A data storage system and method are provided for storing data in non-volatile memory devices. Binary data is received for storage in a non-volatile memory device. The binary data is converted into non-binary data comprising base-X values, where X is an integer greater than two. The non-binary data is encoded to generate a codeword and the codeword is written to a wordline of the non-volatile memory device.

An error correction device includes: a plurality of variable node units each configured to: receive a hard decision bit and a channel reliability value having a first bit-precision; and perform an iteration of a decoding operation on the hard decision bit based on the channel reliability value; a plurality of check node units each configured to: receive one or more reference reliability values having a second bit-precision from one or more variable node units coupled thereto among the plurality of variable node units during the iteration; and transmit, based on the one or more reference reliability values, one or more check reliability values having the second bit-precision to the one or more variable node units coupled thereto, wherein, during the iteration, each of the plurality of variable node units further: receives one or more first check reliability values from one or more check node units coupled thereto among the plurality of check node units; and updates the hard decision bit with reference to the channel reliability value and the one or more first check reliability values by upsizing the first bit-precision of the channel reliability value and the second bit-precision of the one or more first check reliability values.

An error correction device includes a bit reliability value determination circuit configured to determine bit reliability values respectively corresponding to hard decision bits, based on soft decision bit sets respectively corresponding to the hard decision bits; and a decoder including a variable node configured to receive and store the hard decision bits and the bit reliability values, and perform a decoding operation for the hard decision bits by restoring reliability values from the bit reliability values, wherein the reliability values respectively correspond to elements except a decision symbol configured by the hard decision bits, in a Galois field (GF) defined in the variable node.

Provided herein may be an error correction circuit. An error correction circuit for performing error correction decoding based on an iterative decoding scheme using a NB-LDPC code may include a symbol configuration circuit for configuring an initial symbol to be assigned as a variable node value to a variable node, a reliability value initialization circuit for initializing first reliability values of candidate symbols corresponding to the variable node based on the initial symbol assigned to the variable node, and a symbol correction circuit updating the first reliability values of the candidate symbols based on communications received from a check node coupled to the variable node, the candidate symbols having updated first reliability values, respectively, and adjusting the variable node value to one of the candidate symbols based on a comparison with the updated first reliability value of one of the candidate symbols with a first threshold value.

At least a method and an apparatus are presented to decode a signal encoded using an error correcting code. For example, a decoder comprising a check node processing unit is presented. The check node processing unit is configured to receive at least three input messages and to generate at least one output message. A syndrome calculator is configured to determine a set of syndromes from the at least three input messages using at least two elementary check node processors. A decorrelation unit is configured to determine, in association with at least an output message, a set of candidate components from the set of syndromes. A selection unit is configured to determine at least an output message by selecting components comprising distinct symbols from the set of candidate components associated with the at least an output message.

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.

Some embodiments are directed to a method for controlling a check node of a NB-LDPC decoder. The check node receives d.sub.c input lists U.sub.i and delivers and delivers d.sub.c output lists V.sub.i, with i[1 . . . d.sub.c]. Each input list and output list includes n.sub.m elements and each element of the input or output lists includes a reliability value associated to a symbol of a Galois Field GF(q) with q>n.sub.m. The input elements and output elements are sorted according to the reliability values in the lists. The method is a syndrome-based method. The syndromes are sums of d.sub.c elements of input lists U.sub.i. The method includes a step of syndrome calculation, a step of decorrelation and a step for generating the output list.

An error correction device includes: a plurality of variable node units each configured to: receive a hard decision bit and a channel reliability value having a first bit-precision; and perform an iteration of a decoding operation on the hard decision bit based on the channel reliability value; a plurality of check node units each configured to: receive one or more reference reliability values having a second bit-precision from one or more variable node units coupled thereto among the plurality of variable node units during the iteration; and transmit, based on the one or more reference reliability values, one or more check reliability values having the second bit-precision to the one or more variable node units coupled thereto, wherein, during the iteration, each of the plurality of variable node units further: receives one or more first check reliability values from one or more check node units coupled thereto among the plurality of check node units; and updates the hard decision bit with reference to the channel reliability value and the one or more first check reliability values by upsizing the first bit-precision of the channel reliability value and the second bit-precision of the one or more first check reliability values.

Methods and apparatus for code-based asymmetric cryptosystem using Quasi-Cyclic Moderate-Density Parity-Check (QC-MDPC) error correcting codes. Specifically, the method and apparatus generalizes the framework of (QC-MDPC) Code-Based (CB) cryptography from the binary domain (Galois Field of two elements) to an arbitrary size of Galois Field and provides an apparatus for implementing the cryptosystem with a simplified computational complexity of key generation, encryption, and decryption components of the cryptosystems and reduced sizes of the public and private security keys.

A data storage system and method are provided for storing data in non-volatile memory devices. Binary data is received for storage in a non-volatile memory device. The binary data is converted into non-binary data comprising base-X values, where X is an integer greater than two. The non-binary data is encoded to generate a codeword and the codeword is written to a wordline of the non-volatile memory device.