A syndrome calculation circuit includes a matrix product calculation circuit. The matrix product calculation circuit is configured to generate syndrome bits in a composite field by calculating a matrix product of input data bits and a first arithmetic matrix. The first arithmetic matrix is a matrix product of a basis conversion matrix for converting a data string from a Galois field to the composite field and a second arithmetic matrix, which is at least a part of a parity check matrix.

A system comprises a forward error correction decoder comprising syndrome computation circuitry, key-equation solver circuitry, and search and evaluator circuitry. The syndrome computation circuitry may comprise a plurality of syndrome compute units connected in parallel. The syndrome computation circuitry may be dynamically configurable to vary a quantity of the syndrome compute units used for processing of a codeword based on conditions of a channel over which the codeword was received. The syndrome computation circuitry may be operable to use a first quantity of the syndrome compute units for processing of a first codeword received over the channel when the channel is characterized by a first bit error rate and a second quantity of the syndrome compute units for processing of a second codeword received over the channel when the channel is characterized by a second bit error rate.

Error correction is proposed in which a syndrome calculation is carried out in a code domain of a second code and an efficient error correction algorithm is carried out in a code domain of a first code.

A semiconductor device includes a syndrome generation circuit configured to generate a syndrome code based on data and an error correction code corresponding to the data, an error determination circuit configured to detect a 1-bit error in the data based on the syndrome code, and multi-bit error detection circuit configured to determine whether the data detected to have 1-bit error includes a multi-bit error by using an error address of the data detected to have 1-bit error and an error syndrome code of the data detected to have 1-bit error.

A memory system according to an embodiment includes a nonvolatile memory and a memory controller. The nonvolatile memory stores data encoded by using an error correcting code for correcting n-bit errors(n is an integer of 3 or more) or less. The memory controller reads a received word from the nonvolatile memory, calculates a syndrome by using the read received word, estimates the number of bit errors by using the syndrome. When the number of bit errors is 2 or 3, the memory controller calculates an inverse element of a value calculated based on the syndrome, executes, by using the inverse element, variable transformation on a variable of an error locator polynomial corresponding to the number of bit errors and calculation of a root of a transformed polynomial obtained by transforming the error locator polynomial according to the variable transformation, executes variable inverse transformation on the root of the transformed polynomial to obtain the root of the error locator polynomial, and corrects the error in the error location corresponding to the root of the error locator polynomial.

A system is provided to receive, by a controller, a first request to read a first page of data stored in a storage device which comprises a plurality of non-volatile memory units. The system accumulates, by a calculation module, a syndrome associated with the first page of data to obtain a syndrome weight. In response to determining that the syndrome weight is less than a predetermined threshold, the system writes, by the controller, the first page of data to a destination page of the storage device. In response to determining that the syndrome weight is greater than the predetermined threshold and that a current number of retries is less than a predetermined number: the system executes a retry process between the calculation module and a data flip engine of the controller to update the syndrome weight; and the system increments the current number of retries.

An FEC codec module is provided. Code elements, i.e., code words of forward error correction code, are added to the data code stream of each transmission link through the codec module, so that accurate error determination and automatic error correction may be realized at the receiving end. The interleaving process is performed on multi-link data to prevent the occurrence of continuous burst errors in the data link in a transmission process, and the error correction capability of FEC is utilized to improve the data transmission efficiency and anti-interference ability of the system.

A system is provided to receive, by a controller, a first request to read a first page of data stored in a storage device which comprises a plurality of non-volatile memory units. The system accumulates, by a calculation module, a syndrome associated with the first page of data to obtain a syndrome weight. In response to determining that the syndrome weight is less than a predetermined threshold, the system writes, by the controller, the first page of data to a destination page of the storage device. In response to determining that the syndrome weight is greater than the predetermined threshold and that a current number of retries is less than a predetermined number: the system executes a retry process between the calculation module and a data flip engine of the controller to update the syndrome weight; and the system increments the current number of retries.

Disclosed are devices, systems and methods for providing hardware implementations of a quasi-cyclic syndrome decoder. An example method of reducing the complexity of a decoder includes receiving a noisy codeword that is a based on a transmitted codeword generated from a quasi-cyclic linear code; computing a plurality of syndromes based on the noisy codeword; selecting a first syndrome from the plurality of syndromes; generating a memory cell address as a function of the first syndrome; reading, based on the memory cell address, a coset leader corresponding to the first syndrome; and determining, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.

Examples of this disclosure include a method of detecting or correcting one or more errors in data. The method comprises receiving data in a channel, wherein the data comprises one or more symbols, wherein each symbol corresponds to one of a ground state or to one of one or more energized states, and wherein the only transition possible for a symbol in the channel is one in which one of the one or more energized states transitions to the ground state, and wherein the data comprises at least one symbol that corresponds to the ground state. The method also comprises analyzing the one or more symbols to determine if a transition of one or more of the states has occurred and, if a transition of one or more of the states has occurred, detecting or correcting errors in the data.