A device includes a controller, and the controller includes a root detection circuit having multiple sets of multipliers. A method includes configuring the root detection circuit according to a degree of a polynomial. In response to detection of a root of multiple roots of the polynomial, a configuration of the root detection circuit is modified based on a polynomial degree reduction (PDR) scheme. Depending on the particular implementation, the device may be implemented in a data storage device, a communication system (e.g., a wireless communication device or a wired communication device), or another electronic device.

Provided is a BCH decoder in which a folded multiplier is equipped. The BCH decoder may include a key equation solver including a plurality of multipliers. The multiplier includes a plurality of calculation blocks configured to perform a calculation operation. Each of the calculation blocks repeatedly performs a calculation operation of a calculation stage for a plurality of calculation stages, outputs one output value on the basis of at least one input value in each calculation stage, and is connected to at least one another calculation block to transfer an output value of a current calculation stage as an input value of the at least one another calculation block in a next calculation stage.

Low complexity partial parallel architectures for performing a Fourier transform and an inverse Fourier transform over subfields of a finite field are described. For example, circuits to perform the Fourier transforms and the inverse Fourier transform as described herein may have architectures that have simplified multipliers and/or computational units as compared to traditional Fourier transform circuits and traditional inverse Fourier transform circuits that have partial parallel designs. In a particular embodiment, a method includes, in a data storage device including a controller and a non-volatile memory, the controller includes an inverse Fourier transform circuit having a first number of inputs coupled to multipliers, receiving elements of an input vector and providing the elements to the multipliers. The multipliers are configured to perform calculations associated with an inverse Fourier transform operation. The first number is less than a number of inverse Fourier transform results corresponding to the inverse Fourier transform operation.

A syndrome calculation circuit receives input data r(x) including data and a parity bit and having a code length n of (2.sup.m-1) bits at maximum which is represented by a Galois field GF(2.sup.m), and performs syndrome calculation so as to meet

s≡α.sup.i+α.sup.j

z≡(α.sup.i+β).sup.−1+β.sup.−1+(α.sup.j+β).sup.−1+β.sup.1   (A)

thereby calculating syndromes s and z. An error position polynomial coefficient calculation circuit calculates the coefficient of an error position polynomial to obtain s×z by multiplying s and z by one multiplier. After that, 2-bit error data positions i and j are specified. Errors at the error data positions i and j of the input data are corrected.

An apparatus for decoding is disclosed. The apparatus includes a memory and a processor coupled to the memory. The processor is configured to obtain a first codeword comprising one or more information bits and one or more parity bits, obtain a first parameter corresponding to a code rate of the first codeword, and decode the first codeword using a multi-rate decoder to generate a decoded codeword. The multi rate decoder performs a code reconstruction procedure on the first codeword to generate a reconstructed codeword, and decodes the reconstructed codeword. The processor is further configured to output the decoded codeword.

An operating method of a memory controller is provided. The operating method includes receiving a first read data and a second conversion information, the second conversion information including data obtained by converting a second read data based on a linear operation, and the first read data and the second read data including data read from same memory cells; converting the first read data based on the linear operation to generate a first conversion information; performing a logical operation on the first conversion information and the second conversion information to generate an operation information; performing an inverse operation of the linear operation on the operation information to generate a reliability information; and correcting an error of the first read data based on the first read data and the reliability information.

An error correction device according to the technical idea of the present disclosure includes a syndrome generation circuit configured to receive data and generate a plurality of syndromes for the data, a partial coefficient generation circuit configured to generate partial coefficient information on a part of a coefficient of an error location polynomial by using the data while the plurality of syndromes are generated, an error location determination circuit configured to determine the coefficient of the error location polynomial based on the plurality of syndromes and the partial coefficient information, and obtain a location of an error in the data by using the error location polynomial, and an error correction circuit configured to correct the error in the data according to the location of the error.

A read method of a nonvolatile memory device is provided. The method includes storing data sensed from selected memory cells of the nonvolatile memory device into a page buffer, performing an error decoding operation by performing error detection on the sensed data to detect and error, correcting the detected error if the error is detected, and overwriting the page buffer with the corrected data, and de-randomizing data stored in the page buffer by using a seed after the error decoding operation has completed.

The present invention is directed to data communication systems and methods thereof According to various embodiments, the present invention provides a communication with a reconfigurable forward-error-correction (FEC) module. The FEC module processes data received from two or more communication lanes, and depending on the mode of operation, the FEC module can combine data from the two or more communication lanes and perform error correction on the combined data, or the FEC module can processes data from the two communications lanes separately and perform error correction independently for the each of the data communication lanes. There are other embodiments as well.

Disclosed is an erasure-based Reed-Solomon code soft-decision decoding method and device, capable of reducing a decoding time while minimizing the effect on error correction performance. The Reed-Solomon code soft-decision decoding device includes an erasure control circuit configured to determine whether a number of errors in a codeword is odd or even, and to provide a key equation solver circuit with a first erasure pattern or a second erasure pattern according to a result of the determining when a decoding failure is detected by a decoding error detection circuit, the first erasure pattern being provided when the number of errors is odd, the second erasure pattern being provided when the number of errors is even.