Decoding and encoding methods, systems, and devices for wireless communication are described. One method may include receiving a codeword over a wireless channel, the codeword being encoded using a polar code, identifying a set of repeated bit locations in the received codeword, and identifying a set of bit locations of the polar code used for information bits for the encoding. The set of bit locations may be determined based at least in part on recursively partitioning bit-channels of the polar code for each stage of polarization and assigning portions of a number of the information bits to bit-channel partitions of each stage of polarization based on a mutual information transfer function of respective aggregate capacities of the bit-channel partitions. The method may also include decoding the received codeword according to the polar code to obtain an information bit vector at the set of bit locations, and other aspects and features.

A polar-code based encoder is used to perform a transfer of useful data to a polar-code based decoder via a Binary Discrete-input Memory-less Channel. The Divide and Conquer structure consists of a multiplexer having useful data bits and a set of frozen bits as inputs followed by a polarization block of size N=2.sup.L, wherein the polarization block of size N comprises a set of front kernels followed by a shuffler and two complementary polarization sub-blocks of size N/2 with a similar structure as the polarization block of size N but with half its size. A dynamically configurable interleaver is present between the shuffler and one and/or the other of the complementary polarization sub-blocks at each recursion of the Divide and Conquer structure. The configuration of the dynamically configurable interleavers is dynamically modified according to changes detected in the Binary Discrete-input Memory-less Channel.

Disclosed herein are example embodiments of protocols to distill magic states for T-gates. Particular examples have low space overhead and use an asymptotically optimal number of input magic states to achieve a given target error. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T-gate depth of the circuit scale linearly in the logarithm of the target error. Unlike other distillation protocols, examples of the disclosed protocol achieve this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. Embodiments of the protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. Embodiments of the protocol rely on the construction of weakly self-dual Calderbank-Shor-Steane codes (CSS codes) with many logical qubits and large distance, allowing one to implement control-Swaps on multiple qubits. This code is referred to herein as the inner code. The control-Swaps are then used to measure properties of the magic state and detect errors, using another code that is referred to as the outer code. Alternatively, one can use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. Several specific small examples of this protocol are disclosed herein.

Methods, systems, and apparatus for producing CCZ states and T states. In one aspect, a method for distilling a CCZ state includes preparing multiple target qubits, ancilla qubits and stabilizer qubits in a zero state, performing an X gate for each stabilizer qubit on multiple ancilla qubits or multiple ancilla qubits and one of the target qubits using the stabilizer qubit as a control, measuring the stabilizer qubits, performing, on each of the ancilla qubits, a Z.sup.1/4 gate and a Hadamard gate, measuring each of the ancilla qubits, performing, conditioned on each measured ancilla qubit state, a NOT operation on a selected stabilizer qubit, or a NOT operation on the selected stabilizer qubit and a Z gate on one or more respective target qubits, performing, on each target qubit and conditioned on a measured state of a respective stabilizer qubit, a Z gate on the target qubit, and performing an X gate on each of the target qubits.

An encoding method, a decoding method, an electronic device and a storage medium are disclosed. The encoding method includes: acquiring stored data in a storage system, and acquiring nodes corresponding to the stored data to obtain a number of the nodes; dividing the acquired stored data into a sequence of information vectors, and generating an information matrix according to the number of the nodes and a number of the sequence of information vectors; and calculating an encoded block according to each information vector and the information matrix to obtain a sequence of encoded blocks.

The disclosure is directed to a method of transmitting data encoded in Polar Code and an electronic device using the same method. The data transmission method would include not limited to: determining a bit sequence for transmission; dividing the bit sequence into a first bit sequence and a second bit sequence; generating a first transmission bit stream encoded in Polar Code, ordered from most reliable to least reliable, and including the first bit sequence; generating a second transmission bit stream encoded in Polar Code, ordered from most reliable to least reliable, and including the second bit sequence, wherein the first transmission bit stream has the same length as the second transmission bit stream; transmitting the first transmission bit stream; and transmitting the second transmission bit stream.

A processor of an apparatus establishes a wireless communication link with at least one other apparatus via a transceiver of the apparatus. The processor wirelessly communicates with the other apparatus via the wireless communication link by: selecting a first shift-coefficient table from a plurality of shift-coefficient tables; generating a QC-LDPC code using a base matrix and at least a portion of the first shift-coefficient table; selecting a codebook from a plurality of codebooks embedded in the QC-LDPC code; storing the selected codebook in a memory associated with the processor; encoding data using the selected codebook to generate a plurality of modulation symbols of the data; and controlling the transceiver to multiplex, convert, filter, amplify and radiate the modulation symbols as electromagnetic waves through one or more antennas of the apparatus to transmit the modulation symbols of the data to the other apparatus via the wireless communication link.

An error correction and fault, tolerance method and system for an array of disks is presented. The array comprises k+5 disks, where k disks store user data and 5 disks store computed parity. The present invention further comprises a method and a system for reconstituting the original content of each of the k+5 disks, when up to disks have been lost, wherein the number of disks at unknown locations is E and the number of disks wherein the location of the disks is known is Z. All combinations of faulty disks wherein Z+2E4 are reconstituted. Some combinations of faulty disks wherein Z+2E5 are either reconstituted, or errors are limited to a small list.

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.

The embodiments herein provide a system and method for generating a catalog of graphs that acts as a source for creating error correcting codes. A D3 chord index notation is used to describe the graphs. A list of (3, g) Hamiltonian graphs for even girth g is created to satisfy the condition 6?g?16. Each of the lists is infinite and is used for creating LDPC codes of high quality.