Memory controllers and memory systems

An error correction code (ECC) engine comprising:

• an ECC decoder configured to correct Q symbol errors in a codeword read from a memory cell array, Q being a maximum natural number equal to or less than P/2, P being a natural number equal to or greater than four;
• wherein the ECC decoder is configured to:
• generate a syndrome including first through P-th syndrome symbols based on the read codeword by using a parity check matrix;
• and
• perform a first ECC decoding to correct a single symbol error in the read codeword based on the first syndrome symbol and a selected syndrome symbol corresponding to one of the second through P-th syndrome symbols and by using a ratio of the selected syndrome symbol to the first syndrome symbol.

A semiconductor memory device comprising:

• a memory ell array including a plurality of memory cells;
• and
• an error correction code (ECC) engine, the ECC engine including an ECC decoder configured to correct Q symbols errors in a codeword read from the memory cell array, Q being a maximum natural number equal to or less than P/2, P being a natural number equal to or greater than four;
• wherein the ECC decoder is configured to:
• generate a syndrome including first through P-th syndrome symbols based on the read codeword by using a parity check matrix;
• correct a single symbol error in the read codeword by performing a first ECC decoding to estimate an error magnitude and a position of a target symbol in which an error exists based on the first syndrome symbol and a selected syndrome symbol corresponding to one of the second through P-th syndrome symbols;
• correct an error of the target symbol based on the estimated position of the target symbol;
• and
• verify whether the error is corrected

An error correction code (ECC) engine comprising:

• an ECC decoder configured to correct Q symbols errors included a codeword read from a memory cell array, Q being a maximum natural number equal to or less than P/2, P being a natural number equal to or greater than four;
• and
• wherein the ECC decoder is configured to:
• generate a syndrome including first through P-th syndrome symbols based on the read codeword by using a parity check matrix;
• and
• correct a single symbol error in the read codeword by performing a first ECC decoding to estimate an error magnitude and a position of a target symbol in which an error exists based on the first syndrome symbol and a selected syndrome symbol corresponding to one of the second through P-th syndrome symbols;
• and
• correct multi symbol errors in the read codeword based on the first through P-th syndrome symbols by performing a second ECC decoding in parallel with the first ECC decoding.