A method for quasi-cyclic low-density parity-check (QC-LDPC) encoding and decoding of a data packet by a lifted matrix is provided, the method comprising: lifting the QC-LDPC code for maximal code length N.sub.max and maximal circulant size Z.sub.upper of the base matrix; generating a plurality of optimal values r.sub.i for a plurality of circulants Z.sub.1, Z.sub.2, . . . , Z.sub.upper based on the QC-LDPC code lifted for maximal length N.sub.max, 0r.sub.iZ.sub.upper1; saving the generated plurality of optimal values r.sub.i corresponding to the plurality of circulants Z.sub.1, Z.sub.2, . . . , Z.sub.upper and a matrix for the QC-LDPC code lifted for maximal length N.sub.max in the memory; receiving a current circulant Z.sub.current from the plurality of circulants Z.sub.1, Z.sub.2, . . . , Z.sub.upper; selecting a current optimal value r.sub.current from the plurality of optimal values r.sub.i stored in the memory corresponding to the current circulant Z.sub.current; and lifting the base matrix based on the current optimal value r.sub.current.

Techniques are described for encoding information data bits using a low-density parity-check matrix optimized for a Low-Density Parity-Check (LDPC) encoder. In an example, the parity-check matrix includes a first matrix and a second matrix. The second matrix is a square matrix, and is also a block diagonal matrix that includes a set of square submatrices located on the diagonal of the block diagonal matrix. An intermediate vector is generated based on the information data bits and the first matrix, and a parity vector of a codeword is generated based on the intermediate vector and the second matrix.

The method for shuffled decoding of LDPC codes includes calculating check-variable mutual information which is mutual information of a message propagating from a plurality of check nodes to a plurality of variable nodes by a check-variable mutual information calculating unit, calculating variable-check mutual information which is mutual information of a message propagating from the plurality of variable nodes to the plurality of check nodes connected to the plurality of variable nodes based on the check-variable mutual information by a variable-check mutual information calculating unit, and Calculating the entire mutual information which is a sum of variable-check mutual information for each of the plurality of variable nodes and determines an operation order of a variable node having the largest entire mutual information among the plurality of variable nodes to be next, by an operation order determining unit.

According to one embodiment, a memory system comprises an encoder that encodes by a graph code and a data holding unit that holds data to be used in encoding. A check matrix of the graph code includes first to sixth submatrices, and the encoder produces a first vector obtained by multiplying an information word and the first submatrix, produces a second vector obtained by multiplying the information word and the third submatrix, produces a third vector obtained by multiplying the first vector and the fifth submatrix inverted in sign, produces a fourth vector obtained by adding the third vector and the second vector, produces a first parity obtained by multiplying the fourth vector and the data, produces a fifth vector obtained by multiplying the first parity and the second submatrix inverted in sign, and produces a second parity obtained by adding the fifth vector and the first vector.

Techniques are described for encoding information data bits using a low-density parity-check matrix optimized for a Low-Density Parity-Check (LDPC) encoder. In an example, the parity-check matrix includes a first matrix and a second matrix. The second matrix is a square matrix, and is also a block diagonal matrix that includes a set of square submatrices located on the diagonal of the block diagonal matrix. An intermediate vector is generated based on the information data bits and the first matrix, and a parity vector of a codeword is generated based on the intermediate vector and the second matrix.

A method of operating a memory system includes receiving information data corresponding to a second program unit that is a part of a first program unit and a write request for the information data from a host; generating a codeword by performing error correction code (ECC) encoding on the received information data such that a partial parity bit corresponding to the information data among all parity bits of the codeword is updated; and providing a memory device with the generated codeword and a write command regarding the codeword.