A method for decoding a codeword transmitted over a channel demodulates data received over the channel to produce an initial estimate of belief messages for bits of the codeword and decodes the codeword using a belief propagation (BP) decoding that iteratively passes the belief messages between a set of variable nodes representing the bits of the codeword and a set of check nodes representing parity-check constraints on the bits of the codeword until a termination condition is met. The BP decoding selects a look-up table based on a probability of the belief messages and maps, using the look-up table, values of at least two incoming belief messages to values of at least one outgoing belief message that forms an incoming belief message in a subsequent iteration of the BP decoding.

An apparatus and method decode LDPC code. The apparatus includes a memory and a number of LDPC processing elements. The memory is configured to receive a LDPC codeword having a length equal to a lifting factor times a base LDPC code length, wherein the lifting factor is greater than one. The number of LDPC processing elements configured to decode the LDPC codeword, wherein each of the number of LDPC processing elements decode separate portions of the LDPC codeword.

A method of determining a near optimal forward error correction scheme for the transmission of audio data over a lossy packet switched network having preallocated estimated bandwidth, delay and packet losses, between at least a first and second communications devices, the method including the steps of: determining a first coding rate for the audio data; determining a peak redundancy coding rate for redundant versions of the audio data; determining an average redundancy coding rate over a period of time for redundant versions of the audio data; determining an objective function which maximizes a bitrate-perceptual audio quality mapping of the transmitted audio data including a playout function formulation; and optimising the objective function to produce a forward error correction scheme providing a high bitrate perceptual audio quality.

Methods, systems, and apparatus for producing CCZ states and T states. In one aspect, a method for transforming a CCZ state into three T states includes obtaining a first target qubit, a second target qubit and a third target qubit in a CCZ state; performing a X.sup.1/2 gate on the third target qubit; performing an X gate on the first target qubit and the second target qubit using the third target qubit as a control; performing a Z gate on the first target qubit and the second target qubit using the third qubit as a X axis control; performing a Z.sup.1/4 gate on the third target qubit; and performing a Z gate on the first target qubit and the second target qubit using the third qubit as a X axis control to obtain the three T states.

There is provided a method for generating a burst error correction code. The method comprises: setting a mother code; defining a syndrome set corresponding to each burst error pattern for at least two burst error patterns to be corrected based on the mother code; shortening a column of a PCM (parity check matrix) of the mother code so that the defined syndrome sets are relatively prime; and designing an error correction code for the each burst error pattern based on an optimal generator polynomial maximizing a length of the shortened code within a range of a length of a parity bit of the mother code or a syndrome vector included in the syndrome set that is relatively prime.

In some embodiments, a method of encoding a set of information bits to produce a codeword that encodes the set of information bits for reliable communication is provided. The set of information bits is received. The set of information bits are provided to a plurality of permutation layers separated by neural network processing layers. Each permutation layer accepts an input vector and generates a reordered output vector that is a reordering of the input vector. Each neural network processing layer accepts a vector of input values and generates a vector of output values based on a non-linear function of the vector of input values. The reordered output vector of a final permutation layer of the plurality of permutation layers is provided as the codeword. In some embodiments, a corresponding method of decoding a codeword to retrieve a set of information bits is provided.

A system including a first module, a second module and a third module. The first module is configured to generate a first parity check matrix. The second module is configured to append an appended matrix to the first parity check matrix to generate a resultant parity check matrix. The appended matrix includes additional elements. The third module is configured to receive user data and encode the user data based on the resultant parity check matrix.

Provided is an apparatus for designing a quantum code, which includes an analyzing unit for analyzing at least one quantum error generated in a quantum error channel as at least one binary error by using a standard form codeword stabilized quantum (CWS) code, a code generating unit for generating a binary error-correcting code which corrects the at least one binary error, a word operator generating unit for generating at least one word operator of the CWS code by using the at least one binary error-correcting code, and a codeword generating unit for generating at least one codeword including at least one entangled qubit (ebit) by using the at least one word operator.

An encoding method of generating an encoded sequence by performing encoding of a given encoding rate based on a predetermined parity check matrix. The predetermined matrix is either a first parity check matrix or a second parity check matrix. The first parity check matrix corresponds to a low density parity check (LDPC) convolutional code that uses a plurality of parity check polynomials, and the second parity check matrix is generated by performing at least one of row permutation and column permutation on the first parity check matrix. A parity check polynomial satisfying zero of the LDPC convolutional code is expressible by using a specific mathematical expression.

An encoding method of generating an encoded sequence by performing encoding of a given encoding rate based on a predetermined parity check matrix. The predetermined matrix is either a first parity check matrix or a second parity check matrix. The first parity check matrix corresponds to a low density parity check (LDPC) convolutional code that uses a plurality of parity check polynomials, and the second parity check matrix is generated by performing at least one of row permutation and column permutation on the first parity check matrix. A parity check polynomial satisfying zero of the LDPC convolutional code is expressible by using a specific mathematical expression.