33  in the bit string of level- 2  are input to the selectors  22  and  26   a.    
     The bit strings of level- 1  and level- 0  are inputted to the DM processing units  21   a  and  21   b , respectively. The bit string of level- 1  on which DM processing has been performed is inputted to the HD-FEC generation unit  24   y . Furthermore, the bit string of level- 0  on which DM processing has been performed is inputted to the XOR operator  23   y.    
     The XOR operator  23   y  performs an XOR operation on the bit string of level- 0  output from the DM processing unit  21   b  and data selected by the selector  22 . The bit string of level- 0  after being XOR is inputted to the HD-FEC generation unit  24   y  and the SD-FEC generation unit  25   y.    
     The data # 30  to data # 33 , an SD-FEC parity, and an HD-FEC parity in the bit string of level- 2  are inputted to the selector  22 . The selector  22  selects, as an input signal, any one of the data # 30  to data # 33 , the SD-FEC parity, or the HD-FEC parity in the bit string of level- 2  in accordance with the periods Ta to Tc controlled by the operation control unit  20 . 
     The selector  22  selects the data # 30  to data # 33  in the bit string of level- 2  during the period Ta, the SD-FEC parity during the period Tb, and the HD-FEC parity during the period Tc. The selected input signal is inputted to the XOR operator  23   y . For this reason, the SD-FEC parity, the HD-FEC parity, and the data # 30  to data # 33  in the bit string of level- 2  are used for the XOR operation in accordance with the periods Ta to Tc. 
     The SD-FEC generation unit  25   y  is an example of a second generation unit, and generates an SD-FEC parity from the bit string of level- 0  in the periods Ta and Tb or the periods Tb and Tc. The SD-FEC generation unit  25   y  calculates an SD-FEC parity from the data # 1  to data # 4  of level- 0  in the periods Ta and Tb or the data # 5  in the periods Tb and Tc and outputs the SD-FEC parity to the SD delay generation unit  28   b.    
     The SD delay generation unit  28   b , which is an example of a second insertion unit, delays an SD-FEC parity, and inserts the SD-FEC parity into the bit string of level- 2  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0  used to generate the SD-FEC parity. The SD delay generation unit  28   b  delays an SD-FEC parity input from the SD-FEC generation unit  25   y  once every pair of periods Ta and Tb or every pair of periods Tb and Tc by the total of the periods Ta and Tb or the total of the periods Tb and Tc, and outputs the SD-FEC parity to the selectors  22  and  26   a . Note that the total of the periods Ta and Tb and the total of the periods Tb and Tc may be the same or different. 
     The data # 30  to data # 33  and the SD-FEC parity in the bit string of level- 2  are input to the selector  26   a . The selector  26   a  selects, as an input signal, any one of the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 2  in accordance with the periods Ta and Tb controlled by the operation control unit  20 . The selector  26   a  selects the data # 30  to data # 33  in the bit string of level- 2  and outputs it to the HD-FEC generation unit  24   y  during the period Ta, and selects the SD-FEC parity and outputs it to the HD-FEC generation unit  24   y  during the period Tb. 
     The HD-FEC generation unit  24   y  is an example of a first generation unit, and generates an HD-FEC parity from the bit strings of level- 0  to level- 2  for each frame. The bit string of level- 2  alternately includes data # 30  to data # 33  and SD-FEC parities. By switching of the selector  26   a , the HD-FEC generation unit  24   y  generates an HD-FEC parity from the data # 30  to data # 33  during the period Ta, and generates an HD-FEC parity from the SD-FEC parities during the period Tb. 
     The HD-FEC generation unit  24   y  calculates an HD-FEC parity from the values of the bit strings of level- 0  to level- 2  for each frame and outputs the HD-FEC parity to the HD delay generation unit  28   a . Note that, during the period Tc, the HD-FEC generation unit  24   y  receives no input of the bit string of level- 2  from the selector  26   a , and generates an HD-FEC parity from the bit strings of other levels, that is, level- 0  and level- 1 . 
     The HD delay generation unit  28   a , which is an example of a first insertion unit, delays an HD-FEC parity, and inserts the HD-FEC parity into the bit string of level- 2  in a frame subsequent to a frame including the bit string used to generate the HD-FEC parity. The HD delay generation unit  28   a  delays the HD-FEC parity by the period T of the frame, and outputs the HD-FEC parity to the selector  22 . 
     The HD-FEC generation unit  24   y  outputs, to the symbol mapping unit  27   y , the bit strings of level- 1  and level- 0  respectively input from the DM processing units  21   a  and  21   b  as they are. Furthermore, the HD-FEC generation unit  24   y  outputs, to the selector  26   b , the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 2  input from the selector  26   a  as it is. 
     The selector  26   b  selects an input signal in accordance with controls in the periods Ta to Tc by the operation control unit  20 . The selector  26   b  selects the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 2  during the periods Ta and Tb, and selects the HD-FEC parity during the period Tc. The selector  26   b  outputs the selected input signal as a bit string of level- 2  to the symbol mapping unit  27   y.    
     The symbol mapping unit  27   y  is an example of an assigning unit, and assigns, to the bit strings of level- 0  to level- 2 , symbols corresponding to the values of the bit strings of level- 0  to level- 2 , the symbols being among a plurality of symbols in a constellation of 64QAM. The symbol mapping unit  27   y  performs symbol mapping by set-partitioning. The symbol mapping unit  27   y  outputs each of the bit strings of level- 0  to level- 2  as an output signal Sout. 
     Furthermore, to the selector  22 , the HD-FEC parity is input from the HD delay generation unit  28   a , and the SD-FEC parity is input from the SD delay generation unit  28   b . For this reason, the delayed HD-FEC parity and SD-FEC parity are used in the XOR operator  23   y.    
     The PS conversion unit  29   y  is therefore able to convert the bit string of level- 0  using the HD-FEC parity and the SD-FEC parity determined by the HD-FEC generation unit  24   y  and the SD-FEC generation unit  25   y , respectively, in stages following the PS conversion unit  29   y . Thus, the probability distribution of symbol assignment is formed normally. 
     In this way, the symbol mapping unit  27   y  assigns, to the bit strings of level- 0  to level- 2 , symbols corresponding to the values of the bit strings of level- 0  to level- 2 , the symbols being among the plurality of symbols in the constellation of 64QAM. The PS conversion unit  29   y  converts the value of each of the bit strings of level- 0  and level- 1  other than the bit string of level- 2  so that a symbol closer to the center of the constellation is assigned more. 
     The HD-FEC generation unit  24   y  generates an HD-FEC parity from the bit strings of level- 0  to level- 2  for each frame. The HD delay generation unit  28   a  delays an HD-FEC parity, and inserts the HD-FEC parity into the bit string of level- 2  in a frame subsequent to a frame including the bit string used to generate the HD-FEC parity. 
     The SD-FEC generation unit  25   y  generates an SD-FEC parity from the bit string of level- 0  in the periods Ta and Tb or the periods Tb and Tc. The SD delay generation unit  28   b  delays an SD-FEC parity, and inserts the SD-FEC parity into the bit string of level- 2  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0  used to generate the SD-FEC parity. 
     Furthermore, the PS conversion unit  29   y  uses the delayed SD-FEC parity and HD-FEC parity for converting the value of the bit string of level- 0 . 
     According to the configuration described above, the HD-FEC parity and the SD-FEC parity are inserted into the bit string of level- 2  that is not subject to conversion by the PS conversion unit  29   y , that is, not subject to DM processing, so unlike the case of MIC illustrated in  FIG. 9 , the reduction of the amount of data of level- 0  and level- 1  subject to DM processing is suppressed, and the reduction in the effect of noise immunity improvement is suppressed. 
     Furthermore, since the SD-FEC parity is generated only from the bit string of level- 0 , the operation area for the SD-FEC parity becomes narrower than that of the case of BICM illustrated in  FIG. 3 , and power consumption is reduced. Furthermore, since the SD-FEC parity is generated once every pair of the periods Ta and Tb or every pair of periods Tb and Tc that divide the period T of the frame, power consumption is reduced as compared with the case where one SD-FEC parity is generated from the entire frame. 
     Furthermore, the ND-FEC parity is delayed and inserted into the subsequent frame, and the SD-FEC parity is delayed and inserted into periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string used to generate the SD-FEC parity. Since the delayed ND-FEC parity and SD-FEC parity are used for converting the value of the bit string of level- 0 , the probability distribution of symbol assignment is formed normally by using the determined ND-FEC parity and SD-FEC parity. 
     Thus, according to the encoding circuit  120  of the present embodiment, it is possible to reduce power consumption without reducing noise tolerance. 
       FIG. 14  is a configuration diagram illustrating the decoding circuit  121  of the first embodiment. In  FIG. 14 , the same components as those in  FIG. 4  are denoted by the same reference numerals, and description thereof will be omitted. Furthermore, the decoding method of the embodiment is decoding processing of the decoding circuit  121  described below. This will be described below with reference to  FIGS. 12 and 14 . 
     The decoding circuit  121  includes an operation control unit  40 , a soft decision unit  41   y , an SD-FEC decoding unit  42   y , a hard decision unit  43   y , a selector  44   a , an HD-FEC decoding unit  45   y , a bit delay generation unit  46   a , a data delay generation unit  46   b , an HD delay adjustment unit  46   c , and a PS inverse conversion unit  49   y . The PS inverse conversion unit  49   y  includes a selector  44   b , a parity delay generation unit  46   d , an XOR operator  47   y , and IDM processing units  48   a  and  48   b.    
     The operation control unit  40  notifies the hard decision unit  43   y , the soft decision unit  41   y , and the selectors  44   a  and  44   b  of the periods Ta to Tc in accordance with frame synchronization information, for example. The selectors  44   a  and  44   b  select an input signal in accordance with notifications of the periods Ta to Tc. The hard decision unit  43   y  detects an SD-FEC parity from the bit string of level- 2  in accordance with a notification of the period Tb, and the soft decision unit  41   y  detects an HD-FEC parity from the bit string of level- 2  in accordance with a notification of the period Tc. 
     An input signal Sin′ is input from an analog-digital conversion unit  13  to the soft decision unit  41   y  and the hard decision unit  43   y  separately. The data delay generation unit  46   b  that delays the input signal Sin′ is provided in a stage preceding the hard decision unit  43   y.    
     The soft decision unit  41   y  is an example of a second decision unit, and performs soft decision on the value of the bit string of level- 0  in a frame to which symbols in the constellation of 64QAM are assigned, on the basis of the symbols. At this time, the soft decision unit  41   y  performs symbol demapping by set-partitioning. On the basis of the result of the soft decision, the soft decision unit  41   y  extracts an SD-FEC parity from the bit string of level- 2  in the period Ta, and outputs the SD-FEC parity to the SD-FEC decoding unit  42   y.    
     Furthermore, on the basis of the result of the soft decision, the soft decision unit  41   y  outputs the data # 1  to data # 5  in the bit string of level- 0  to the SD-FEC decoding unit  42   y . The bit delay generation unit  46   a  that delays the bit string of level- 0  is provided in a stage preceding the SD-FEC decoding unit  42   y.    
     The bit delay generation unit  46   a  is an example of a second delay unit, and delays the bit string of level- 0  by the periods Ta and Tb or the periods Tb and Tc. For this reason, the bit string of level- 0  is input to the SD-FEC decoding unit  42   y  in synchronization with the SD-FEC parity delayed by the total of the periods Ta and Tb or the total of the periods Tb and Tc in the encoding circuit  120 . 
     The SD-FEC decoding unit  42   y  is an example of a second correction unit, and corrects an error in a result of the soft decision by the soft decision unit  41   y  on the basis of the SD-FEC parity inserted into the bit string of level- 2  during the periods Ta and Tb or the periods Tb and Tc. For example, the SD-FEC decoding unit  42   y  decodes the bit string of level- 0  on the basis of the SD-FEC parity. 
     The timing at which the SD-FEC parity is input to the SD-FEC decoding unit  42   y  is synchronized with the timing at which the bit string of level- 0  is input to the SD-FEC decoding unit  42   y . For this reason, the SD-FEC decoding unit  42   y  may correct the value of the bit string of level- 0  by using the SD-FEC parity in periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string of level- 0 . In the case of the example of  FIG. 12 , the data # 1  in the bit string of level- 0  is corrected by using the SD-FEC parity # 1 , and the data # 2  in the bit string of level- 0  is corrected by using the SD-FEC parity # 2 . 
     The bit string of level- 0  is input from the SD-FEC decoding unit  42   y  to the hard decision unit  43   y  and the HD delay adjustment unit  46   c . The SD-FEC parity is input from the SD-FEC decoding unit  42   y  to the selector  44   a.    
     Furthermore, the hard decision unit  43   y  is an example of the second decision unit, and performs hard decision on the value of each of the bit strings of level- 1  and level- 2  among the bit strings of level- 0  to level- 2  to which symbols in the constellation of 64QAM are assigned, on the basis of the symbols. For example, the hard decision unit  43   y  is used for hard decision of the bit string of level- 0  input from the SD-FEC decoding unit  42   y . At this time, the hard decision unit  43   y  performs symbol demapping by set-partitioning. The bit string of level- 2  is input from the hard decision unit  43   y  to the selector  44   a , and the bit string of level- 1  is input from the hard decision unit  43   y  to the HD delay adjustment unit  46   c . Furthermore, on the basis of the result of the hard decision, the hard decision unit  43   y  extracts an HD-FEC parity from the bit string of level- 2 , and outputs the HD-FEC parity to the HD-FEC decoding unit  45   y.    
     Furthermore, the data delay generation unit  46   b  delays each bit string input to the hard decision unit  43   y . The data delay generation unit  46   b  delays each bit string for the same amount of time as the bit delay generation unit  46   a . The timing at which the bit strings of level- 1  and level- 2  and the HD-FEC parity are input to the hard decision unit  43   y  and the timing at which the bit string and the SD-FEC parity are input to the SD-FEC decoding unit  42   y  are therefore aligned. 
     The selector  44   a  selects an input signal corresponding to the periods Ta to Tc from the data # 30  to data # 33  and the SD-FEC parity in the bit string of level- 2  in accordance with controls in the periods Ta to Tc by the operation control unit  20 . The selector  44   a  outputs the data # 30  to data # 33  in the bit string of level- 2  to the HD delay adjustment unit  46   c  during the period Ta, and outputs the SD-FEC parity to the HD delay adjustment unit  46   c  during the period Tb. 
     The HD delay adjustment unit  46   c , which is an example of a first delay unit, delays each of the bit strings of level- 0  to level- 2  by the period T of the frame, and outputs the delayed bit string to the HD-FEC decoding unit  45   y.    
     The HD-FEC decoding unit  45   y  is an example of a first correction unit, and corrects an error in a result of the hard decision by the hard decision unit  43   y  on the basis of the HD-FEC parity inserted into the bit string of level- 2  for each frame. Since each of the bit strings of level- 0  to level- 2  is input after being delayed by the period T of the frame, the HD-FEC decoding unit  45   y  may correct each of the bit strings of level- 0  to level- 2  on the basis of the HD-FEC parity inserted into the bit string of level- 2  in a subsequent frame. The HD-FEC decoding unit  45   y  outputs each of the bit strings of level- 0  to level- 2  to the PS inverse conversion unit  49   y.    
     The PS inverse conversion unit  49   y  is an example of an inverse conversion unit, and performs inverse conversion of the value of each of the bit strings of level- 0  and level- 1  converted by DM processing. The HD-FEC parity is input to the parity delay generation unit  46   d , and the bit string of level- 1  is input to the IDM processing unit  48   a , Furthermore, the bit string of level- 0  is input to the XOR operator  47   y , and the bit string of level- 2  is input to the selector  44   b.    
     The parity delay generation unit  46   d  delays the HD-FEC parity in accordance with the period Tc within the period T of the frame, and outputs the HD-FEC parity to the selector  44   b.    
     The selector  44   b  selects an input signal from the bit string of level- 2  and the HD-FEC parity in accordance with controls in the periods Ta to Tc by the operation control unit  40 . The selector  44   b  selects the bit string of level- 2  and outputs it to the XOR operator  47   y  during the periods Ta and Tb, and selects the delayed HD-FEC parity and outputs it to the XOR operator  47   y  during the period Tc. 
     The XOR operator  47   y  XORs the bit string of level- 0  with the bit string of level- 2  or the HD-FEC parity from the selector  44   b . Since the HD-FEC parity is input to the XOR operator  47   y  with a delay in accordance with the period Tc, the bit string of level- 0  is normally converted to the value before XOR by the XOR operator  23   y  of the encoding circuit  120 . The bit string of level- 0  after XOR is input to the IDM processing unit  48   b.    
     The bit string of level- 2  and the bit strings of level- 1  and level- 0  respectively subjected to inverse-DM processing by the IDM processing units  48   a  and  48   b  are output to the framer chip  11  as an output signal Sout′ by parallel-serial conversion, for example. 
     In this way, the HD delay adjustment unit  46   c  delays each of the bit strings of level- 0  to level- 2  so that an error in a result of the hard decision of each of the bit strings of level- 1  and level- 2  is corrected on the basis of the HD-FEC parity inserted into the bit string of level- 2  in a subsequent frame. For this reason, the delay time given to the HD-FEC parity by the HD delay generation unit  28   a  of the encoding circuit  120  is reduced. 
     Furthermore, the bit delay generation unit  46   a  delays the bit string of level- 0  so that an error in a result of the soft decision of the bit string of level- 0  is corrected on the basis of the SD-FEC parity inserted into the bit string of level- 2  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0 . For this reason, the delay time given to the SD-FEC parity by the SD delay generation unit  28   b  of the encoding circuit  120  is reduced. 
     Thus, the decoding circuit  121  may correct errors in the hard decision and the soft decision of each of the bit strings of level- 0  to level- 2  on the basis of the HD-FEC parity and SD-FEC parity delayed by the encoding circuit  120 . 
     The PS inverse conversion unit  49   y  performs inverse conversion of the value of the bit string of level- 0  delayed by the bit delay generation unit  46   a  and the HD delay adjustment unit  46   c  by using the HD-FEC parity inserted into the bit string of level- 2  in a subsequent frame, and the SD-FEC parity inserted into periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string of level- 2 . 
     For this reason, the PS inverse conversion unit  49   y  may normally perform inverse conversion of the bit string of level- 0  using the HD-FEC parity and SD-FEC parity delayed by the encoding circuit  120 . 
     Thus, the decoding circuit  121  may decode a bit string on the basis of encoding by the encoding circuit  120 , thereby reducing power consumption without reducing noise tolerance. 
     Second Embodiment 
       FIG. 15  is a diagram illustrating a frame format of an output signal Sout output from an encoding circuit  120  of a second embodiment. In the present embodiment, a case where 256QAM is used as a multi-level modulation scheme will be described. In this case, a frame includes bit strings of level- 0  to level- 3 . Note that, in  FIG. 15 , the same components as those in  FIG. 12  are denoted by the same reference numerals, and description thereof will be omitted. 
     The bit strings of level- 1  and level- 2  in frame #n respectively include data # 10  and data # 20  subject to DM processing in any of periods Ta to Tc. 
     The bit string of level- 3  in the frame #n includes data # 30  to data # 33  not subject to DM processing, SD-FEC parities (SD-PTY) # 0  to # 4 , and an HD-FEC parity (HD-PTY) # 0 . The SD-FEC parities # 0  to # 4  are each inserted into the bit string in a period Tb, and the data # 30  to data # 33  are each included in the bit string in a period Ta. Furthermore, the HD-PTY # 0  is inserted into the bit string in the period Tc at the end. 
     In comparison with the frame format of the first embodiment, the frame format of the present embodiment additionally has a bit string of level- 2  including data # 120  on which DM processing has been performed, but has the same configuration for other levels, that is, level- 0 , level- 1 , and level- 3 . For this reason, according to the frame format of the present embodiment, an effect similar to that of the frame format of the first embodiment may be obtained. 
     For this reason, the encoding circuit  120  and a decoding circuit  121  respectively perform encoding processing and the decoding processing on the basis of the frame format described above, thereby reducing power consumption without reducing noise tolerance. Each of the configurations of the encoding circuit  120  and the decoding circuit  121  will be described below. 
       FIG. 16  is a configuration diagram illustrating the encoding circuit  120  of the second embodiment. The encoding method of the embodiment is encoding processing of the encoding circuit  120  described below. This will be described below with reference to  FIGS. 15 and 16 . 
     The encoding circuit  120  includes an operation control unit  70 , a PS conversion unit  79 , an HD-FEC generation unit  74 , an SD-FEC generation unit  75 , a symbol mapping unit  77 , selectors (SELs)  72   b  and  72   c , an HD delay generation unit  78   a , and an SD delay generation unit  78   b . The PS conversion unit  79  includes DM processing units  71   a  to  71   c , a selector (SEL)  72   a , and XOR operators  73   a  and  73   b.    
     Each of the bit strings of level- 0  to level- 3  is input to the PS conversion unit  79 . The PS conversion unit  79  is an example of the conversion unit, and converts the value of each of the bit strings of level- 0  to level- 2  other than level- 3  so that a symbol closer to the center of a constellation is assigned more. The data # 30  to data # 33  in the bit string of level- 3  are input to the selectors  72   a  and  72   b.    
     The bit strings of level- 2 , level- 1 , and level- 0  are input to the DM processing units  71   a  to  71   c , respectively. The DM processing units  71   a  to  71   c  perform DM processing similarly to the DM processing units  21   a  and  21   b . The bit string of level- 2  on which DM processing has been performed is input to the HD-FEC generation unit  74 . Furthermore, the bit strings of level- 1  and level- 0  on which DM processing has been performed are input to the XOR operators  73   a  and  73   b , respectively. 
     The XOR operator  73   a  performs an XOR operation on the bit string of level- 1  output from the DM processing unit  71   b  and the bit string of level- 2  output from the DM processing unit  71   a . The bit string of level- 1  after XOR is input to the HD-FEC generation unit  74 . 
     The XOR operator  73   b  performs an XOR operation on the bit string of level- 0  output from the DM processing unit  71   c  and data selected by the selector  72   a . The bit string of level- 0  after performed XOR is input to the HD-FEC generation unit  74  and the SD-FEC generation unit  75 . 
     The data # 30  to data # 33 , an SD-FEC parity, and an HD-FEC parity in the bit string of level- 3  are input to the selector  72   a . The selector  72   a  selects, as an input signal, any one of the bit string of level- 3 , the SD-FEC parity, or the HD-FEC parity in accordance with the periods Ta to Tc controlled by the operation control unit  70 . 
     The selector  72   a  selects the bit string of level- 3  during the period Ta, selects the SD-FEC parity during the period Tb, and selects the HD-FEC parity during the period Tc. The selected input signal is input to the XOR operator  73   b . For this reason, the SD-FEC parity, the HD-FEC parity, and the data # 30  to data # 33  in the bit string of level- 3  are used for the XOR operation in accordance with the periods Ta to Tc. 
     The SD-FEC generation unit  75  is an example of the second generation unit, and generates an SD-FEC parity from the bit string of level- 0  in the periods Ta and Tb or the periods Tb and Tc. The SD-FEC generation unit  75  calculates an SD-FEC parity from data # 1  to data # 4  of level- 0  in the periods Ta and Tb or data # 5  in the periods Tb and Tc and outputs the SD-FEC parity to the SD delay generation unit  78   b.    
     The SD delay generation unit  78   b , which is an example of the second insertion unit, delays an SD-FEC parity, and inserts the SD-FEC parity into the bit string of level- 2  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0  used to generate the SD-FEC parity. The SD delay generation unit  78   b  delays an SD-FEC parity input from the SD-FEC generation unit  75  once every pair of periods Ta and Tb or every pair of periods Tb and Tc by the total of the periods Ta and Tb or the total of the periods Tb and Tc, and outputs the SD-FEC parity to the selectors  72   a  and  72   b.    
     The data # 30  to data # 33  and the SD-FEC parity in the bit string of level- 3  are input to the selector  72   b . The selector  72   b  selects, as an input signal, any one of the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 3  in accordance with the periods Ta and Tb controlled by the operation control unit  70 . The selector  72   b  selects the data # 30  to data # 33  in the bit string of level- 3  and outputs it to the HD-FEC generation unit  74  during the period Ta, and selects the SD-FEC parity and outputs it to the HD-FEC generation unit  74  during the period Tb. 
     The HD-FEC generation unit  74  is an example of the first generation unit, and generates an HD-FEC parity from the bit strings of level- 0  to level- 3  for each frame. The bit string of level- 3  alternately includes data # 30  to data # 33  and SD-FEC parities, By switching of the selector  72   b , the HD-FEC generation unit  74  generates an HD-FEC parity from the data # 30  to data # 33  during the period Ta, and generates an HD-FEC parity from the SD-FEC parities during the period Tb. 
     The HD-FEC generation unit  74  calculates an HD-FEC parity from values of the bit strings of level- 0  to level- 3  for each frame and outputs the HD-FEC parity to the HD delay generation unit  78   a . Note that, during the period Tc, the HD-FEC generation unit  74  receives no input of the bit string of level- 3  from the selector  72   b , and generates an HD-FEC parity from the bit strings of other levels, that is, level- 0  to level- 2 . 
     The HD delay generation unit  78   a , which is an example of the first insertion unit, delays an HD-FEC parity, and inserts the HD-FEC parity into the bit string of level- 2  in a frame subsequent to a frame including the bit string used to generate the HD-FEC parity. The HD delay generation unit  78   a  delays the HD-FEC parity by a period T of a frame, and outputs the HD-FEC parity to the selectors  72   a  and  72   c.    
     The HD-FEC generation unit  74  outputs, to the symbol mapping unit  77 , the bit strings of level- 2 , level- 1 , and level- 0  respectively input from the DM processing units  71   a  to  71   c  as they are. Furthermore, the HD-FEC generation unit  74  outputs, to the selector  72   c , the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 3  input from the selector  72   b  as it is. 
     The selector  72   c  selects an input signal in accordance with controls in the periods Ta to Tc by the operation control unit  70 . The selector  72   c  selects the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 3  during the periods Ta and Tb, and selects the HD-FEC parity during the period Tc. The selector  72   c  outputs the selected input signal as a bit string of level- 3  to the symbol mapping unit  77 . 
     The symbol mapping unit  77  is an example of the assigning unit, and assigns, to the bit strings of level- 0  to level- 3 , symbols corresponding to the values of the bit strings of level- 0  to level- 3 , the symbols being among a plurality of symbols in a constellation of 256QAM. The symbol mapping unit  77  performs symbol mapping by set-partitioning. The symbol mapping unit  77  outputs each of the bit strings of level- 0  to level- 3  as an output signal Sout. 
     Furthermore, to the selector  72   a , the HD-FEC parity is input from the HD delay generation unit  78   a , and the SD-FEC parity is input from the SD delay generation unit  78   b . For this reason, the delayed HD-FEC parity and SD-FEC parity are used in the XOR operator  73   b.    
     The PS conversion unit  79  is therefore able to convert the bit string of level- 0  using the HD-FEC parity and the SD-FEC parity determined by the HD-FEC generation unit  74  and the SD-FEC generation unit  75 , respectively, in stages following the PS conversion unit  79 . Thus, the probability distribution of symbol assignment is formed normally. 
     In this way, the symbol mapping unit  77  assigns, to the bit strings of level- 0  to level- 3 , symbols corresponding to the values of the bit strings of level- 0  to level- 3 , the symbols being among the plurality of symbols in the constellation of 256QAM. The PS conversion unit  79  converts the value of each of the bit strings of level- 0  to level- 2  other than the bit string of level- 3  so that a symbol closer to the center of the constellation is assigned more. 
     The HD-FEC generation unit  74  generates an HD-FEC parity from the bit strings of level- 0  to level- 3  for each frame. The HD delay generation unit  78   a  delays an HD-FEC parity, and inserts the HD-FEC parity into the bit string of level- 3  in a frame subsequent to a frame including the bit string used to generate the HD-FEC parity. 
     The SD-FEC generation unit  75  generates an SD-FEC parity from the bit string of level- 0  in the periods Ta and Tb or the periods Tb and Tc. The SD delay generation unit  78   b  delays an SD-FEC parity, and inserts the SD-FEC parity into the bit string of level- 2  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0  used to generate the SD-FEC parity. 
     Furthermore, the PS conversion unit  79  uses the delayed SD-FEC parity and HD-FEC parity for converting the value of the bit string of level- 0 . 
     In comparison with the configuration of the first embodiment, the configuration described above is the same except for a difference due to the addition of the bit string of level- 2  on which DM processing has been performed. Thus, according to the encoding circuit  120  of the present embodiment, it is possible to reduce power consumption without reducing noise tolerance. 
       FIG. 17  is a configuration diagram illustrating the decoding circuit  121  of the second embodiment. The decoding method of the embodiment is decoding processing of the decoding circuit  121  described below. This will be described below with reference to  FIGS. 15 and 17 . 
     The decoding circuit  121  includes an operation control unit  80 , a soft decision unit  81 , an SD-FEC decoding unit  82 , a hard decision unit  83 , a selector  84   a , an HD-FEC decoding unit  85 , a bit delay generation unit  86   a , a data delay generation unit  86   b , an HD delay adjustment unit  86   c , and a PS inverse conversion unit  89 . The PS inverse conversion unit  89  includes a selector  84   b , a parity delay generation unit  86   d , XOR operators  87   a  and  87   b , and IDM processing units  88   a  to  88   c.    
     The operation control unit  80  notifies the hard decision unit  83 , the soft decision unit  81 , and the selectors  84   a  and  84   b  of the periods Ta to Tc in accordance with frame synchronization information, for example. The selectors  84   a  and  84   b  select an input signal in accordance with notifications of the periods Ta to Tc. The hard decision unit  83  detects an SD-FEC parity from the bit string of level- 2  in accordance with a notification of the period Tb, and the soft decision unit  81  detects an HD-FEC parity from the bit string of level- 2  in accordance with a notification of the period Tc. 
     An input signal Sin∝ is input from an analog-digital conversion unit  13  to the soft decision unit  81  and the hard decision unit  83  separately. The data delay generation unit  86   b  that delays the input signal Sin′ is provided in a stage preceding the hard decision unit  83 . 
     The soft decision unit  81  is an example of the second decision unit, and performs soft decision on the value of the bit string of level- 0  in a frame to which symbols in the constellation of 256QAM are assigned, on the basis of the symbols. At this time, the soft decision unit  81  performs symbol demapping by set-partitioning. On the basis of the result of the soft decision, the soft decision unit  81  extracts an SD-FEC parity from the bit string of level- 2  in the period Ta, and outputs the SD-FEC parity to the SD-FEC decoding unit  82 . 
     Furthermore, on the basis of the result of the soft decision, the soft decision unit  81  outputs the data # 1  to data # 5  in the bit string of level- 0  to the SD-FEC decoding unit  82 . The bit delay generation unit  86   a  that delays the bit string of level- 0  is provided in a stage preceding the SD-FEC decoding unit  82 . 
     The bit delay generation unit  86   a  is an example of the second delay unit, and delays the bit string of level- 0  by the periods Ta and Tb or the periods Ta and Tc. For this reason, the bit string of level- 0  is input to the SD-FEC decoding unit  82  in synchronization with the SD-FEC parity delayed by the total of the periods Ta and Tb or the total of the periods Ta and Tc in the encoding circuit  120 . 
     The SD-FEC decoding unit  82  is an example of the second correction unit, and corrects an error in a result of the soft decision by the soft decision unit  81  on the basis of the SD-FEC parity inserted into the bit string of level- 2  during the periods Ta and Tb or the periods Tb and Tc. For example, the SD-FEC decoding unit  82  decodes the bit string of level- 0  on the basis of the SD-FEC parity. 
     The timing at which the SD-FEC parity is input to the SD-FEC decoding unit  82  is synchronized with the timing at which the bit string of level- 0  is input to the SD-FEC decoding unit  82 . For this reason, the SD-FEC decoding unit  82  may correct the value of the bit string of level- 0  by using the SD-FEC parity in periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string of level- 0 . In the case of the example of  FIG. 15 , the data # 1  in the bit string of level- 0  is corrected by using the SD-FEC parity # 1 , and the data # 2  in the bit string of level- 0  is corrected by using the SD-FEC parity # 2 . 
     The bit string of level- 0  is input from the SD-FEC decoding unit  82  to the hard decision unit  83  and the HD delay adjustment unit  86   c . The SD-FEC parity is input from the SD-FEC decoding unit  82  to the selector  84   a.    
     Furthermore, the hard decision unit  83  is an example of the second decision unit, and performs hard decision on the value of each of the bit strings of level- 1  to level- 3  among the bit strings of level- 0  to level- 3  to which symbols in the constellation of 256QAM are assigned, on the basis of the symbols. For example, the hard decision unit  83  is used for hard decision of the bit string of level- 0  input from the SD-FEC decoding unit  82 . At this time, the hard decision unit  83  performs symbol demapping by set-partitioning. The bit string of level- 3  is input from the hard decision unit  83  to the selector  84   a , and the bit strings of level- 1  and level- 2  are input from the hard decision unit  83  to the HD delay adjustment unit  86   c . Furthermore, on the basis of the result of the hard decision, the hard decision unit  83  extracts an HD-FEC parity from the bit string of level- 3 , and outputs the HD-FEC parity to the HD-FEC decoding unit  85 . 
     Furthermore, the data delay generation unit  86   b  delays each bit string input to the hard decision unit  83 . The data delay generation unit  86   b  delays each bit string for the same amount of time as the bit delay generation unit  86   a , The timing at which the bit strings of level- 1  and level- 2  and the HD-FEC parity are input to the hard decision unit  83  and the timing at which the bit string and the SD-FEC parity are input to the SD-FEC decoding unit  82  are therefore aligned. 
     The selector  84   a  selects an input signal from the data # 30  to data # 33  and the SD-FEC parity in the bit string of level- 3  in accordance with controls in the periods Ta to Tc by the operation control unit  80 . The selector  84   a  outputs the data # 30  to data # 33  in the bit string of level- 2  to the HD delay adjustment unit  86   c  during the period Ta, and outputs the SD-FEC parity to the HD delay adjustment unit  86   c  during the period Tb. 
     The HD delay adjustment unit  86   c , which is an example of the first delay unit, delays each of the bit strings of level- 0  to level- 3  by the period T of the frame, and outputs the delayed bit string to the HD-FEC decoding unit  85 . 
     The HD-FEC decoding unit  85  is an example of the first correction unit, and corrects an error in a result of the hard decision by the hard decision unit  83  on the basis of the HD-FEC parity inserted into the bit string of level- 3  for each frame. Since each of the bit strings of level- 0  to level- 3  is input after being delayed by the period T of the frame, the HD-FEC decoding unit  85  may correct each of the bit strings of level- 0  to level- 3  on the basis of the HD-FEC parity inserted into the bit string of level- 3  in a subsequent frame. The HD-FEC decoding unit  85  outputs each of the bit strings of level- 0  to level- 3  to the PS inverse conversion unit  89 . 
     The PS inverse conversion unit  89  is an example of the inverse conversion unit, and performs inverse conversion of the value of each of the bit strings of level- 0  to level- 2  converted by DM processing. The HD-FEC parity is input to the parity delay generation unit  86   d , and the bit string of level- 2  is inputted to the IDM processing unit  88   a  and the XOR operator  87   a . Furthermore, the bit strings of level- 1  and level- 0  are input to the XOR operators  87   a  and  87   b , respectively, and the bit string of level- 3  is inputted to the selector  84   b.    
     The parity delay generation unit  86   d  delays the HD-FEC parity in accordance with the period Tc within the period T of the frame, and outputs the HD-FEC parity to the selector  84   b.    
     The selector  84   b  selects an input signal from the bit string of level- 3  and the HD-FEC parity in accordance with controls in the periods Ta to Tc by the operation control unit  80 . The selector  84   b  selects the bit string of level- 3  and outputs it to the XOR operator  87   b  during the periods Ta and Tb, and selects the delayed HD-FEC parity and outputs it to the XOR operator  87   b  during the period Tc. 
     The XOR operator  87   a  performs XOR operation the bit string of level- 1  with the bit string of level- 2 . The bit string of level- 1  after performed XOR operation is inputted to the IDM processing unit  88   b.    
     The XOR operator  87   b  performs XOR operation the bit string of level- 0  with the bit string of level- 3  or the HD-FEC parity from the selector  84   b . Since the HD-FEC parity is input to the XOR operator  87   b  with a delay in accordance with the period Tc, the bit string of level- 0  is normally converted to the value before performing XOR operation by the XOR operator  73   b  of the encoding circuit  120 . The bit string of level- 0  after XOR is input to the IDM processing unit  88   c.    
     The bit string of level- 3  and the bit strings of level- 2 , level- 1 , and level- 0  respectively subjected to inverse-DM processing by the IDM processing units  88   a  to  88   c  are output to the framer chip  11  as an output signal Sour by parallel-serial conversion, for example. 
     In this way, the HD delay adjustment unit  86   c  delays each of the bit strings of level- 0  to level- 3  so that an error in a result of the hard decision of each of the bit strings of level- 1  to level- 3  is corrected on the basis of the HD-FEC parity inserted into the bit string of level- 2  in a subsequent frame. For this reason, the delay time given to the HD-FEC parity by the HD delay generation unit  78   a  of the encoding circuit  120  is reduced. 
     Furthermore, the bit delay generation unit  86   a  delays the bit string of level- 0  so that an error in a result of the soft decision of the bit string of level- 0  is corrected on the basis of the SD-FEC parity inserted into the bit string of level- 2  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0 . For this reason, the delay time given to the SD-FEC parity by the SD delay generation unit  78   b  of the encoding circuit  120  is reduced. 
     Thus, the decoding circuit  121  may correct errors in the hard decision and the soft decision of each of the bit strings of level- 0  to level- 3  on the basis of the HD-FEC parity and SD-FEC parity delayed by the encoding circuit  120 . 
     The PS inverse conversion unit  89  performs inverse conversion of the value of the bit string of level- 0  delayed by the bit delay generation unit  86   a  and the HD delay adjustment unit  86   c  by using the HD-FEC parity inserted into the bit string of level- 3  in a subsequent frame, and the SD-FEC parity inserted into periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string of level- 3 . 
     For this reason, the PS inverse conversion unit  89  may normally perform inverse conversion of the bit string of level- 0  using the HD-FEC parity and SD-FEC parity delayed by the encoding circuit  120 . 
     Thus, the decoding circuit  121  may decode a bit string on the basis of encoding by the encoding circuit  120 , thereby reducing power consumption without reducing noise tolerance. 
     Third Embodiment 
       FIG. 18  is a diagram illustrating example of a frame format of an output signal Sout output from an encoding circuit  120  of a third embodiment. In the present embodiment, a case where 16QAM is used as a multi-level modulation scheme will be described. In this case, a frame includes bit strings of level- 0  and level- 1 . Note that, in  FIG. 18 , the same components as those in FIG.  12  are denoted by the same reference numerals, and description thereof will be omitted. 
     The bit string of level- 1  in frame #n includes data # 30  to data # 33  not subject to DM processing, SD-FEC parities (SD-PTY) # 0  to # 4 , and an HD-FEC parity (HD-PTY) # 0 , The SD-FEC parities # 0  to # 4  are each inserted into the bit string in a period Tb, and the data # 30  to data # 33  are each included in the bit string in a period Ta. Furthermore, the HD-PTY # 0  is inserted into the bit string in a period Tc at the end. 
     In comparison with the frame format of the first embodiment, the frame format of the present embodiment does not have a bit string including data # 10  on which DM processing has been performed, but has the same configuration for other levels, that is, level- 0  and level- 1 . For this reason, according to the frame format of the present embodiment, an effect similar to that of the frame format of the first embodiment may be obtained. 
       FIG. 19  is a configuration diagram illustrating the encoding circuit  120  of the third embodiment. The encoding method of the embodiment is encoding processing of the encoding circuit  120  described below. This will be described below with reference to  FIGS. 18 and 19 . 
     The encoding circuit  120  includes an operation control unit  50 , a PS conversion unit  59 , an HD-FEC generation unit  54 , an SD-FEC generation unit  55 , a symbol mapping unit  57 , selectors (SELs)  56   a  and  56   b , an HD delay generation unit  58   a , and an SD delay generation unit  58   b . The PS conversion unit  59  includes a DM processing unit  51 , a selector (SEL)  52 , and an XOR operator  53 . 
     Each of the bit strings of level- 0  and level- 1  is input to the PS conversion unit  59 . The PS conversion unit  59  is an example of the conversion unit, and converts the value of the bit string of level- 0  other than level- 1  so that a symbol closer to the center of a constellation is assigned more. The data # 30  to data # 33  in the bit string of level- 1  are input to the selectors  52  and  56   a.    
     The bit string of level- 0  is inputted to the DM processing unit  51 . The DM processing unit  51  performs DM processing similarly to the DM processing units  21   a  and  21   b . The bit string of level- 0  on which DM processing has been performed is inputted to the XOR operator  53 . 
     The XOR operator  53  performs an XOR operation on the bit string of level- 0  output from the DM processing unit  51  and data selected by the selector  52 . The bit string of level- 0  after performing XOR operation is input to the HD-FEC generation unit  54  and the SD-FEC generation unit  55 . 
     The data # 30  to data # 33 , an SD-FEC parity, and an HD-FEC parity in the bit string of level- 1  are inputted to the selector  52 . The selector  52  selects, as an input signal, any one of the data # 30  to data # 33 , the SD-FEC parity, or the FID-FEC parity in the bit string of level- 1  in accordance with the periods Ta to Tc controlled by the operation control unit  50 . 
     The selector  52  selects the data # 30  to data # 33  in the bit string of level- 1  during the period Ta, the SD-FEC parity during the period Tb, and the HD-FEC parity during the period Tc. The selected input signal is inputted to the XOR operator  53 . For this reason, the SD-FEC parity, the HD-FEC parity, and the data # 30  to data # 33  in the bit string of level- 1  are used for the XOR operation in accordance with the periods Ta to Tc. 
     The SD-FEC generation unit  55  is an example of the second generation unit, and generates an SD-FEC parity from the bit string of level- 0  in the periods Ta and Tb or the periods Tb and Tc. The SD-FEC generation unit  55  calculates an SD-FEC parity from the data # 1  to data # 4  of level- 0  in the periods Ta and Tb or the data # 5  in the periods Tb and Tc and outputs the SD-FEC parity to the SD delay generation unit  58   b.    
     The SD delay generation unit  58   b , which is an example of the second insertion unit, delays an SD-FEC parity, and inserts the SD-FEC parity into the bit string of level- 1  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0  used to generate the SD-FEC parity. The SD delay generation unit  58   b  delays an SD-FEC parity input from the SD-FEC generation unit  55  once every pair of periods Ta and Tb or every pair of periods Tb and Tc by the total of the periods Ta and Tb or the total of the periods Tb and Tc, and outputs the SD-FEC parity to the selectors  52  and  56   a.    
     The data # 30  to data # 33  and the SD-FEC parity in the bit string of level- 1  are input to the selector  56   a . The selector  56   a  selects, as an input signal, any one of the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 1  in accordance with the periods Ta and Tb controlled by the operation control unit  50 . The selector  56   a  selects the data # 30  to data # 33  in the bit string of level- 1  and outputs it to the HD-FEC generation unit  54  during the period Ta, and selects the SD-FEC parity and outputs it to the HD-FEC generation unit  54  during the period Tb. 
     The HD-FEC generation unit  54  is an example of the first generation unit, and generates an HD-FEC parity from the bit strings of level- 0  and level- 1  for each frame. The bit string of level- 1  alternately includes data # 30  to data # 33  and SD-FEC parities. By switching of the selector  56   a , the HD-FEC generation unit  54  generates an HD-FEC parity from the data # 30  to data # 33  during the period Ta, and generates an HD-FEC parity from the SD-FEC parities during the period Tb. 
     The HD-FEC generation unit  54  calculates an HD-FEC parity from values of the bit strings of level- 0  and level- 1  for each frame and outputs the HD-FEC parity to the HD delay generation unit  58   a , Note that, during the period Tc, the HD-FEC generation unit  54  receives no input of the bit string of level- 1  from the selector  56   a , and generates an HD-FEC parity from the bit string of another level, that is, level- 0 . 
     The HD delay generation unit  58   a , which is an example of the first insertion unit, delays an HD-FEC parity, and inserts the HD-FEC parity into the bit string of level- 1  in a frame subsequent to a frame including the bit string used to generate the HD-FEC parity. The HD delay generation unit  58   a  delays the HD-FEC parity by a period T of a frame, and outputs the HD-FEC parity to the selectors  52  and  56   b.    
     The HD-FEC generation unit  54  outputs, to the symbol mapping unit  57 , the bit string of level- 0  input from the DM processing unit  51  as it is. Furthermore, the HD-FEC generation unit  54  outputs, to the selector  56   b , the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 1  input from the selector  56   a  as it is. 
     The selector  56   b  selects an input signal in accordance with controls in the periods Ta to Tc by the operation control unit  50 . The selector  56   b  selects the data # 30  to data # 33  or the SD-FEC parity in the bit string of level- 1  during the periods Ta and Tb, and selects the HD-FEC parity during the period Tc. The selector  56   b  outputs the selected input signal as a bit string of level- 1  to the symbol mapping unit  57 . 
     The symbol mapping unit  57  is an example of the assigning unit, and assigns, to the bit strings of level- 0  and level- 1 , symbols corresponding to the values of the bit strings of level- 0  and level- 1 , the symbols being among a plurality of symbols in a constellation of 16QAM. The symbol mapping unit  57  performs symbol mapping by set-partitioning. The symbol mapping unit  57  outputs each of the bit strings of level- 0  and level- 1  as an output signal Sout. 
     Furthermore, to the selector  52 , the HD-FEC parity is input from the HD delay generation unit  58   a , and the SD-FEC parity is input from the SD delay generation unit  58   b . For this reason, the delayed HD-FEC parity and SD-FEC parity are used in the XOR operator  53 . 
     The PS conversion unit  59  is therefore able to convert the bit string of level- 0  using the HD-FEC parity and the SD-FEC parity determined by the HD-FEC generation unit  54  and the SD-FEC generation unit  55 , respectively, in stages following the PS conversion unit  59 . Thus, the probability distribution of symbol assignment is formed normally. 
     In this way, the symbol mapping unit  57  assigns, to the bit strings of level- 0  and level- 1 , symbols corresponding to the values of the bit strings of level- 0  and level- 1 , the symbols being among the plurality of symbols in the constellation of 16QAM. The PS conversion unit  59  converts the value of the bit string of level- 0  other than the bit string of level- 1  so that a symbol closer to the center of the constellation is assigned more. 
     The HD-FEC generation unit  54  generates an HD-FEC parity from the bit strings of level- 0  and level- 1  for each frame. The HD delay generation unit  58   a  delays an HD-FEC parity, and inserts the HD-FEC parity into the bit string of level- 1  in a frame subsequent to a frame including the bit string used to generate the HD-FEC parity. 
     The SD-FEC generation unit  55  generates an SD-FEC parity from the bit string of level- 0  in the periods Ta and Tb or the periods Tb and Tc. The SD delay generation unit  58   b  delays an SD-FEC parity, and inserts the SD-FEC parity into the bit string of level- 1  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0  used to generate the SD-FEC parity. 
     Furthermore, the PS conversion unit  59  uses the delayed SD-FEC parity and HD-FEC parity for converting the value of the bit string of level- 0 . 
     In comparison with the configuration of the first embodiment, the configuration described above is the same except for a difference due to the addition of the bit string of data # 10  on which DM processing has been performed. Thus, according to the encoding circuit  120  of the present embodiment, it is possible to reduce power consumption without reducing noise tolerance. 
       FIG. 20  is a configuration diagram illustrating a decoding circuit  121  of the third embodiment. The decoding method of the embodiment is decoding processing of the decoding circuit  121  described below. This will be described below with reference to  FIGS. 18 and 20 . 
     The decoding circuit  121  includes an operation control unit  60 , a soft decision unit  61 , an SD-FEC decoding unit  62 , a hard decision unit  63 , a selector  64   a , an HD-FEC decoding unit  65 , a bit delay generation unit  66   a , a data delay generation unit  66   b , an HD delay adjustment unit  66   c , and a PS inverse conversion unit  69 . The PS inverse conversion unit  69  includes a selector  64   b , a parity delay generation unit  66   d , an XOR operator  67 , and IDM processing units  68   a  and  68   b.    
     The operation control unit  60  notifies the hard decision unit  63 , the soft decision unit  61 , and the selectors  64   a  and  64   b  of the periods Ta to Tc in accordance with frame synchronization information, for example. The selectors  64   a  and  64   b  select an input signal in accordance with notifications of the periods Ta to Tc. The hard decision unit  63  detects an SD-FEC parity from the bit string of level- 1  in accordance with a notification of the period Tb, and the soft decision unit  61  detects an HD-FEC parity from the bit string of level- 1  in accordance with a notification of the period Tc. 
     An input signal Sin′ is input from an analog-digital conversion unit  13  to the soft decision unit  61  and the hard decision unit  63  separately. The data delay generation unit  66   b  that delays the input signal Sin′ is provided in a stage preceding the hard decision unit  63 . 
     The soft decision unit  61  is an example of the second decision unit, and performs soft decision on the value of the bit string of level- 0  in a frame to which symbols in the constellation of 16QAM are assigned, on the basis of the symbols. At this time, the soft decision unit  61  performs symbol demapping by set-partitioning. On the basis of the result of the soft decision, the soft decision unit  61  extracts an SD-FEC parity from the bit string of level- 2  in the period Ta, and outputs the SD-FEC parity to the SD-FEC decoding unit  62 . 
     Furthermore, on the basis of the result of the soft decision, the soft decision unit  61  outputs the data # 1  to data # 5  in the bit string of level- 0  to the SD-FEC decoding unit  62 . The bit delay generation unit  66   a  that delays the bit string of level- 0  is provided in a stage preceding the SD-FEC decoding unit  62 . 
     The bit delay generation unit  66   a  is an example of the second delay unit, and delays the bit string of level- 0  by the periods Ta and Tb or the periods Ta and Tc. For this reason, the bit string of level- 0  is inputted to the SD-FEC decoding unit  62  in synchronization with the SD-FEC parity delayed by the total of the periods Ta and Tb or the total of the periods Ta and Tc in the encoding circuit  120 . 
     The SD-FEC decoding unit  62  is an example of the second correction unit, and corrects an error in a result of the soft decision by the soft decision unit  61  on the basis of the SD-FEC parity inserted into the bit string of level- 2  during the periods Ta and Tb or the periods Tb and Tc. For example, the SD-FEC decoding unit  62  decodes the bit string of level- 0  on the basis of the SD-FEC parity. 
     The timing at which the SD-FEC parity is input to the SD-FEC decoding unit  62  is synchronized with the timing at which the bit string of level- 0  is input to the SD-FEC decoding unit  62 . For this reason, the SD-FEC decoding unit  62  may correct the value of the bit string of level- 0  by using the SD-FEC parity in periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string of level- 0 . In the case of the example of  FIG. 18 , the data # 1  in the bit string of level- 0  is corrected by using the SD-FEC parity # 1 , and the data # 2  in the bit string of level- 0  is corrected by using the SD-FEC parity # 2 . 
     The bit string of level- 0  is input from the SD-FEC decoding unit  62  to the hard decision unit  63  and the HD delay adjustment unit  66   c . The SD-FEC parity is input from the SD-FEC decoding unit  62  to the selector  64   a.    
     Furthermore, the hard decision unit  63  is an example of the second decision unit, and performs hard decision on the value of the bit string of level- 1  among the bit strings of level- 0  to level- 3  to which symbols in the constellation of 16QAM are assigned, on the basis of the symbols. For example, the hard decision unit  63  is used for hard decision of the bit string of level- 0  input from the SD-FEC decoding unit  62 . At this time, the hard decision unit  63  performs symbol demapping by set-partitioning. The bit string of level- 1  is inputted from the hard decision unit  63  to the selector  64   a , and the bit string of level- 1  is input from the hard decision unit  63  to the HD delay adjustment unit  66   c . Furthermore, on the basis of the result of the hard decision, the hard decision unit  63  extracts an HD-FEC parity from the bit string of level- 1 , and outputs the HD-FEC parity to the HD-FEC decoding unit  65 . 
     Furthermore, the data delay generation unit  66   b  delays each bit string input to the hard decision unit  63 . The data delay generation unit  66   b  delays each bit string for the same amount of time as the bit delay generation unit  66   a . The timing at which the bit string of level- 1  and the HD-FEC parity are input to the hard decision unit  63  and the timing at which the bit string and the SD-FEC parity are input to the SD-FEC decoding unit  62  are therefore aligned. 
     The selector  64   a  selects an input signal from the data # 30  to data # 33  and the SD-FEC parity in the bit string of level- 3  in accordance with controls in the periods Ta to Tc by the operation control unit  60 . The selector  64   a  outputs the data # 30  to data # 33  in the bit string of level- 2  to the HD delay adjustment unit  66   c  during the period Ta, and outputs the SD-FEC parity to the HD delay adjustment unit  66   c  during the period Tb. 
     The HD delay adjustment unit  66   c , which is an example of the first delay unit, delays each of the bit strings of level- 0  and level- 1  by the period T of the frame, and outputs the delayed bit string to the HD-FEC decoding unit  65 . 
     The HD-FEC decoding unit  65  is an example of the first correction unit, and corrects an error in a result of the hard decision by the hard decision unit  63  on the basis of the HD-FEC parity inserted into the bit string of level- 1  for each frame. Since each of the bit strings of level- 0  and level- 1  is input after being delayed by the period T of the frame, the HD-FEC decoding unit  65  may correct each of the bit strings of level- 0  to level- 3  on the basis of the HD-FEC parity inserted into the bit string of level- 1  in a subsequent frame. The HD-FEC decoding unit  65  outputs each of the bit strings of level- 0  and level- 1  to the PS inverse conversion unit  69 . 
     The PS inverse conversion unit  69  is an example of the inverse conversion unit, and performs inverse conversion of the value of each of the bit strings of level- 0  and level- 1  converted by DM processing. The FID-FEC parity is input to the parity delay generation unit  66   d , and the bit string of level- 1  is input to the IDM processing unit  68   a  and the selector  64   b . Furthermore, the bit string of level- 0  is input to the XOR operator  67 . 
     The parity delay generation unit  66   d  delays the HD-FEC parity in accordance with the period Tc within the period T of the frame, and outputs the HD-FEC parity to the selector  64   b.    
     The selector  64   b  selects an input signal from the bit string of level- 1  and the HD-FEC parity in accordance with controls in the periods Ta to Tc by the operation control unit  60 . The selector  64   b  selects the bit string of level- 1  and outputs it to the XOR operator  67  during the periods Ta and Tb, and selects the delayed HD-FEC parity and outputs it to the XOR operator  67  during the period Tc. 
     The XOR operator  67  performs XOR operation the bit string of level- 0  with the bit string of level- 1  or the HD-FEC parity from the selector  64   b . Since the HD-FEC parity is input to the XOR operator  67  with a delay in accordance with the period Tc, the bit string of level- 0  is normally converted to the value before XOR by the XOR operator  53  of the encoding circuit  120 . The bit string of level- 0  after performing XOR operation is inputted to the IDM processing unit  68   b.    
     The bit strings of level- 1  and level- 0  subjected to inverse-DM processing by the IDM processing units  68   a  and  68   b , respectively, are output to the framer chip  11  as an output signal Sout by parallel-serial conversion, for example. 
     In this way, the HD delay adjustment unit  66   c  delays each of the bit strings of level- 0  and level- 1  so that an error in a result of the hard decision of the bit string of level- 1  is corrected on the basis of the HD-FEC parity inserted into the bit string of level- 1  in a subsequent frame. For this reason, the delay time given to the HD-FEC parity by the HD delay generation unit  58   a  of the encoding circuit  120  is reduced. 
     Furthermore, the bit delay generation unit  66   a  delays the bit string of level- 0  so that an error in a result of the soft decision of the bit string of level- 0  is corrected on the basis of the SD-FEC parity inserted into the bit string of level- 1  in periods Ta and Tb or periods Tb and Tc that are later than periods Ta and Tb or periods Tb and Tc of the bit string of level- 0 . For this reason, the delay time given to the SD-FEC parity by the SD delay generation unit  58   b  of the encoding circuit  120  is reduced. 
     Thus, the decoding circuit  121  may correct errors in the hard decision and the soft decision of each of the bit strings of level- 0  and level- 1  on the basis of the HD-FEC parity and SD-FEC parity delayed by the encoding circuit  120 . 
     The PS inverse conversion unit  69  performs inverse conversion of the value of the bit string of level- 0  delayed by the bit delay generation unit  66   a  and the HD delay adjustment unit  66   c  by using the HD-FEC parity inserted into the bit string of level- 1  in a subsequent frame, and the SD-FEC parity inserted into periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string of level- 1 . 
     For this reason, the PS inverse conversion unit  69  may normally perform inverse conversion of the bit string of level- 0  using the HD-FEC parity and SD-FEC parity delayed by the encoding circuit  120 . 
     Thus, the decoding circuit  121  may decode a bit string on the basis of encoding by the encoding circuit  120 , thereby reducing power consumption without reducing noise tolerance. 
     In the encoding circuit  120  of the first to third embodiments, the HD delay generation units  28   a ,  78   a , and  58   a  delay an HD-FEC parity, and insert the HD-FEC parity into the bit string of the most significant level in a frame next to a frame including the bit string used to generate the HD-FEC parity. 
     For this reason, the HD delay generation units  28   a ,  78   a , and  58   a  just have to retain an HD-FEC parity for one frame, and this enables minimization of the capacity of a memory for storing the HD-FEC parity, for example. Note that the HD delay generation units  28   a ,  78   a , and  58   a  are not limited to this, and may insert the HD-FEC parity into any one of subsequent frames. In this case, a capacity for storing HD-FEC parities for the number of frames corresponding to the amount of delay is desired. 
     Furthermore, in the encoding circuit  120  of the first to third embodiments, the SD delay generation units  28   b ,  78   b , and  58   b  delay an SD-FEC parity, and insert the SD-FEC parity into the bit string of the most significant level in periods Ta and Tb or periods Ta and Tc that are next to periods Ta and Tb or periods Ta and Tc during which the SD-FEC parity has been generated. 
     For this reason, the SD delay generation units  28   b ,  78   b , and  58   b  just have to retain SD-FEC parities for the total of the periods Ta and Tb or the total of the periods Ta and Tc, and this enables minimization of the capacity of a memory for storing the SD-FEC parities, for example. Note that the SD delay generation units  28   b ,  78   b , and  58   b  are not limited to this, and may insert the SD-FEC parity into the bit string of periods Ta and Tb or periods Ta and Tc that are later than periods Ta and Tb or periods Ta and Tc of the bit string used to generate the SD-FEC parity. In this case, a capacity for storing SD-FEC parities for the periods corresponding to the amount of delay is desired. 
     Furthermore, the transponders  1   a  and  1   b  and the optical transmission system of the present embodiment include the encoding circuit  120  and the decoding circuit  121  of the first to third embodiments. For this reason, operational effects similar to those described above may be obtained. Note that, the encoding circuit  120  and the decoding circuit  121  described above may be, for example, circuits including hardware such as a field programmable gate array (FPGA) or an application specified integrated circuit (ASIC). 
     The embodiments described above are preferred examples. However, the embodiments are not limited to this, and a variety of modifications may be made without departing from the scope of the embodiments. 
     All examples and conditional language provided herein are intended for the pedagogical purposes of aiding the reader in understanding the invention and the concepts contributed by the inventor to further the art, and are not to be construed as limitations to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although one or more embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.