Patent Publication Number: US-2023163786-A1

Title: Error correction code circuit, memory device including error correction code circuit, and operation method of error correction code circuit

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority under 35 U.S.C. § 119 to Korean Patent Application Nos. 10-2021-0164956 filed on Nov. 25, 2021, and 10-2022-0049463 filed on Apr. 21, 2022, in the Korean Intellectual Property Office, the disclosures of which are incorporated by reference herein in their entireties. 
     BACKGROUND 
     1. Field 
     Embodiments relate to a semiconductor memory, and more particularly, relate to an error correction code circuit, a memory device including the same, and an error correction code decoding method of the memory device. 
     2. Description of the Related Art 
     A semiconductor memory device may be classified as a volatile memory device, in which stored data disappear when a power supply is turned off, such as a static random access memory (SRAM) or a dynamic random access memory (DRAM), or a nonvolatile memory device, in which stored data are retained even when a power supply is turned off, such as a flash memory device, a phase-change RAM (PRAM), a magnetic RAM (MRAM), a resistive RAM (RRAM), or a ferroelectric RAM (FRAM). 
     SUMMARY 
     An embodiment is directed to a memory device that includes a memory cell array that stores first data and first parity data, an error correction code (ECC) circuit that performs ECC decoding based on the first data and the first parity data and outputs error-corrected data and a decoding status flag, and an input/output circuit that provides the error-corrected data and the decoding status flag to a memory controller. The ECC circuit includes a syndrome generator that generates a syndrome based on the first data and the first parity data, a syndrome decoding circuit that decodes the syndrome to generate an error vector, a correction logic circuit that generates the error-corrected data based on the error vector and the first data, and a fast decoding status flag (DSF) generator that generates the decoding status flag based on the syndrome, without the error vector. 
     An embodiment is directed to an error correction code (ECC) circuit configured to correct first data based on the first data and first parity data stored in a memory device that includes a syndrome generator that generates a syndrome based on the first data and the first parity data from the memory device, a syndrome decoding circuit that compares the syndrome and each column of a parity-check matrix to generate an error vector, a correction logic circuit that generates error-corrected data based on the error vector and the first data, and a fast decoding status flag (DSF) generator that generates a decoding status flag indicating a decoding status for the error-corrected data based on the syndrome and a simplified-parity-check matrix, and the simplified-parity-check matrix includes only information about a part of the parity-check matrix. 
     An embodiment is directed to an operation method of an error correction code (ECC) circuit configured to correct first data based on the first data and first parity data stored in a memory device that includes generating a syndrome based on the first data and the first parity data, generating an error vector based on the syndrome and a parity-check matrix, performing a logical operation on the error vector and the first data to generate error-corrected data, generating a decoding status flag based on the syndrome and a simplified-parity-check matrix, and sending the error-corrected data and the decoding status flag to a memory controller, and the generating of the error vector and the generating of the decoding status flag are performed in parallel. 
     An embodiment is directed to a method of configuring an error correction code (ECC) circuit that includes determining a parity-check matrix to be used for syndrome decoding of the ECC circuit, generating a simplified-parity-check matrix based on a repetitive characteristic of the parity-check matrix, and configuring a fast decoding status flag generator of the ECC circuit such that a logical operation is performed on information about a part of the syndrome based on information about each column of the simplified-parity-check matrix, and the simplified-parity- check matrix includes only information about a part of the parity-check matrix. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       Features will become apparent to those of skill in the art by describing in detail example embodiments with reference to the attached drawings in which: 
         FIG.  1    is a block diagram illustrating a memory system according to an example embodiment. 
         FIG.  2    is a block diagram illustrating a memory device of  FIG.  1   . 
         FIG.  3    is a block diagram for describing an operation of an error correction code (ECC) circuit of  FIG.  2   . 
         FIG.  4 A  is a flowchart illustrating a reference example of an operation of an ECC decoder. 
         FIG.  4 B  is a block diagram illustrating an ECC decoder configured to perform an operation according to the flowchart of  FIG.  4 A . 
         FIG.  4 C  is a block diagram illustrating a DSF generator of  FIG.  4 B . 
         FIG.  5 A  is a flowchart illustrating an operation of an ECC decoder of  FIG.  3    according to an example embodiment. 
         FIG.  5 B  is a block diagram illustrating an ECC decoder configured to perform an operation according to the flowchart of  FIG.  5 A . 
         FIG.  5 C  is a block diagram illustrating a fast DSF generator of  FIG.  5 B . 
         FIGS.  6 ,  7 A,  7 B, and  7 C  are diagrams illustrating some examples of an H-matrix used by a syndrome decoding circuit of  FIG.  5 A . 
         FIG.  8    is a diagram illustrating some examples of a simplified H-matrix generated based on an H-matrix of  FIGS.  6  to  7 C . 
         FIG.  9    is a diagram illustrating an example of a syndrome checking circuit of a fast DSF generator of  FIG.  5 C . 
         FIG.  10    is a diagram illustrating an example of a parity error checking circuit of a fast DSF generator of  FIG.  5 C . 
         FIG.  11    is a diagram illustrating an example of a fast data error checking circuit of a fast DSF generator of  FIG.  5 C . 
         FIG.  12    is a diagram illustrating an example of a DSF detecting circuit of a fast DSF generator of  FIG.  5 C . 
         FIG.  13    is a block diagram illustrating an example of a DSF detecting circuit of a fast DSF generator of  FIG.  5 C . 
         FIG.  14    is a diagram illustrating an example of a simplified H-matrix according to an example embodiment. 
         FIG.  15    is a diagram illustrating an example of a fast data error checking circuit using a simplified H-matrix for UE determination of  FIG.  14   . 
         FIG.  16    is a diagram illustrating an example of a DSF detecting circuit using data error information from a fast data error checking circuit of  FIG.  15   . 
         FIG.  17    is a diagram illustrating an example of a DSF detecting circuit using data error information from a fast data error checking circuit of  FIG.  15   . 
         FIG.  18    is a flowchart for describing a method of generating a simplified H-matrix according to an example embodiment. 
         FIG.  19    is a block diagram illustrating a memory system according to an example embodiment. 
         FIG.  20    is a block diagram illustrating a memory system according to an example embodiment. 
         FIG.  21    is a block diagram illustrating a memory system according to an example embodiment. 
         FIG.  22    is a diagram illustrating an example of a memory package according to an example embodiment. 
         FIG.  23    is a diagram illustrating an example of a memory package according to an example embodiment. 
         FIG.  24    is a block diagram illustrating a memory module to which a memory device according to the present disclosure is applied. 
         FIG.  25    is a diagram illustrating a system to which a storage device according to an example embodiment is applied. 
     
    
    
     DETAILED DESCRIPTION 
       FIG.  1    is a block diagram illustrating a memory system according to an example embodiment. 
     Referring to  FIG.  1   , a memory system  10  may include a memory controller  11  and a memory device  100 . The memory system  10  may be one of information processing devices, which are configured to process a variety of information and to store the processed information, such as a personal computer (PC), a laptop, a server, a workstation, a smartphone, a tablet PC, a digital camera, and a black box. 
     The memory controller  11  may store data in the memory device  100  or may read data stored in the memory device  100 . For example, the memory controller  11  may send a clock signal CK and a command/address signal CA to the memory device  100 , and may exchange a data signal DQ and a data strobe signal DQS with the memory device  100 . Through the data signal DQ and the data strobe signal DQS, data “DATA” may be sent from the memory controller  11  to the memory device  100  or may be sent from the memory device  100  to the memory controller  11 . The memory controller  11  and the memory device  100  may communicate with each other based on the DDR interface or the LPDDR interface. 
     The memory device  100  may operate under control of the memory controller  11 . The memory device  100  may be a dynamic random access memory (DRAM). The memory device  100  may include a volatile memory such as an SRAM or a nonvolatile memory such as a PRAM and/or an RRAM. 
     The memory device  100  may include an error correction code (ECC) circuit  110 . The ECC circuit  110  may be configured to detect and correct an error of data that are present in the memory device  100 . The ECC circuit  110  may generate parity data by performing ECC decoding on first data received from the memory controller  11 . The memory device  100  may store the first data received from the memory controller  11  together with the parity data generated by the ECC circuit  110 . During an operation of the memory device  100 , an error may occur in the first data present in the memory device  100  due to various factors. When a read request for the first data is made by the memory controller  11 , the ECC circuit  110  may correct an error of the first data by performing ECC decoding based on the first data and the corresponding parity data. The memory device  100  may send the corrected first data to the memory controller  11 . 
     The ECC circuit  110  may correct an error within the given error correction capability. When an error detected from data exceeds the error correction capability of the ECC circuit  110 , the error detected from the data may not be corrected normally. 
     The memory controller  11  requires information for determining whether read data received from the memory device  100  include an uncorrectable error or are normal data (e.g., error-free data or data in which an error is normally corrected). Thus, the ECC circuit  110  may send a decoding status flag DSF to the memory controller  11 . The decoding status flag DSF refers to information indicating whether data sent to the memory controller  11  are UE data or normal data. The decoding status flag DSF may be generated in the ECC decoding process of the ECC circuit  110 . 
     According to an example embodiment, the ECC circuit  110  may generate the decoding status flag DSF using only a syndrome that is generated in the ECC decoding process. In this case, a time taken to generate the decoding status flag DSF may be shortened. A configuration of the ECC circuit  110  and a way to generate the decoding status flag DSF will be described in detail below. 
       FIG.  2    is a block diagram illustrating a memory device of  FIG.  1   . 
     Referring to  FIGS.  1  and  2   , the memory device  100  may include the ECC circuit  110 , a memory cell array  120 , a CA buffer  130 , an address decoder  140 , a command decoder  150 , a sense amplifier and write driver  160 , and an input/output circuit  170 . 
     The ECC circuit  110  may generate parity data by performing ECC encoding on data to be stored in the memory cell array  120 . In another implementation, the ECC circuit  110  may correct an error of data read from the memory cell array  120  by performing ECC decoding based on the read data and parity data. A configuration and an operation of the ECC circuit  110  will be described in detail below. 
     The memory cell array  120  may include a plurality of memory cells. The plurality of memory cells may be connected with a plurality of word lines and a plurality of bit lines. The plurality of word lines may be driven by an X-decoder (or row decoder) X-DEC, and the plurality of bit lines may be driven by a Y-decoder (or column decoder) Y-DEC. 
     The CA buffer  130  may be configured to receive command/address signals CA, and to temporarily store or buffer the received signals. 
     The address decoder  140  may decode address signals ADDR stored in the CA buffer  130 . The address decoder  140  may control the X-decoder and the Y-decoder based on a decoding result. 
     The command decoder  150  may decode a command CMD stored in the CA buffer  130 . 
     The command decoder  150  may control the components of the memory device  100  based on a decoding result. 
     For example, in the case where the command signal stored in the CA buffer  130  corresponds to a write command (i.e., in the case where a command received from the memory controller  11  is the write command), the command decoder  150  may control the ECC circuit  110  such that the data “DATA” received through the input/output circuit  170  are written in the memory cell array  120  (i.e., may perform ECC encoding), and may control an operation of the sense amplifier and write driver  160  (i.e., may activate the write driver). 
     As another example, in the case where the command signal stored in the CA buffer  130  corresponds to a read command (i.e., in the case where the command received from the memory controller  11  is the read command), the command decoder  150  may control the ECC circuit  110  such that data stored in the memory cell array  120  are read out (i.e., may perform ECC decoding), and may control an operation of the sense amplifier and write driver  160  (i.e., may activate the sense amplifier). 
     Under control of the command decoder  150 , the sense amplifier and write driver  160  may read data stored in the memory cell array  120  through the plurality of bit lines or may write data in the memory cell array  120  through the plurality of bit lines. 
     Based on the data signal DQ and the data strobe signal DQS, the input/output circuit  170  may receive the data “DATA” from the memory controller  11  or may send the data “DATA” to the memory controller  11 . The input/output circuit  170  may receive the decoding status flag DSF from the ECC circuit  110 , and may send the decoding status flag DSF to the memory controller  11 . The decoding status flag DSF may be sent to the memory controller  11  through the data signal DQ. In another implementation, the decoding status flag DSF may be sent to the memory controller  11  through various control signals (e.g., DBI and DMI) or any other dedicated signal. 
       FIG.  3    is a block diagram for describing an operation of an ECC circuit of  FIG.  2   . 
     For brevity of drawing and for convenience of description, components that are unnecessary to describe an operation of the ECC circuit  110  are omitted. 
     Referring to  FIGS.  2  and  3   , the ECC circuit  110  may include an ECC encoder and an ECC decoder. The ECC encoder may generate parity data PRT by performing ECC encoding on write data WDT to be stored in the memory array  120 . For example, with regard to the write data WDT having  128   b  (i.e., 128 bits), the ECC encoder may generate the parity data PRT having  8   b  (i.e., 8 bits) using a basis bit BB having  8   b . The write data WDT and the parity data PRT may be stored in the memory cell array  120  through the sense amplifier and write driver  160 . 
     The ECC decoder may output error-corrected read data RDT_cor by performing ECC decoding based on read data RDT and the parity data PRT read from the memory cell array  120 . For example, with regard to the write data RDT having  128   b , the ECC decoder may generate the error-corrected read data RDT_cor having  128   b  using the parity data PRT having  8   b . The ECC decoder may generate the decoding status flag DSF indicating an ECC decoding result or a status of the error-corrected read data RDT_cor. 
     Below, descriptions may be given as some data values or bit values are at a specific level or a specific bit level (e.g., “1” or “0”). However, it will be understood that various data values or bit values that are described by way of example may be variously modified or changed. 
       FIG.  4 A  is a flowchart illustrating a reference example of an operation of an ECC decoder.  FIG.  4 B  is a block diagram illustrating an ECC decoder configured to perform an operation according to the flowchart of  FIG.  4 A .  FIG.  4 C  is a block diagram illustrating a DSF generator of  FIG.  4 B . 
     In the reference example, referring to  FIGS.  4 A,  4 B, and  4 C , in operation S 10 , the ECC decoder ECC-DEC may generate a syndrome SYD based on the read data RDT and the parity data PRT. For example, in the read operation of the memory device  100 , the sense amplifier and write driver  160  may read the read data RDT and the parity data PRT stored in the memory cell array  120 , and may transfer the read data RDT and the parity data PRT to the ECC decoder ECC-DEC. Referring to  FIG.  4 B , the ECC decoder ECC-DEC may include a syndrome generator  111 . The syndrome generator  111  may generate the syndrome SYD based on the read data RDT having  128   b  and the parity data PRT having  8   b . The syndrome generator  111  may generate a check bit having  8   b  based on the read data RDT having  128   b , and may generate the syndrome SYD having  8   b  by performing a logical operation on the check bit having  8   b  and the parity data PRT having  8   b.    
     In operation S 20 , the ECC decoder ECC-DEC may generate an error vector ERV by performing decoding on the syndrome SYD. For example, referring to  FIG.  4 B , the ECC decoder ECC-DEC may further include a syndrome decoding circuit  112 . The syndrome decoding circuit  112  may receive the syndrome SYD from the syndrome generator  111 , and may generate the error vector ERV by decoding the received syndrome SYD. 
     The syndrome decoding circuit  112  may detect a position of the read data RDT, at which an error occurs, by comparing a parity-check matrix (hereinafter referred to as an “H-matrix”) and the syndrome SYD. The error vector ERV may have a size having  128   b  (which may be equal in size to the read data RDT), and may include information about the position of the read data RDT, at which the error occurs. 
     In operation S 31 , the ECC decoder ECC-DEC may correct an error of the read data RDT using the error vector ERV. For example, referring to  FIG.  4 B , the ECC decoder ECC-DEC may further include a correction logic circuit  113 . The correction logic circuit  113  may output error-corrected read data RDT cor by performing a bitwise XOR operation on the error vector ERV and the read data RDT read by the sense amplifier and write driver  160 . 
     In operation S 32 , the ECC decoder ECC-DEC may generate the decoding status flag DSF based on the error vector ERV and the syndrome SYD. For example, referring to  FIG.  4 B , the ECC decoder ECC-DEC may further include a DSF generator  114 . The DSF generator  114  may receive the syndrome SYD having  8   b  from the syndrome decoding circuit  112  and the error vector ERV having  128   b , and may generate the decoding status flag DSF based on the received information. 
     In detail, referring to  FIG.  4 C , the DSF generator  114  may include a data error checking circuit  114   a , a syndrome checking circuit  114   b , a parity error checking circuit  114   c , and a decoding status flag detecting circuit  114   d.    
     The data error checking circuit  114   a  may generate error information ERR indicating whether an error is included in the read data RDT, based on the error vector ERV. The data error checking circuit  114   a  may generate the error information ERR by determining whether at least one of bits of the error vector ERV is “1”. In an implementation, a value of the error information ERR of “0” may indicate that an error is absent from the read data RDT; and a value of the error information ERR of “1” may indicate that an error is present in the read data RDT. 
     The syndrome checking circuit  114   b  may generate syndrome information SYD_ 0 , based on that all the bits of the syndrome SYD are “0”. The case where all the bits of the syndrome SYD are “0” (i.e., the case where the syndrome SYD is all-zero) may indicate that an error is absent from the read data RDT and the parity data PRT; the case where at least one of the bits of the syndrome SYD are “1” (i.e., the case where the syndrome SYD is non-zero) may indicate that an error is present in the read data RDT or the parity data PRT. In an implementation, the syndrome information SYD_ 0  may be “1” when all the bits of the syndrome SYD are “0” (i.e., when the syndrome SYD is all-zero), and may be “0” when at least one of the bits of the syndrome SYD is “1” (i.e., when the syndrome SYD is non-zero). 
     The parity error checking circuit  114   c  may output parity error information PRT_ERR based on that an error is present in the parity data PRT. The parity error checking circuit  114   c  may generate the parity error information PRT_ERR based on an n×n identity matrix and the syndrome SYD (n indicating a size (i.e., the number of bits) of the syndrome SYD). 
     The decoding status flag detecting circuit  114   d  may generate the decoding status flag DSF based on the error information ERR, the syndrome information SYD_ 0 , and the parity error information PRT_ERR. For example, the decoding status flag DSF may indicate a decoding status for the ECC decoding result. The decoding status may indicate whether final data (i.e., the error-corrected read data RDT_cor) output by the ECC decoder ECC-DEC are normal data (i.e., whether an error is absent from the read data RDT or whether a correctable error is included in the read data RDT) or whether an uncorrectable error is included in the read data RDT. 
     Below, for convenience of description, a decoding result or a decoding status corresponding to the case where an uncorrectable error is included in the read data RDT is called an uncorrectable error (UE); a decoding result or a decoding status corresponding to the case where a correctable error is included in the read data RDT is called a correctable error (CE); a decoding result or a decoding status corresponding to the case where an error is not included in the read data RDT is called a non-error (NE). 
     In an example, when the error information ERR is “0” (i.e., an error is absent from the read data RDT), the syndrome information SYD_ 0  is “0” (i.e., an error is present in the read data RDT or the parity data PRT), and the parity error information PRT_ERR is “0” (i.e., an error is absent from the parity data PRT), the decoding status may be the “UE”. In this case, the decoding status flag DSF may be set to “1”. 
     As another example, in an information combination different from the above case, the error-corrected read data RDT_cor output from the ECC decoder ECC-DEC may be in a state of normal data (i.e., in a state where there is no error or an error is normally corrected). In this case (i.e., in the case of the “NE” or “CE”), the decoding status flag DSF may be set to “0”. 
     Returning to  FIG.  4 A , in operation S 40 , the ECC decoder ECC-DEC may output the error-corrected read data RDT_cor and the decoding status flag DSF. The error-corrected read data RDT_cor and the decoding status flag DSF may be provided to the memory controller  11  through the input/output circuit  170  (refer to  FIG.  2   ). The memory controller  11  may determine a decoding status of the received data based on the decoding status flag DSF and may perform various operations based on a determination result. For example, with regard to the “UE”, the memory controller  11  may perform a read retry operation, a reset operation, or various other maintenance operations. 
     The operation of the ECC decoder ECC-DEC described with reference to  FIGS.  4 A to  4 C  may be implemented through various logical operation circuits. For example, operation S 10  (i.e., the syndrome generating operation) may be performed through a 6-stage XOR operation circuit, operation S 20  (i.e., the syndrome decoding operation) may be performed through a 3-stage AND operation circuit, and operation S 31  (i.e., the error correction operation) may be performed through a 1-stage XOR operation circuit. 
     With regard to operation S 32  (i.e., the operation of generating the decoding status flag DSF), the data error checking circuit  114   a  may be implemented through a 7-stage XOR operation circuit, the syndrome checking circuit  114   b  may be implemented through a 4-stage AND operation circuit, the parity error checking circuit  114   c  may be implemented through a 3-stage AND operation circuit and a 3-stage OR operation circuit, and the decoding status flag detecting circuit  114   d  may be implemented through a 1-stage OR operation circuit, a 1-stage invert (INV) operation circuit, and a 1-stage AND operation circuit. 
     As described above in the reference example of  FIGS.  4 A- 4 C , the error vector ERV is used in operation S 32 . Thus, operation S 32  is performed after operation S 20  is completed. In this case, a latency from a time point when the syndrome decoding operation is completed to a time point when operation S 32  is completed may be a latency corresponding to a total of 7 OR operation circuit stages, one INV operation circuit stage, and one AND operation circuit stage. 
     Also, operation S 32  is performed in parallel with operation S 31 . In this case, because the latency for operation S 32  is relatively long compared to the latency for operation S 31 , it will be understood that operation S 32  (i.e., the operation of generating the decoding status flag DSF) may cause a reduction of performance. 
     On the other hand, as will now be described with reference to  FIGS.  5 A to  5 C , a latency in generating the decoding status flag DSF according to an example embodiment may be decreased. 
       FIG.  5 A  is a flowchart illustrating an operation of an ECC decoder of  FIG.  3    according to an example embodiment.  FIG.  5 B  is a block diagram illustrating an ECC decoder configured to perform an operation according to the flowchart of  FIG.  5 A .  FIG.  5 C  is a block diagram illustrating a fast DSF generator of  FIG.  5 B . Below, for convenience of description, additional description associated with the components described above will be omitted to avoid redundancy. 
     In the present example embodiment, referring to  FIGS.  3 ,  5 A,  5 B, and  5 C , in operation S 110 , an ECC decoder ECC-DEC- 1  may generate the syndrome SYD based on the read data RDT and the parity data PRT. For example, referring to  FIG.  5 B , the ECC decoder ECC-DEC- 1  may include the syndrome generator  111 . Operation S 110  of  FIG.  5 A  is similar to operation S 10  of  FIG.  4 A , and an operation of the syndrome generator  111  of  FIG.  5 B  may be similar to the operation of the syndrome generator  111  of  FIG.  4 B . 
     In operation S 120 , the ECC decoder ECC-DEC- 1  may generate the error vector ERV by performing decoding on the syndrome SYD. For example, referring to  FIG.  5 B , the ECC decoder ECC-DEC- 1  may include the syndrome decoding circuit  112 . Operation S 120  of  FIG.  5 A  is similar to operation S 20  of  FIG.  4 A , and an operation of the syndrome decoding circuit  112  of  FIG.  5 B  may be similar to the operation of the syndrome decoding circuit  112  of  FIG.  4 B . 
     In operation S 130 , the ECC decoder ECC-DEC- 1  may correct an error of the read data RDT based on the error vector ERV. For example, referring to  FIG.  5 B , the ECC decoder ECC-DEC- 1  may include the correction logic circuit  113 . Operation S 130  of  FIG.  5 A  is similar to operation S 31  of  FIG.  4 A , and an operation of the correction logic circuit  113  of  FIG.  5 B  may be similar to the operation of the correction logic circuit  113  of  FIG.  4 B . 
     In operation S 140 , the ECC decoder ECC-DEC- 1  may generate the decoding status flag DSF based on a syndrome and a simplified H-matrix. As compared to the ECC decoder ECC_DEC of the reference example in  FIG.  4 B , the ECC decoder ECC-DEC- 1  of the present example embodiment may further include a fast DSF generator  115 , as shown in  FIG.  5 B . This is further described below. 
     Referring again to  FIG.  5 A , in operation S 150 , the ECC decoder ECC-DEC- 1  may output the error-corrected read data RDT_cor and the decoding status flag DSF. Operation S 150  of  FIG.  5 A  may be similar to operation S 40  of  FIG.  4 A . 
     In the present example embodiment, referring to  FIGS.  5 A to  5 C , the fast DSF generator  115  may generate the decoding status flag DSF using only the syndrome SYD without the error vector ERV. Because the fast DSF generator  115  does not use the error vector ERV for the purpose of generating the decoding status flag DSF, the fast DSF generator  115  may operate in parallel with the syndrome decoding operation (e.g., operation S 120 ) or at the same time with the syndrome decoding operation (e.g., operation S 120 ). 
     For example, referring to  FIG.  5 B , the ECC decoder ECC-DEC- 1  may include the fast DSF generator  115 . The fast DSF generator  115  may receive the syndrome SYD from the syndrome decoding circuit  112 , and may generate the decoding status flag DSF based on the received syndrome SYD. 
     Referring to  FIG.  5 C , the fast DSF generator  115  may include a syndrome checking circuit  115   b , a parity error checking circuit  115   c , a DSF detecting circuit  115   d , and a fast data error checking circuit  115   e.    
     Operations of the syndrome checking circuit  115   b , the parity error checking circuit  115   c , the DSF detecting circuit  115   d  of  FIG.  5 C  may be similar to the operations of the syndrome checking circuit  114   b , the parity error checking circuit  114   c , and the DSF detecting circuit  114   d  of  FIG.  4 C . 
     The DSF generator  114  of the reference example of  FIG.  4 C  generates the error information ERR about the read data RDT using the error vector ERV. In contrast, the fast data error checking circuit  115   e  of  FIG.  5 C  may generate error information DT_ERR about the read data RDT using the syndrome SYD. 
     In detail, the fast data error checking circuit  115   e  may generate the error information DT_ERR about the read data RDT by comparing at least some of the bits of the syndrome SYD with the simplified H-matrix. The simplified H-matrix may be generated based on the H-matrix that is used by the syndrome decoding circuit  112  or based on partial information of the H-matrix. 
     A configuration of the simplified H-matrix will be described in detail with reference to  FIGS.  6  to  8   . 
     As described above, the fast DSF generator  115  according to the present example embodiment may generate the decoding status flag DSF using only the syndrome SYD without the error vector ERV. Accordingly, the operation of generating the decoding status flag DSF may be performed in parallel with the syndrome decoding operation, and the whole latency of the ECC decoder ECC-DEC- 1  may be decreased. 
     Stated another way, as described for the reference example of  FIGS.  4 A to  4 C , the latency that is required to generate the decoding status flag DSF from a time point when the syndrome decoding operation is completed may be the latency corresponding to a total of 7 OR operation circuit stages, one INV operation circuit stage, and one AND operation circuit stage. On the other hand, as described with reference to the present example embodiment of  FIGS.  5 A to  5 C , the latency that is required to generate the decoding status flag DSF from a time point when the syndrome decoding operation is completed may be a latency corresponding to a total of 4 OR operation circuit stages, one INV operation circuit stage, and one AND operation circuit stage, through the fast DSF generator  115 . 
     Thus, according to the present example embodiment, the decoding status flag DSF may be generated using only the syndrome SYD without the error vector ERV, such that the whole latency of the ECC decoder ECC-DEC- 1  may be decreased. 
       FIGS.  6 ,  7 A,  7 B, and  7 C  are diagrams illustrating some examples of an H-matrix used by a syndrome decoding circuit of  FIG.  5 A .  FIG.  8    is a diagram illustrating some examples of a simplified H-matrix generated based on an H-matrix of  FIGS.  6  to  7 C . 
     In the present example embodiment, a simplified H-matrix sim-H-mat may be used by the fast data error checking circuit  115   e.    
     In an example embodiment, an H-matrix H-mat, which will now be described with reference to  FIGS.  6  to  8   , is an example of the H-matrix for single bit error correction (SEC). However, it will be understood that the H-matrix and the simplified H-matrix may be variously modified or changed depending an implementation of the memory device  100  or the ECC circuit  110 . 
     Referring to  FIGS.  6 ,  7 B, and  7 C , the H-matrix H-mat may include a plurality of sub-matrices SM 00  to SM 15  and a parity matrix PR. Referring to  FIG.  7 A , the parity matrix PR may be an identity matrix IM of dimension 8×8. The eight rows of the parity matrix PR may respectively correspond to each of eight bits (e.g., S 0 , S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , and S 7 ) of the syndrome SYD. 
     Each of the plurality of sub-matrices SM 00  to SM 15  may be of dimension 8×8. The plurality of sub-matrices SM 00  to SM 15  may be arranged to have a given rule. 
     For example, three rows of each of the plurality of sub-matrices SM 00  to SM 15  may constitute an A 0 -th segment matrix A 0 . That is, the three rows of each of the plurality of sub-matrices SM 00  to SM 15  may have the same repetitive pattern. 
     Also, the remaining rows, that is, five rows of each of the plurality of sub-matrices SM 00  to SM 15 , may constitute each of B 00 -th to B 15 -th sub-matrices B 00  to B 15 . The B 00 -th to B 15 -th sub-matrices B 00  to B 15  may be configured to have the same column. 
     In detail, referring to  FIG.  7 B , each row of the H-matrix H-mat may correspond to each of the bits (e.g., S 0 , S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , and S 7 ) of the syndrome SYD. 
     For example, the A 0 -th sub-matrix A 0  may be placed at first sub-rows (e.g., rows corresponding to S 0 , S 1 , and S 2 ) of the 00-th sub-matrix SM 00  of the H-matrix H-mat, and the A 0 -th sub-matrix A 0  may be placed at first sub-rows (e.g., rows corresponding to S 0 , S 1 , and S 2 ) of the 01-th sub-matrix SM 01 . Likewise, the A 0 -th sub-matrix A 0  may be placed at first sub-rows (e.g., rows corresponding to S 0 , S 1 , and S 2 ) of the 15-th sub-matrix SM 15 . 
     Similarly, the B 00 -th sub-matrix B 00  may be placed at second sub-rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the 00-th sub-matrix SM 00  of the H-matrix H-mat, and the B 01 -th sub-matrix B 01  may be placed at second sub-rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the 01-th sub-matrix SM 01 . Likewise, the B 15 -th sub-matrix B 15  may be placed at second sub-rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the 15-th sub-matrix SM 15 . 
     In the first sub-rows 1st sub-Rows (e.g., rows corresponding to S 0 , S 1 , and S 2 ) of the H-matrix H-mat, a pattern in which the same matrix is repeatedly placed in units of a sub-matrix (i.e., in units of 8 columns) may be formed. In the second sub-rows 2nd sub-Rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the H-matrix H-mat, a pattern in which different sub-matrices are placed in units of a sub-matrix (i.e., in units of 8 columns) may be formed. Likewise, columns in the same sub-matrix of the second sub-rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the H-matrix H-mat may be identical to each other. 
     The H-matrix H-mat may be implemented referring to  FIG.  7 C , by combining the parity matrix PR and the sub-matrices SM 00  to SM 15  described with reference to  FIGS.  7 A and  7 B . The syndrome decoding circuit  112  may generate the error vector ERV having  128   b  by comparing each bit of the syndrome SYD and each column of the H-matrix H-mat of  FIG.  7 C . For example, the syndrome decoding circuit  112  may search for a column that is identical to the syndrome SYD from among the plurality of columns of the H-matrix H-mat, and may generate the error vector ERV based on a found result. A bit of a position corresponding to the column that is identical to the syndrome SYD from among the plurality of columns of the H-matrix H-mat may indicate an error detected from the read data RDT. 
     The case where all the bits of the syndrome SYD are not “0” (i.e., the syndrome SYD is non-zero) but a column that is identical to the syndrome SYD is absent from the plurality of columns of the H-matrix H-mat may indicate that an uncorrectable error is included in the read data RDT. 
     The H-matrix H-mat illustrated in  FIG.  7 C  is merely an example of the H-matrix for the single bit error correction (SEC), and may be varied. 
     In the case where the H-matrix H-mat is implemented referring to  FIG.  7 C , the simplified H-matrix sim-H-mat may be generated or determined referring to  FIG.  8   . 
     For example, referring to  FIG.  8   , the simplified H-matrix sim-H-mat may be generated based on information of some rows (e.g., the second sub-rows 2nd sub-Rows) of the H-matrix H-mat. 
     In detail, as described above, the H-matrix H-mat may be divided into the plurality of sub-matrices SM 00  to SM 15 . Each of the plurality of sub-matrices SM 00  to SM 15  may be of dimension 8×8. That is, the H-matrix H-mat may be divided in units of 8 columns (i.e., in units having  8   b ). In this case, in each of the plurality of sub-matrices SM 00  to SM 15 , the first sub-rows 1st sub-Rows (e.g., rows corresponding to S 0 , S 1 , and S 2 ) may constitute the same A 0 -th sub-matrix A 0 , and each column of the A 0 -th sub-matrix A 0  indicates all combinations capable of being expressed by three bits. In other words, this means that the first sub-rows 1st sub-Rows (e.g., rows corresponding to S 0 , S 1 , and S 2 ) of the H-matrix H-mat have a function of specifying an accurate position of an error through comparison with the bits S 0 , S 1 , and S 2  of the syndrome SYD, and are not used to determine whether an error occurs. 
     In contrast, the second sub-rows 2nd sub-Rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the H-matrix H-mat are composed of the B 00 -th to B 15 -th sub-matrices B 00  to B 15 . In this case, the B 00 -th to B 15 -th sub-matrices B 00  to B 15  have different bit columns. However, in the same sub-matrix, that is, in each of the B 00 -th to B 15 -th sub-matrices B 00  to B 15 , all the columns have the same value. In this case, the second sub-rows 2nd sub-Rows (e.g. rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the H-matrix H-mat may be expressed by 16 bit columns. In this case, the number of bit columns capable of being combined through 5 bits is 32. That is, in the second sub-rows 2nd sub-Rows (e.g., rows corresponding to S 3 , S 4 , S 5 , S 6 , and S 7 ) of the H-matrix H-mat, whether an error is detected may be determined by comparing each column with the bits S 3 , S 4 , S 5 , S 6 , and S 7  of the syndrome SYD. 
     According to an example embodiment, the simplified H-matrix sim-H-mat may be generated using information about some of the remaining rows other than rows necessary to specify an error position in the H-matrix H-mat, as described above. In this case, referring to  FIG.  8   , the simplified H-matrix sim-H-mat may include the B 00 -th to B 15 -th sub-matrices B 00  to B 15  and the parity matrix PR. The B 00 -th to B 15 -th sub-matrices B 00  to B 15  may have a structure in which bit columns such as [11000] T , [00111] T , [10100] T , [01011] T , [01100] T , [10011] T , [11100] T , [00011] T , [10011] T , [01101] T , [01010] T , [10101] T , [11010] T , [00101] T , [00110] T , and [11001] T  are repeated in units of 8 (i.e., in units having  8   b ). The parity matrix PR may be the identity matrix IM of dimension 8×8. 
     The simplified H-matrix sim-H-mat illustrated in  FIG.  8    may be generated from the H-matrix H-mat described with reference to  FIGS.  6  to  7 C , and whether an error is included in the read data RDT may be checked by comparing some bits S 3 , S 4 , S 5 , S 6 , and S 7  of the syndrome SYD with each column of the simplified H-matrix sim-H-mat. 
     The simplified H-matrix sim-H-mat illustrated in  FIG.  8    is merely an example, and may be varied. For example, the simplified H-matrix sim-H-mat illustrated in  FIG.  8    may be composed of a plurality of sub-matrices obtained by dividing the H-matrix H-mat in units having  8   b , but the H-matrix H-mat may be divided in units of  4   b , 16 bits, or N bits (N being a natural number) depending on an implementation of the H-matrix H-mat. In this case, the dimension of the simplified H-matrix may be variously changed or modified. 
     Some examples of components (e.g., the syndrome checking circuit  115   b , the parity error checking circuit  115   c , the fast data error checking circuit  115   e , and the DSF detecting circuit  115   d ) of a fast DSF generator configured to generate the decoding status flag DSF based on the simplified H-matrix sim-H-mat of  FIG.  8    will now be described with reference to  FIGS.  9  to  13   . However, a configuration of logical operation circuits associated with the components of the fast DSF generator may be variously changed or modified depending on a way to implement or depending on the H-matrix or the simplified H-matrix. 
     Also, for brevity of drawing and for convenience of description, in the following drawings, the syndrome SYD has a size having  8   b , and bits of the syndrome SYD are respectively marked by S 0 , S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , and S 7 , and bits (hereinafter referred to as “complementary bits”) that correspond to inverted versions of the bits S 0 , S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , and S 7  of the syndrome SYD are respectively marked by SO 0 B, S 1 B, S 2 B, S 3 B, S 4 B, S 5 B, S 6 B, and S 7 B. 
       FIG.  9    is a diagram illustrating an example of a syndrome checking circuit of a fast DSF generator of  FIG.  5 C . 
     The syndrome checking circuit  115   b  may check whether all the bits S 0  to S 7  of the syndrome SYD are “0” (i.e., are all-zero). For example, the syndrome checking circuit  115   b  may be configured to perform the AND operation on the bits S 0 B, S 1 B, S 2 B, S 3 B, S 4 B, S 5 B, S 6 B, and S 7 B complementary to the bits (e.g., S 0  to S 7 ) of the syndrome SYD. In this case, as AND gates with two input terminals are connected in a 3-stage structure, whether all the bits S 0  to S 7  of the syndrome SYD are “0” (i.e., are all-zero) may be checked. 
     When all the bits S 0  to S 7  of the syndrome SYD are “0”, a value of “1” may be output from the syndrome checking circuit  115   b  of  FIG.  9    as the syndrome information SYD_ 0 . When at least one (e.g., at least one of S 0  to S 7 ) of the syndrome SYD is “1”, a value of “0” may be output from the syndrome checking circuit  115   b  of  FIG.  9    as the syndrome information SYD_ 0 . 
     In other words, that the syndrome information SYD_ 0  output from the syndrome checking circuit  115   b  of  FIG.  9    has a value of “1” means that an error is absent from the read data RDT and the parity data PRT corresponding to the syndrome SYD; on the other hand, that the syndrome information SYD_ 0  output from the syndrome checking circuit  115   b  of  FIG.  9    has a value of “0” means that an error is present in the read data RDT and the parity data PRT corresponding to the syndrome SYD. 
       FIG.  10    is a diagram illustrating an example of a parity error checking circuit of a fast DSF generator of  FIG.  5 C . 
     Referring to  FIGS.  5 C and  10   , the parity error checking circuit  115   c  may detect whether an error is present in the parity data PRT, based on the syndrome SYD. 
     In an example embodiment, whether an error is present in the parity data PRT may be detected based on the parity data PRT of the simplified H-matrix sim-H-mat described with reference to  FIG.  8    or the parity data PRT described with reference to  FIG.  7 A . For example, the parity error checking circuit  115   c  may include a plurality of AND operation circuits AS 10  to AS 17 . The plurality of AND operation circuits AS 10  to AS 17  may be respectively connected with the bits S 0  to S 7  of the syndrome SYD based on the parity data PRT of the simplified H-matrix sim-H-mat described with reference to  FIG.  8    or the parity data PRT described with reference to  FIG.  7 A . 
     In detail, the 10th AND operation circuit AS 10  may be connected with bits of the syndrome SYD so as to correspond to the 0-th column (i.e., [1, 0, 0, 0, 0, 0, 0, 0] T ) of the parity matrix PR. That is, the 10th AND operation circuit AS 10  receives the bits S 0 , S 1 B, S 2 B, S 3 B, S 4 B, S 5 B, S 6 B, and S 7 B as inputs. When all the inputs are “1”, the 10th AND operation circuit AS 10  outputs “1”; if not (i.e., when at least one of the inputs is “0”), the 10th AND operation circuit AS 10  outputs “0”. The 11th AND operation circuit AS 11  may be connected with bits of the syndrome SYD so as to correspond to the 1st column (i.e., [0, 1, 0, 0, 0, 0, 0, 0] T ) of the parity matrix PR. That is, the 11th AND operation circuit AS 11  receives the bits S 0 B, S 1 , S 2 B, S 3 B, S 4 B, S 5 B, S 6 B, and S 7 B as inputs. When all the inputs are “1”, the 11th AND operation circuit AS 11  outputs “1”; if not (i.e., when at least one of the inputs is “0”), the 11th AND operation circuit AS 11  outputs “0”. Likewise, each of the 12th to 17th AND operation circuits AS 12  to AS 17  may be connected with bits of the syndrome SYD so as to correspond to each of the 2nd to 7th columns of the parity matrix PR. When all the inputs are “1”, each of the 12th to 17th AND operation circuits AS12 to AS17 outputs “1”; if not (i.e., when at least one of the inputs is “0”), each of the 12th to 17th AND operation circuits AS 12  to AS 17  outputs “0”. 
     That is, through the plurality of AND operation circuits AS 10  to AS 17 , the syndrome SYD may be compared with each column of the parity matrix PR, and whether a column of the same bits (or bit pattern) as the syndrome SYD exists may be determined. That a column having the same bit pattern as the syndrome SYD is present in the columns of the parity matrix PR means that an error is present in the parity data PRT. 
     The parity error checking circuit  115   c  may include a plurality of OR gates. The parity error checking circuit  115   c  may output the parity error information PRT_ERR by performing an OR operation on outputs of the plurality of AND operation circuits AS 10  to AS 17  through the plurality of OR gates. In an example embodiment, when at least one of the outputs of the plurality of AND operation circuits AS 10  to AS 17  is “1”, the parity error information PRT_ERR has a value of “1”; when all the outputs of the plurality of AND operation circuits AS 10  to AS 17  are “0”, the parity error information PRT_ERR has a value of “0”. 
     In an example embodiment, that the parity error information PRT ERR has a value of “1” indicates that an error is present in the parity data PRT corresponding to the syndrome SYD; that the parity error information PRT_ERR has a value of “0” indicates that an error is absent from the parity data PRT corresponding to the syndrome SYD. 
       FIG.  11    is a diagram illustrating an example of a fast data error checking circuit of a fast DSF generator of  FIG.  5 C . 
     Referring to  FIGS.  5 C and  11   , the fast data error checking circuit  115   e  may determine whether an error is present in data, based on the syndrome SYD. 
     In an example embodiment, whether an error is present in data may be detected based on the simplified H-matrix sim-H-mat described with reference to  FIG.  8   . 
     For example, the fast data error checking circuit  115   e  may include a plurality of AND operation circuits AS 20  to AS 2   f . The plurality of AND operation circuits AS 20  to AS 2   f  may be connected with some bits S 3  to S 7  of the syndrome SYD based on the simplified H-matrix sim-H-mat described with reference to  FIG.  8   . 
     In detail, the 20th AND operation circuit AS 20  may be connected with some bits S 3  to S 7  of the syndrome SYD so as to correspond to a column (i.e., [1, 1, 0, 0, 0] T ) of the B 00 -th sub-matrix B 00  of the simplified H-matrix sim-H-mat of  FIG.  8   . That is, the 20th AND operation circuit AS 20  receives the bits S 3 , S 4 , S 5 B, S 6 B, and S 7 B as inputs. When all the inputs are “1”, the 20th AND operation circuit AS 20  outputs “1”; if not (i.e., when at least one of the inputs is “0”), the 20th AND operation circuit AS 20  outputs “0”. 
     Also, the 21st AND operation circuit AS 21  may be connected with some bits S 3  to S 7  of the syndrome SYD so as to correspond to a column (i.e., [0, 0, 1, 1, 1] T ) of the B 01 -th sub-matrix B 01  of the simplified H-matrix sim-H-mat of  FIG.  8   . That is, the 21st AND operation circuit AS 21  receives the bits S 3 B, S 4 B, S 5 , S 6 , and S 7  as inputs. When all the inputs are “1”, the 21st AND operation circuit AS 21  outputs “1”; if not (i.e., when at least one of the inputs is “0”), the 21st AND operation circuit AS 21  outputs “0”. 
     Likewise, the 22nd to 2f-th AND operation circuits AS 22  to AS 2   f  may be connected with some bits S 3  to S 7  of the syndrome SYD so as to correspond to columns of the B 02 -th to B 15 -th sub-matrices B 02  to B 15  of the simplified H-matrix sim-H-mat of  FIG.  8   , respectively. Each of the 22nd to 2f-th AND operation circuits AS 22  to AS 2   f  outputs “1” when all the inputs are “1” and outputs “0” if not (when at least one of the inputs is “0”). 
     As some bits (e.g., S 3  to S 7 ) of the syndrome SYD are compared with each column of the simplified H-matrix sim-H-mat or each sub-matrix through the plurality of AND operation circuits AS 20  to AS 2   f , whether a column having the same bit pattern as some bits S 3  to S 7  of the syndrome SYD is present in the simplified H-matrix sim-H-mat may be determined. That a column having the same bit pattern as some bits S 3  to S 7  of the syndrome SYD is present in the simplified H-matrix sim-H-mat may mean that an error is present in the read data RDT. However, because only some bits of the syndrome SYD are compared through the simplified H-matrix sim-H-mat, a position of an error is not detected. 
     The fast data error checking circuit  115   e  may further include a plurality of OR gates. The fast data error checking circuit  115   e  may output the data error information DT_ERR by performing an OR operation on outputs of the plurality of AND operation circuits AS 20  to AS 2   f  through the plurality of OR gates. In an example embodiment, when at least one of the outputs of the plurality of AND operation circuits AS 20  to AS 2   f  is “1”, the data error information DT_ERR has a value of “1”; when all the outputs of the plurality of AND operation circuits AS 20  to AS 2   f  are “0”, the data error information DT_ERR has a value of “0”. In an example embodiment, that the data error information DT_ERR indicates “1” may indicate that an error is present in the read data RDT; that the data error information DT_ERR indicates “0” may indicate that an error is absent from the read data RDT. 
       FIG.  12    is a diagram illustrating an example of a DSF detecting circuit of a fast DSF generator of  FIG.  5 C . 
     Referring to  FIGS.  5 C and  12   , the DSF detecting circuit  115   d  may receive the syndrome information SYD_ 0  from the syndrome checking circuit  115   b , may receive the parity error information PRT_ERR from the parity error checking circuit  115   c , and may receive the data error information DT_ERR from the fast data error checking circuit  115   e.    
     The DSF detecting circuit  115   d  may include an OR gate, INV gates, and an AND gate. The DSF detecting circuit  115   d  may generate the decoding status flag DSF by performing an OR operation on the syndrome information SYD_ 0  and the parity error information PRT_ERR through the OR gate, the INV gates, and the AND gate, and by performing an AND operation on an inverted value of an output of the OR gate and an inverted value of the data error information DT_ERR. 
     That the decoding status flag DSF thus generated has a value of “1” indicates that the decoding status is the “UE”; that the decoding status flag DSF thus generated has a value of “0” indicates that the decoding status is the “CE” or “NE”. 
     For example, according to the DSF detecting circuit  115   d  illustrated in  FIG.  12   , when the syndrome information SYD_ 0  indicates “0”, the parity error information PRT_ERR indicates “0”, and the data error information DT_ERR indicates “0”, the decoding status flag DSF may have a value of “1”. In other words, in the case where that an error is present in the read data RDT or the parity data PRT is determined by the syndrome information SYD_ 0  of “0”, that an error is absent from the parity data PRT is determined by the parity error information PRT_ERR of “0”, and that an error is absent from data is determined by the data error information DT_ERR of “0”, because conditions for pieces of information do not coincide with each other, the error-corrected read data RDT_cor that are finally output may be determined as including an uncorrectable error (i.e., the decoding status may be determined to be the “UE”). In the remaining cases other than the above case, the decoding status flag DSF may have a value of “0”; in this case, the error-corrected read data RDT_cor may be determined to be normal data or data whose error is normally corrected (i.e., the decoding status may be determined to be the “CE” or “NE”). 
     As described above, according to the present example embodiment, the fast DSF generator  115  may generate the decoding status flag DSF using the syndrome SYD without the error vector ERV generated by the syndrome decoding circuit  112 . In this case, because the fast DSF generator  115  operates in parallel with the syndrome decoding circuit  112 , the whole latency of the ECC decoder ECC-DEC- 1  may be shortened. Also, because the fast data error checking circuit  115   e  of the fast DSF generator  115  compares some bits of the syndrome SYD with the simplified H-matrix sim-H-mat, the complexity of lines for respective bits of the syndrome SYD may be decreased. 
       FIG.  13    is a block diagram illustrating an example of a DSF detecting circuit of a fast DSF generator of  FIG.  5 C . 
     For convenience of description, thus, additional description associated with the components described with reference to  FIG.  12    will be omitted to avoid redundancy. 
     Referring to  FIGS.  5 C and  13   , a DSF detecting circuit  115   d - 1  may generate the decoding status flag DSF of  1   b  through a method similar to that described with reference to  FIG.  12   . In this case, a first decoding status flag DSF 1  is similar to the decoding status flag DSF described with reference to  FIG.  12   . That is, that the first decoding status flag DSF 1  indicates “1” means that the decoding status is the “UE”; that the first decoding status flag DSF 1  indicates “0” means that the decoding status is the “NE” or “CE”. 
     The DSF detecting circuit  115   d - 1  of  FIG.  13    may invert the syndrome information SYD _ 0  to generate a second decoding status flag DSF 2  of  1   b . That is, that the second decoding status flag DSF 2  indicates “1” means that an error is present in the read data RDT or the parity data PRT; that the second decoding status flag DSF 2  indicates “0” means that an error is absent from the read data RDT and the parity data PRT. In other words, that the second decoding status flag DSF 2  indicates “1” means that the decoding status is the “CE” or “UE”; that the second decoding status flag DSF 2  indicates “0” means that the decoding status is the “NE”. 
     The DSF detecting circuit  115   d - 1  of  FIG.  13    may combine the first and second decoding status flags DSF 1  and DSF 2  to output the decoding status flag DSF of  2   b . For example, in the decoding status flag DSF, a least significant bit LSB may correspond to the first decoding status flag DSF 1 , and a most significant bit MSB may correspond to the second decoding status flag DSF 2 . In this case, that the decoding status flag DSF of  2   b  are “00” may mean that the decoding status is the “NE”; that the decoding status flag DSF of  2   b  are “10” may mean that the decoding status is the “CE”; that the decoding status flag DSF of  2   b  are “11” may mean that the decoding status is the “UE”. 
     The above bit combination and bit values are only examples, and it will be understood that the above bit combination and bit values may be variously changed or modified depending on an implementation of the ECC decoder. 
     As described above, the DSF detecting circuit  115   d - 1  may notify the decoding result (e.g., UE, CE, and NE) to the memory controller  11  by generating the decoding status flag DSF of  2   b.    
       FIG.  14    is a diagram illustrating an example of a simplified H-matrix according to an example embodiment. 
     The simplified H-matrix sim-H-mat described with reference to  FIG.  8    is generated based on information of the second sub-rows 2nd sub-Rows of the H-matrix H-mat that is used by the syndrome decoding circuit  112 . In contrast, a simplified H-matrix sim-H-mat_UE for UE determination illustrated in  FIG.  14    may be generated based on information that is not included in the information of the second sub-rows 2nd sub-Rows of the H-matrix H-mat. 
     For example, as described with reference to  FIG.  8   , the B 00 -th to B 15 -th sub-matrices B 00  to B 15  included in the simplified H-matrix sim-H-mat may be implemented with columns of [11000] T , [00111] T , [10100] T , [01011] T , [01100] T , [10011] T , [11100] T , [00011] T , [10011] T , [01101] T , [01010] T , [10101] T , [11010] T , [00101] T , [00110] T , and [11001] T , and the B 00 -th to B 15 -th sub-matrices B 00  to B 15  may constitute the second sub-rows 2nd sub-Rows of the H-matrix H-mat. 
     In contrast, referring to  FIG.  14   , the simplified H-matrix sim-H-mat UE for UE determination may be implemented with sub-matrices such as [01001] T , [01110] T , [10001] T , [10110] T , [10000] T , [01000] T , [00100] T , [00010] T , [00001] T , [01111] T , [10111] T , [11011] T , [11101] T , [11110] T , and [11111] T . Sub-matrices of the simplified H-matrix sim-H-mat UE for UE determination may be implemented with columns different from those of sub-matrices (i.e., B 00  to B 15 ) included in the second sub-rows 2nd sub-Rows of the H-matrix H-mat. 
     In detail, columns included in the second sub-rows 2nd sub-Rows of the H-matrix H-mat described with reference to  FIGS.  7 A to  7 C  are divided by “16” (except for the parity matrix PR). In this case, because the number of combinations of bit columns through 5 bits is 32 (=2 5 ), the number of bit columns that do not correspond to the sub-matrices (i.e., B 00  to B 15 ) is 16 (=32−16). In this case, a column of [00000] T  corresponds to the case where the syndrome SYD is all-zero (i.e., the case where there is no error), and the column of [00000] T  is excluded from the simplified H-matrix sim-H-mat_UE for UE determination. Accordingly, the simplified H-matrix sim-H-mat_UE for UE determination may include sub-matrices and the parity matrix PR each composed of 10 columns. 
     In this case, that a column of the third to seventh bits S 3  to S 7  of the syndrome SYD coincides with at least one of the columns of the simplified H-matrix sim-H-mat_UE for UE determination means that the syndrome SYD does not coincide with all the columns of the H-matrix H-mat (i.e., that an error position is not determined). Accordingly, whether an error is present in data may be determined by comparing the column of the third to seventh bits S 3  to S 7  of the syndrome SYD with the columns of the simplified H-matrix sim-H-mat_UE for UE determination. 
     In another implementation, in some cases, whether an uncorrectable error is included in data, that is, the decoding result is the “UE”, may be determined by comparing the column of the third to seventh bits S 3  to S 7  of the syndrome SYD with the columns of the simplified H-matrix sim-H-mat_UE for UE determination. 
       FIG.  15    is a diagram illustrating an example of a fast data error checking circuit using a simplified H-matrix for UE determination of  FIG.  14   . 
     Referring to  FIG.  15   , a fast data error checking circuit  115   e - 2  may include a plurality of AND operation circuits AS 30  to AS 3   e . The plurality of AND operation circuits AS 30  to AS 3   e  may be connected with some bits S 3  to S 7  of the syndrome SYD based on the simplified H-matrix sim-H-mat_UE for UE determination described with reference to  FIG.  14   . 
     For example, the 30th AND operation circuit AS 30  may be connected with some bits S 3  to S 7  of the syndrome SYD so as to correspond to [01001] T  of the simplified H-matrix sim-H-mat_UE for UE determination of  FIG.  14   . That is, the 30th AND operation circuit AS 30  receives the bits S 3 B, S 4 , SSB, S 6 B, and S 7  as inputs. When all the inputs are “1”, the 30th AND operation circuit AS 30  outputs “1”; if not (when at least one of the inputs is “″ 0 ”), the 30th AND operation circuit AS 30  outputs “0”. 
     The remaining AND operation circuits AS 31  to AS 3   e  are connected with some bits S 3  to S 7  of the syndrome SYD so as to respectively correspond to the remaining columns of the simplified H-matrix sim-H-mat_UE for UE determination described with reference to  FIG.  14   . Each of the AND operation circuits AS 31  to AS 3   e  outputs “1” when all the inputs are “1” and outputs “0” if not (when at least one of the inputs is “0”). 
     The fast data error checking circuit  115   e - 2  may include a plurality of OR gates. The fast data error checking circuit  115   e - 2  may output data error information DT_ERR- 2  by performing an OR operation on outputs of the plurality of AND operation circuits AS 30  to AS 3   e  through the plurality of OR gates. In an example embodiment, when at least one of the outputs of the plurality of AND operation circuits AS 30  to AS 3   e  is “1”, the data error information DT_ERR- 2  has a value of “1”; if not (i.e., when all the outputs are “0”), the data error information DT_ERR- 2  has a value of “0”. 
     In an example embodiment, that the data error information DT_ERR- 2  indicates “1” may mean that an error is present in the read data RDT or an error is present in the parity data PRT. That the data error information DT_ERR- 2  indicates “0” may mean that an error is not present in the read data RDT or a correctable error exists. 
       FIG.  16    is a diagram illustrating an example of a DSF detecting circuit using data error information from a fast data error checking circuit of  FIG.  15   . 
     Referring to  FIGS.  14 ,  15 , and  16   , a DSF detecting circuit  115   d - 2  may include first and second INV gates and first and second AND gates. The DSF detecting circuit  115   d - 2  may invert the syndrome information SYD _ 0  through the first INV gate. The DSF detecting circuit  115   d - 2  may perform an AND operation on an output (i.e., inverted syndrome information) of the first INV gate and the data error information DT_ERR- 2  through the first AND gate. The DSF detecting circuit  115   d - 2  may invert the parity error information PRT_ERR through the second INV gate. The parity error information PRT_ERR may be calculated by the parity error checking circuit  115   c  described with reference to  FIG.  10   , and refers to information indicating whether a 1-bit error is present in the parity data PRT. 
     The DSF detecting circuit  115   d - 2  may generate the decoding status flag DSF by performing an AND operation on an output of the first AND gate and an output of the second INV gate, through the second AND gate. That the decoding status flag DSF has a value of “1” indicates that the decoding status is the “UE”; that the decoding status flag DSF has a value of “0” indicates that the decoding status is the “NE” or “CE”. 
     For example, the decoding status flag DSF having a value “1” corresponds to the case where the syndrome information SYD_ 0  indicates “0”, the data error information DT_ERR- 2  indicates “1”, and the parity error information PRT_ERR indicates “0”. This means the case where that an error is present in data and parity data is detected by the syndrome information SYD_ 0 , that an uncorrectable error is present in data or parity data is detected by the data error information DT_ERR- 2 , and that an error is absent from the parity matrix PR is detected by the parity error information PRT_ERR. In this case, the ECC decoding result may be the “UE”. In contrast, in the remaining cases other than the above case, the ECC decoding result may be the “NE” or “CE”. 
     In an example embodiment, in the case where that an error is present in data and parity data is detected by the syndrome information SYD_ 0 , that an uncorrectable error is present in data or parity data is detected by the data error information DT_ERR- 2 , and that an error is present from the parity matrix PR is detected by the parity error information PRT_ERR, the decoding status flag DSF may indicate “0” (i.e., NE or CE). In this case, an error included in the parity data PRT may be a 1-bit error. In the case where the number of bits of an error included in the parity data PRT exceeds “1” or the error exceeds the error correction capability of the ECC engine, the decoding status flag DSF may be determined to be “1” (i.e., UE). 
       FIG.  17    is a diagram illustrating an example of a DSF detecting circuit using data error information from a fast data error checking circuit of  FIG.  15   . 
     Referring to  FIGS.  15  and  17   , a DSF detecting circuit  115   d - 3  may generate the first decoding status flag DSF 1  through an operation (i.e., /SYD_ 0 *DT_ERR- 2 )*/PRT_ERR) similar to that described with reference to  FIG.  16   . The DSF detecting circuit  115   d - 3  may invert the syndrome information SYD_ 0  to generate the second decoding status flag DSF 2 . The DSF detecting circuit  115   d - 3  may combine the first and second decoding status flags DSF 1  and DSF 2  to generate the decoding status flag DSF of  2   b . The meanings of the first decoding status flag DSF 1  of  1   b , the second decoding status flag DSF 2  of  1   b , and each value of the decoding status flag DSF of  2   b  are similar to those described with reference to  FIG.  13   , and thus, additional description will be omitted to avoid redundancy. 
     As described above, the ECC decoder according to the present example embodiment may generate the decoding status flag DSF based on information that is not included in some columns of the H-matrix. The fast data error checking circuit  115   e - 2  described with reference to  FIGS.  14  and  15    may determine whether an error is present in the read data RDT, by comparing some bits of the syndrome SYD with each column of the simplified H-matrix sim-H-mat_UE for UE determination. In this case, because each column of the simplified H-matrix sim-H-mat_UE for UE determination is based on information not included in the H-matrix H-mat, that some bits of the syndrome SYD coincide with any one of the columns of the simplified H-matrix sim-H-mat_UE for UE determination (i.e., that the data error information DT_ERR- 2  has a value of “1”) may mean that an uncorrectable error is included in the read data RDT (i.e., that the decoding status is the “UE”), and that some bits of the syndrome SYD are different from all the columns of the simplified H-matrix sim-H-mat_UE for UE determination (i.e., that the data error information DT_ERR- 2  has a value of “0”) may mean that an error is not included in the read data RDT or a correctable error is included therein (i.e., that the decoding status is the “NE” or “CE”). That is, the data error information DT_ERR- 2  generated by the fast data error checking circuit  115   e - 2  described with reference to  FIGS.  14  and  15    may be used as the decoding status flag DSF without passing through the DSF detecting circuits  115   d - 2  or  115   d - 3  described with reference to  FIG.  16  or  17   . 
     The configuration of the above ECC decoder may be implemented based on a specific 
     H-matrix. Accordingly, the configuration of the ECC decoder (in particular, the decoding status flag generator) according to the present disclosure may be variously changed or modified depending on a structure and a characteristic of the H-matrix. 
       FIG.  18    is a flowchart for describing a method of generating a simplified H-matrix according to an example embodiment. 
     The flowchart of  FIG.  18    may be performed by the design tool for an ECC circuit in the process of manufacturing the memory device  100 . 
     Referring to  FIG.  18   , in operation S 210 , the H-matrix may be determined. For example, the H-matrix that is used in the ECC circuit  110  (in particular, the ECC decoder) may be determined depending on the reliability or performance of the memory device  100 . The H-matrix may be determined in various manners in consideration of various factors such as a way to implement the ECC circuit  110 , an error correction algorithm, or an error correction capability. 
     In operation S 220 , the simplified H-matrix sim-H-mat may be extracted based on a characteristic or partial information of the H-matrix. For example, it is assumed that the H-matrix is the H-matrix H-mat described with reference to  FIGS.  6 ,  7 A,  7 B, and  7 C . In this case, in the H-matrix, the first sub-rows (i.e., rows corresponding to S 0  to S 2 ) are implemented with the same sub-matrices repeated in units of 8 columns, and second sub-rows (i.e., rows corresponding to S 3  to S 7 ) are implemented with a plurality of sub-matrices each including the same columns. In this case, whether the syndrome SYD is matched with at least one of the columns of the H-matrix H-mat (i.e., whether an error is present in the read data RDT or the parity data PRT) may be determined by comparing only partial information (i.e., rows corresponding to the second sub-rows S 3  to S 7 ) with some bits S 3  to S 7  of the syndrome SYD instead of comparing the syndrome SYD with all the columns of the H-matrix H-mat. That is, whether an error is present in the read data RDT or the parity data PRT may be determined by comparing some bits of the syndrome SYD with only partial information of the H-matrix H-mat. 
     The above characteristic of the H-matrix H-mat may vary depending on an implementation of the H-matrix H-mat, and the H-matrix H-mat may not be divided by a specific row unit or a specific column unit. For example, in the simplified H-matrix sim-H-mat generated based on a part of the H-matrix H-mat, values corresponding to S 0 , S 1 , S 2 , S 3 , and S 4  of the syndrome SYD may be compared with respect to the 0-th column thereof, and values corresponding to S 0 , S 2 , S 4 , S 6 , and S 7  of the syndrome SYD may be compared with respect to the first column. In an example embodiment, in the case where a size of the read data RDT is  128   b , the H-matrix may be of dimension 8×128+8×8 (parity matrix). In this case, in the case where the H-matrix is divided in units having  8   b , the simplified H-matrix sim-H-mat may be of dimension (8-log 2 8)×(128/8). In another implementation, in the case where the H-matrix is divided in units having  16   b  (or 16 bits), the simplified H-matrix sim-H-mat may be of dimension (8-log 2 16)×(128/16). 
     The simplified H-matrix sim-H-mat may be generated based on partial information included in the H-matrix. In another implementation, the simplified H-matrix sim-H-mat may be generated based on information different from the partial information included in the H-matrix (i.e., partial information not included in the H-matrix). In this case, the simplified H-matrix sim-H-mat may be extracted or generated as described with reference to  FIGS.  14  and  15   , and whether the decoding status is the “UE” may be determined through the simplified H-matrix sim-H-mat. 
     The simplified H-matrix sim-H-mat may be generated by performing various logical operations (i.e., OR, AND, and XOR operations) on specific columns of the H-matrix. 
     In operation S 230 , the fast DSF generator may be configured based on the simplified H-matrix sim-H-mat. For example, as described with reference to  FIGS.  1  to  17   , the fast data error checking circuit may be implemented based on the simplified H-matrix sim-H-mat, and the fast DSF generator may generate the decoding status flag DSF based on an output of the fast data error checking circuit (i.e., data error information), an output of a syndrome checking circuit (i.e., syndrome information), and an output of a parity error checking circuit (i.e., parity error information). 
     In example embodiments, it will be understood that the H-matrix may be variously changed or modified depending on an implementation of the ECC circuit or the performance of the ECC circuit, and a way to implement the simplified H-matrix may also be variously changed or modified depending on the modification of the H-matrix. 
       FIG.  19    is a block diagram illustrating a memory system according to an example embodiment. 
     Referring to  FIG.  19   , a memory system  1000  may include a memory controller  1100 , an ECC circuit  1200 , and a memory device  1300 . 
     The memory controller  1100  may store data in the memory device  1300  or may read data stored in the memory device  1300 . 
     The ECC circuit  1200  may be placed on a data path between the memory controller  1100  and the memory device  1300 . The ECC circuit  1200  may be configured to correct an error of data that are exchanged between the memory controller  1100  and the memory device  1300 . The ECC circuit  1200  may generate the decoding status flag DSF based on the operation method described with reference to  FIGS.  1  to  18   . 
       FIG.  20    is a block diagram illustrating a memory system according to an example embodiment. 
     Referring to  FIG.  20   , a memory system  2000  may include a memory controller  2100  and a memory device  2200 . 
     The memory controller  2100  may include an ECC circuit  2110 . The ECC circuit  2110  may be configured to generate parity data for data to be stored in the memory device  2200  or to correct an error of data read from the memory device  2200  based on the read data and parity data read from the memory device  2200 . The ECC circuit  2110  may generate the decoding status flag DSF based on the method described with reference to  FIGS.  1  to  18   . 
       FIG.  21    is a block diagram illustrating a memory system according to an example embodiment. 
     Referring to  FIG.  21   , a memory system  3000  may include a memory controller  3100  and a memory device  3200 . 
     The memory controller  3100  may include a controller ECC circuit  3110 . The controller ECC circuit  3110  may generate first parity for write data to be stored in the memory device  3200  and may correct an error of read data based on the read data and the first parity received from the memory device  3200 . 
     The memory device  3200  may include a memory ECC circuit  3210 . The memory ECC circuit  3210  may generate second parity for write data and the first parity received from the memory controller  3100 , and may correct an error of read data and the first parity based on the read data, the first parity, and the second parity read from the memory device  3200 . 
     In an example embodiment, each of the controller ECC circuit  3110  and the memory ECC circuit  3210  may generate the decoding status flag DSF based on the method described with reference to  FIGS.  1  to  18   . 
     In the example embodiments described with reference to  FIGS.  1  to  18   , the description is given as the ECC circuit  110  is an on-die ECC (OD-ECC) circuit included in the memory device  100 , but, referring to  FIGS.  19  to  21   , an ECC circuit may be inside or outside a memory device or may be placed within a memory controller. 
       FIG.  22    is a diagram illustrating an example of a memory package according to an example embodiment. 
     Referring to  FIG.  22   , a memory package  4000  may include a plurality of memory dies  4110  to  4140  and a buffer die  4200 . Each of the plurality of memory dies  4110  to  4140  may be a DRAM device. The plurality of memory dies  4110  to  4140  and the buffer die  4200  may be implemented in a stacked structure, may be electrically connected with each other through TSV (through silicon via), and may communicate with each other. 
     The memory package  4000  may be provided as one semiconductor package through packaging by the following: package on package (PoP), ball grid arrays (BGAs), chip scale packages (CSPs), plastic leaded chip carrier (PLCC), plastic dual in-line package (PDIP), die in waffle pack, die in wafer form, chip on board (COB), ceramic dual in-line package (CERDIP), plastic metric quad flat pack (MQFP), small outline (SOIC), shrink small outline package (SSOP), thin small outline (TSOP), thin quad flatpack (TQFP), system in package (SIP), multi-chip package (MCP), wafer-level fabricated package (WFP), or wafer-level processed stack package (WSP). 
     The buffer die  4200  may communicate with an external host device (or a memory controller). The buffer die  4200  may be configured to temporarily store data to be written in the plurality of memory dies  4110  to  4140  or to temporarily store data read from the plurality of memory dies  4110  to  4140 . The buffer die  4200  may include an ECC circuit  4210 . The ECC circuit  4210  may generate parity for data to be stored in the memory dies  4110  to  4140  or may correct an error of data read from the memory dies  4110  to  4140 . The ECC circuit  4210  may be implemented based on the method described with reference to  FIGS.  1  to  18    or may generate the decoding status flag DSF. 
       FIG.  23    is a diagram illustrating an example of a memory package according to an example embodiment. 
     Referring to  FIG.  23   , a memory package  5000  may include a plurality of memory dies  5110  to  5140  and a host die  5200 . The plurality of memory dies  5110  to  5140  may be electrically connected with each other through micro bumps MCB, may have a stacked structure, and may be directly stacked on the host die  5200 . The host die  5200  may be a SoC, a CPU, or a GPU. In an example embodiment, each of the plurality of memory dies  5110  to  5140  or the host die  5200  may include the ECC circuit described with reference to  FIGS.  1  to  18   . 
       FIG.  24    is a block diagram illustrating a memory module  6000  to which a memory device according to the present disclosure is applied. 
     Referring to  FIG.  24   , the memory module  6000  may include a register clock driver (RCD)  6100 , a plurality of memory devices  6210  to  6290 , and a plurality of data buffers DB. 
     The RCD  6100  may receive the command/address CA and the clock signal CK from an external device (e.g., a host or a memory controller). In response to the received signals, the RCD  6100  may send the command/address CA to the plurality of memory devices  6210  to  6290  and may control the plurality of data buffers DB. 
     The plurality of memory devices  6210  to  6290  may be respectively connected with the plurality of data buffers DB through data lines MDQ. 
     In an example embodiment, each of the plurality of memory devices  6210  to  6290  may include the ECC circuit described with reference to  FIGS.  1  to  18   , and may generate the decoding status flag DSF based on the method described with reference to  FIGS.  1  to  18   . The plurality of data buffers DB may send and receive data to and from an external device (e.g., a host or a memory controller) through a plurality of data lines DQ. 
     The memory module  6000  illustrated in  FIG.  24    may have the form factor of a load-reduced dual in-line memory module (LRDIMM) or, e.g., the memory module  6000  may have the form factor of a registered DIMM (RDIMM) in which the plurality of data buffers DB are not included. 
     The memory module  6000  may further include an ECC circuit placed outside the plurality of memory devices  6210  to  6290 , and may be configured to generate the decoding status flag DSF based on the method described with reference to  FIGS.  1  to  18   . 
     In an example embodiment, at least one of the plurality of memory devices  6210  to  6290  may be configured to store parity data. The parity data may be provided from the external device (e.g., a host or a memory controller); in this case, the external device may include the ECC circuit described with reference to  FIGS.  1  to  18    or may generate the decoding status flag DSF based on the method described with reference to  FIGS.  1  to  18   . 
       FIG.  25    is a diagram of a system  7000  to which a storage device is applied, according to an example embodiment. 
     The system  7000  of  FIG.  25    may be a mobile system (such as a portable communication terminal (e.g., a mobile phone), a smartphone, a tablet personal computer (PC), a wearable device, a healthcare device, or an Internet of things (IoT) device) or, e.g., a PC, a laptop computer, a server, a media player, or an automotive device (e.g., a navigation device). 
     Referring to  FIG.  25   , the system  7000  may include a main processor  7100 , memories (e.g.,  7200   a  and  7200   b ), and storage devices (e.g.,  7300   a  and  7300   b ). In addition, the system  7000  may include at least one of an image capturing device  7410 , a user input device  7420 , a sensor  7430 , a communication device  7440 , a display  7450 , a speaker  7460 , a power supplying device  7470 , and a connecting interface  7480 . 
     The main processor  7100  may control all operations of the system  7000 , more specifically, operations of other components included in the system  7000 . The main processor  7100  may be implemented as a general-purpose processor, a dedicated processor, or an application processor. 
     The main processor  7100  may include at least one CPU core  7110  and further include a controller  7120  configured to control the memories  7200   a  and  7200   b  and/or the storage devices  7300   a  and  7300   b . In some example embodiments, the main processor  7100  may further include an accelerator  7130 , which is a dedicated circuit for a high-speed data operation, such as an artificial intelligence (AI) data operation. The accelerator  7130  may include a graphics processing unit (GPU), a neural processing unit (NPU) and/or a data processing unit (DPU) and be implemented as a chip that is physically separate from the other components of the main processor  7100 . 
     The memories  7200   a  and  7200   b  may be used as main memory devices of the system  7000 . Although each of the memories  7200   a  and  7200   b  may include a volatile memory, such as static random access memory (SRAM) and/or dynamic RAM (DRAM), each of the memories  7200   a  and  7200   b  may include non-volatile memory, such as a flash memory, phase-change RAM (PRAM) and/or resistive RAM (RRAM). The memories  7200   a  and  7200   b  may be implemented in the same package as the main processor  7100 . 
     The memories  7200   a  and  7200   b  may be the memory device including the ECC circuit described with reference to  FIGS.  1  to  18   , and generate the decoding status flag (DSF) based on the method described with reference to  FIGS.  1  to  18   . 
     The storage devices  7300   a  and  7300   b  may serve as non-volatile storage devices configured to store data regardless of whether power is supplied thereto, and have larger storage capacity than the memories  7200   a  and  7200   b . The storage devices  7300   a  and  7300   b  may respectively include storage controllers (STRG CTRL)  7310   a  and  7310   b  and NVM (Non-Volatile Memory)s  7320   a  and  7320   b  configured to store data via the control of the storage controllers  7310   a  and  7310   b . Although the NVMs  7320   a  and  7320   b  may include flash memories having a two-dimensional (2D) structure or a three-dimensional (3D) V-NAND structure, the NVMs  7320   a  and  7320   b  may include other types of NVMs, such as PRAM and/or RRAM. 
     The storage devices  7300   a  and  7300   b  may be physically separated from the main processor  7100  and included in the system  7000  or implemented in the same package as the main processor  7100 . In addition, the storage devices  7300   a  and  7300   b  may have types of solid-state devices (SSDs) or memory cards and be removably combined with other components of the system  700  through an interface, such as the connecting interface  7480  that will be described below. The storage devices  7300   a  and  7300   b  may be devices to which a standard protocol, such as a universal flash storage (UFS), an embedded multi-media card (eMMC), or a non-volatile memory express (NVMe), is applied. 
     The image capturing device  7410  may capture still images or moving images. The image capturing device  7410  may include a camera, a camcorder, and/or a webcam. 
     The user input device  7420  may receive various types of data input by a user of the system  7000  and include a touch pad, a keypad, a keyboard, a mouse, and/or a microphone. 
     The sensor  7430  may detect various types of physical quantities, which may be obtained from the outside of the system  7000 , and convert the detected physical quantities into electric signals. The sensor  7430  may include a temperature sensor, a pressure sensor, an illuminance sensor, a position sensor, an acceleration sensor, a biosensor, and/or a gyroscope sensor. 
     The communication device  7440  may transmit and receive signals between other devices outside the system  7000  according to various communication protocols. The communication device  7440  may include an antenna, a transceiver, and/or a modem. 
     The display  7450  and the speaker  7460  may serve as output devices configured to respectively output visual information and auditory information to the user of the system  7000 . 
     The power supplying device  7470  may appropriately convert power supplied from a battery (not shown) embedded in the system  7000  and/or an external power source, and supply the converted power to each of components of the system  7000 . 
     The connecting interface  7480  may provide connection between the system  7000  and an external device, which is connected to the system  7000  and capable of transmitting and receiving data to and from the system  7000 . The connecting interface  7480  may be implemented using various interface schemes, such as advanced technology attachment (ATA), serial ATA (SATA), external SATA (e-SATA), small computer small interface (SCSI), serial attached SCSI (SAS), peripheral component interconnection (PCI), PCI express (PCIe), NVMe, IEEE 1394, a universal serial bus (USB) interface, a secure digital (SD) card interface, a multi-media card (MMC) interface, an eMMC interface, a UFS interface, an embedded UFS (eUFS) interface, and a compact flash (CF) card interface. 
     By way of summation and review, a DRAM may be used as a system memory of a mobile device or a computer device. The DRAM device may include an error correction code (ECC) circuit for improving the reliability of data stored therein. The ECC circuit may correct an error of data stored in the DRAM device. When the error of data exceeds an error correction capability of the ECC circuit, the error is incapable of being normally corrected. In this case, a means for notifying a decoding result of the ECC circuit to an external device (e.g., a memory controller) is required. Thus, the ECC circuit may provide the memory controller with a decoding status flag DSF including information about the ECC decoding result. According to an example embodiment, the ECC circuit may generate the decoding status flag DSF using only a syndrome that is generated in the ECC decoding process. In this case, a time taken to generate the decoding status flag DSF may be shortened. 
     As described above, example embodiments may provide a memory device with improved reliability and improved performance and an error correction code decoding method of the memory device. 
     According to an example embodiment, an ECC circuit included in a memory device may generate a decoding status flag using only syndrome information. In this case, because the decoding status flag is generated together with a syndrome decoding operation, the whole latency of ECC decoding may be decreased. Accordingly, a memory device with improved performance and improved reliability and an error correction code decoding method of the memory device are provided. 
     Example embodiments have been disclosed herein, and although specific terms are employed, they are used and are to be interpreted in a generic and descriptive sense only and not for purpose of limitation. In some instances, as would be apparent to one of ordinary skill in the art as of the filing of the present application, features, characteristics, and/or elements described in connection with a particular embodiment may be used singly or in combination with features, characteristics, and/or elements described in connection with other embodiments unless otherwise specifically indicated. Accordingly, it will be understood by those of skill in the art that various changes in form and details may be made without departing from the spirit and scope of the present invention as set forth in the following claims.