The present invention is directed to communication systems and methods. In a specific embodiment, the present invention provides a receiver that includes an error correction module. A syndrome value, calculated based on received signals, may be used to enable the error correction module. The error correction module includes an error generator, a Nyquist error estimator, and a decoder. The decoder uses error estimation generated by the Nyquist error estimator to correct the decoded data. There are other embodiments as well.

Provided herein may be an electronic device using an artificial neural network. The electronic device may include a training data generator configured to determine an input vector corresponding to a trapping set, detected during error correction decoding corresponding to a codeword, and a target vector corresponding to the input vector, and a training component configured to train an artificial neural network based on supervised learning by inputting the input vector to an input layer of the artificial neural network and by inputting the target vector to an output layer of the artificial neural network.

A memory system includes a nonvolatile semiconductor memory, and a controller configured to maintain a plurality of log likelihood ratio (LLR) tables for correcting data read from the nonvolatile semiconductor memory, determine an order in which the LLR tables are referred to, based on a physical location of a target unit storage region of a read operation, and carry out correcting of data read from the target unit storage region, using one of the LLR tables selected according to the determined order.

A method for determining the n best positions of frozen bits in a channel decoder for a noisy communication channel. A decoding method and decoding processing unit for implementing the channel having frozen bits at the n worst positions. A method and system that iteratively, for each bit i from the n bits, determines a probability vector for the bit i by traversing a logical graph using contraction identities simplified to specific values, indexes the specific values from the contraction identities newly computed during the determination of the probability vector for subsequent reference during a following iteration based on corresponding contraction identities, fixes the bit i from the probability vector and moving to bit i+1 until all n bits are fixed.

A method of decoding polar codes based on belief propagation includes conventional belief propagation to decode the polar codes first; when a number of iterations exceeds a predefined upper limit and a cyclic redundancy check fails, the method selects log-likelihood ratio vectors of a plurality of R or L messages from a plurality of log-likelihood ratio vectors generated in each of the iterations and generates another set of log-likelihood ratio vectors (referred to as candidate vector group) to be used as initial values of the R or L messages for a subsequent belief propagation to perform belief propagation decoding iterations and cyclic redundancy check again. Whenever a decoding result passes the cyclic redundancy check, the method exits; otherwise, the method iterates the above procedure until a maximum number of candidate vector groups has been reached.

The present disclosure relates to a device for generating a polar code x.sub.N of length N and dimension K on the basis of a transformation matrix G.sub.N of size N×N, wherein the transformation matrix G.sub.N is based on a first matrix G.sub.N, of size N.sub.r×N, and on a second matrix G.sub.N.sub.d of size N.sub.d×N.sub.d, wherein N=N.sub.r.Math.N.sub.d, and wherein the polar code x.sub.N is given by x.sub.N=u.sub.N.Math.G.sub.N, wherein u.sub.N=(u.sub.0, . . . u.sub.N-1) is a vector of size N, an element u.sub.i, i=0, . . . N−1, of the vector corresponding to an information bit if i∈I, I being a set of K information bit indices, and u.sub.i=0, if i∈F, F being a set of N−K frozen bit indices.

A receiving unit (2020) generates a received frame from a modulated optical signal. The modulated optical signal is generated such that a transmission symbol is generated by mapping an encoded bit string obtained by encoding a transmission bit string to an m-dimensional symbol space, a transmission frame is generated by mapping the transmission symbol to an n-dimensional frame space (n<m), and an optical carrier wave is modulated by using the transmission frame. A converting unit (2040) generates candidate vectors (m-dimensional vectors belonging to a partial symbol space within the symbol space) by using a received frame. A first computing unit (2070) computes a probability of that the transmission symbol belonging to the partial symbol space is transmitted for each partial symbol space. A second computing unit (2080) computes a log-likelihood ratio of each bit of the encoded bit string by using the probability.

Disclosed are devices, systems and methods for precoding and decoding polar codes using local feedback are described. One example method for improving an error correction capability of a decoder includes receiving a noisy codeword vector of length n, the codeword having been generated based on a concatenation of a convolutional encoding operation and a polar encoding operation and provided to a communication channel prior to reception by the decoder, performing a successive-cancellation decoding operation on the noisy codeword vector to generate a plurality of polar decoded symbols (n), generating a plurality of information symbols (k) by performing a convolutional decoding operation on the plurality of polar decoded symbols, wherein k/n is a rate of the concatenation of the convolutional encoding operation and the polar encoding operation, and performing a bidirectional communication between the successive-cancellation decoding operation and the convolutional decoding operation.

Provided is a method for decoding a staircase code. The method includes following steps: soft information updating is performed on S initial encoding blocks in a staircase code to obtain a first information block, and last S−T encoding blocks in the first information block and T newly-added encoding blocks are updated to obtain a second information block; decoding is performed on first T encoding blocks in the first information block and first S−T encoding blocks in the second information block to obtain a third information block; and following operations are repeatedly performed: S−T information blocks are selected, the soft information updating is performed to obtain S updated information blocks, and the S updated information blocks are used as a new second information block; and decoding is performed to obtain a new third information block, and information of first T blocks is outputted as the output of the decoder.

A decoder decodes a soft information input vector represented by an input vector that is binary and that is constructed from the soft information input vector. The decoder stores even parity error vectors that are binary and odd parity error vectors that are binary for L least reliable bits (LRBs) of the input vector. The decoder computes a parity check of the input vector, and selects as error vectors either the even parity error vectors or the odd parity error vectors based at least in part on the parity check. The decoder hard decodes test vectors, representing respective sums of the input vector and respective ones of the error vectors, based on the L LRBs, to produce codewords that are binary for corresponding ones of the test vectors, and metrics associated with the codewords. The decoder updates the soft information input vector based on the codewords and the metrics.