Abstract: A digital signal processor includes K first electronic circuits. The first inputs receive K groups of G successive coefficients of a polynomial. The polynomial are of degree N with N+1 coefficients, where K is a sub-multiple of N+1 greater than or equal to two and G is equal to (N+1)/K. The first electronic circuits are configured to simultaneously implement K respective Horner methods and deliver K output results. A second electronic circuit includes a first input configured to successively receive the output results of the first electronic circuits starting with the output result of the first electronic circuit having processed the highest rank coefficient of the coefficients. A second input is configured to receive a variable X and the second electronic circuit is configured to implement a Horner method and deliver a value of the polynomial for the variable X on the output of the second electronic circuit.

Abstract: A Reed-Solomon decoder circuit includes: a syndrome calculator circuit to compute syndrome values for a first codeword and a second codeword sequentially supplied to the syndrome calculator circuit, where last symbols of the first codeword overlap with first symbols of the second codeword during an overlap clock cycle between: a first plurality of non-overlap clock cycles during which the first codeword is supplied to the syndrome calculator circuit; and a second plurality of non-overlap clock cycles during which the second codeword is supplied to the syndrome calculator circuit; an error locator and error evaluator polynomial calculator circuit; an error location and error value calculator circuit; an error counter; and an error corrector circuit to correct the errors in the first codeword and the second codeword based on error counts and the error magnitudes computed by an error evaluator circuit.

Abstract: A soft-decision decoding computes a first syndrome polynomial in accordance with a received word, computes a second syndrome polynomial by multiplying the first syndrome polynomial by a locator polynomial based on locations of erasures within the received word, finds a basis and private solution to an affine space of polynomials that solve key equations based on the second syndrome polynomial, determines a weak set of a locations of symbols in the received word with confidence below a certain confidence level, computes a matrix from the basis, the private solution and the weak set, determines sub-matrices in the matrix whose rank is equal to a rank of the matrix, determines error locator polynomial (ELP) candidates from the sub-matrices, the basis, and the private solution, and corrects the received word using a selected one of the ELP candidates.

Abstract: A function is decomposed into a plurality of function shares. The function returns a Boolean result based on whether an input y satisfies a query on a data set. The function shares hide the function from non-collaborating entities that separately execute the function shares. Each of the functions shares are sent to one of a plurality of servers having a same data set. The function shares are executed on the data set at the servers to obtain a respective plurality of shares. A conditional disclosure of secrets operation is simulated on the shares and the input y. The conditional disclosure of secrets operation uses a secret known to at least one of the servers, and further uses a source of randomness shared between the servers. A Boolean value corresponding to the Boolean result is returned based on the conditional disclosure of secrets operation returning the secret.

Abstract: Described herein are embodiments related to first-pass continuous read level calibration (cRLC) operations on memory cells of memory systems. A processing device determines that a first programming pass of a programming operation has been performed on a memory cell of a memory component. The processing device then adjusts a read level threshold of the memory cell to be centered between a first programming distribution and a second programming distribution before the second programming pass of the programming operation is performed on the memory cell.

Abstract: An operating method of a memory device includes storing position information regarding a codeword including an erasure and erasure information including position information regarding the erasure in a memory region, loading the position information regarding the codeword to a row decoder and a column decoder, determining whether a read address corresponding to a read instruction is identical to the position information regarding the codeword including the erasure, in response to the read instruction from a host, transmitting the position information of the erasure to an error correction code (ECC) decoder, when the read address is identical to the position information regarding the codeword including the erasure, and correcting, by the ECC decoder, an error in a codeword received from a memory cell array using the position information regarding the erasure.

Abstract: An apparatus for wireless communication is provided. The apparatus may be a receiver device that includes an error correction decoder, such as a low-density parity check (LDPC) decoder. The apparatus may achieve power savings and/or operation cycle savings by disabling the error correction decoder in scenarios where bits of a codeword in a signal transmission are received without errors. The apparatus obtains a first set of bits of a codeword, wherein the codeword includes the first set of bits and a second set of bits, and wherein the second set of bits is punctured. The apparatus recovers the second set of bits based on at least the first set of bits and determines whether to operate an error correction decoder based on a result of an error detection operation performed on the codeword using the first set of bits and the second set of bits.

Abstract: A processing circuit is configured to: construct a first locator polynomial for a Reed-Solomon codeword to identify locations of erasures in the Reed-Solomon codeword; determine a first syndrome of the Reed-Solomon codeword; calculate a first error evaluator polynomial from the first syndrome and the first locator polynomial; and perform error detection based on the first error evaluator polynomial to determine presence of errors in the Reed-Solomon codeword. When presence of errors in the Reed-Solomon codeword is not detected in the error detection, the processing circuit bypasses updating the first locator polynomial and proceeds to completing decoding of the Reed-Solomon codeword, but when presence of errors in the Reed-Solomon codeword is detected in the error detection, the system first updates the first locator polynomial to a second locator polynomial in a process with reduced complexity compared to the common one, before completing decoding of the Reed-Solomon codeword.

Abstract: An intermediate representation construction method is provided. The method includes: obtaining a first intermediate representation (IR), where the first IR includes a computing statement, the computing statement includes a tensor and an operator, an operation represented by the operator is performed by a computing unit, the computing unit is configured to perform an operation on data that is migrated through a first storage location and a second storage location sequentially, and the tensor is data that is used when the operation represented by the operator is performed; and generating a second IR based on the computing statement, where the second IR includes first data migration information and data segmentation information.

Abstract: A computer-implemented technique for correcting a data packet having a payload and cyclic redundancy check information provides a list of possible packet errors by algorithmic operations of forcing and cancelling bits at certain positions using a generator polynomial, while maintaining the equivalence relationship with the original syndrome, performed explicitly using arithmetic operations or implicitly using a table representative of such arithmetic operations. Correction can then be implemented using an error chosen from the list.

Abstract: Embodiments herein describe a FEC codec for generating a check byte for a message. The FEC codec includes a port encoder having a storage unit, a Galois field multiplier, and a sum unit. The storage unit stores a first staged result, which is accumulated based on previous sets of input bytes of the message for all clock cycles from a first clock cycle to a clock cycle immediately prior to the current clock cycle. The Galois field multiplier performs a Galois field multiplication of the first staged result and a power of the alpha to generate a Galois field product. The sum unit performs a Galois field addition on an internal input based on a consolidated byte for the current clock cycle and the Galois field product to generate a second staged result for subsequent use to generate the check byte. Other embodiments may be described and/or claimed.

Abstract: An error detection and correction method for a flash memory includes: a setting step, setting selection information to select a first error detection and correction function for performing 1-bit error detection and correction or a second error detection and correction function for performing multiple-bit error detection and correction; and an executing step, performing the first error detection and correction function or the second error detection and correction function based on the selection information during a read operation or a write operation.

Abstract: A Reed-Solomon decoder circuit includes: a syndrome calculator circuit to compute syndrome values for a first codeword and a second codeword sequentially supplied to the syndrome calculator circuit, where last symbols of the first codeword overlap with first symbols of the second codeword during an overlap clock cycle between: a first plurality of non-overlap clock cycles during which the first codeword is supplied to the syndrome calculator circuit; and a second plurality of non-overlap clock cycles during which the second codeword is supplied to the syndrome calculator circuit; an error locator and error evaluator polynomial calculator circuit; an error location and error value calculator circuit; an error counter; and an error corrector circuit to correct the errors in the first codeword and the second codeword based on error counts and the error magnitudes computed by an error evaluator circuit.

Abstract: 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.

Abstract: An context-based encoding mechanism uses a predetermined number of bytes in a previous segment of a file to determine a context for the current segment. The current segment is encoded using a dictionary that corresponds to the determined context. An example method includes determining, for a first segment in a data file, a first context state based on a first context segment within the data file that precedes the first segment, identifying a first indexed dictionary from a plurality of indexed dictionaries based on the first context state, and encoding the first segment using the identified first indexed dictionary.

Abstract: A multi-port, multi-mode Reed Solomon (RS) forward error correction system includes a plurality of data in lines, each associated with a data port. The system includes a syndrome block (SDM) that has a plurality of syndrome slices and a SDM switching logic. An input of a SDM slice couples with a data in line from the plurality of data in lines. The switching logic couples with an interface port width (IFW) line a mode line. The IFW line identifies a number of data in lines tied together and the mode line to identify a RS mode. A reformulated inversionless Berlekamp-Massey (RiBM) block has a plurality of RiBM slices and a RiBM switching logic. A Chien Forney (ChFr) block has a plurality of ChFr slices. An error evaluation magnitude (ErEval) block has a plurality of ErEval slices. A plurality of adders couple with an output of a corresponding ErEval slice.

Abstract: A method for error correction comprises receiving data at a first device, and decoding, by decoder circuitry of the first device, the data. Decoding the data comprises determining a first error location within the data, and determining a first error magnitude within the data in parallel with determining the first error location. Decoding the data further comprises performing error correction to generate the decoded data based on the first error location and the first error magnitude. The method further comprises transmitting the decoded data to a second device.

Abstract: A fleet management system dispatches autonomous electric vehicles (AEVs) as on-demand power sources. The fleet management system receives a request for a power source including a location and data describing the amount of power requested. The fleet management system selects an AEV of the fleet to service the request based on the relative locations of the AEVs to the requested location, and based on the amount of power requested. The fleet management system instructs the selected AEV to drive to the location and supply power. The fleet management system instructs the selected AEV to disconnect and return to the charging station, and may instruct another AEV to continue fulfilling the request if additional power is needed.

Abstract: An example methods for interleaved BCH codes can include encoding a plurality of portions of data using a first generator polynomial to obtain a plurality of respective BCH codewords. The method can include encoding an additional BCH codeword based at least in part on a second plurality of portions of data and the plurality of BCH codewords using a second generator polynomial. The method can include outputting the plurality of respective BCH codewords and the additional BCH codeword.

Abstract: An apparatus is described. The apparatus includes a memory chip. The memory chip has an error correction code (ECC) encoder logic circuit and an ECC decoder logic circuit. The ECC decoder logic circuit is to place an additional one or more errors that result from incorrect error correction applied to a read code word into a same block of multiple blocks of the read code word's raw data bit portion where original errors in the read code word existed before the read code word was decoded by the ECC decoder logic circuit.

Abstract: Estimation of read parameters for a read channel of a solid-state storage device using a machine learning apparatus. The machine learning apparatus may be provided with signal count metrics from multiple regions of the memory cell signal space and syndrome weights from an error correction code. Other inputs may also be provided comprising metrics of the memory or read operations. In an example, the read parameters may include one or more reference threshold voltage values for read voltages applied to a memory cell and/or log-likelihood ratio (LLR) values for the memory cell.

Abstract: Methods for converting an n-bit number into an m-bit number for situations where n>m and also for situations where n<m, where n and m are integers. The methods use truncation or bit replication followed by the calculation of an adjustment value which is applied to the replicated number.

Abstract: A memory system controls a shift register memory and writes encoded data including a plurality of error correction code frames into a block of the shift register memory. The memory system is configured to store, into a location corresponding to a first layer in a first data storing shift string, first data included in a first error correction code frame, to store, into a location corresponding to a second layer in the first data storing shift string, second data included in a second error correction code frame, and to store, into a location corresponding to the second layer in a second data storing shift string, third data included in the first error correction code frame.

Abstract: A data processing method includes performing first equalization processing on a data stream that comprises a plurality of sub-data stream segments, performing segment de-interleaving on the data stream, separately performing first forward error correction (FEC) decoding on each sub-data stream segment in a data stream, performing, according to an equalization termination state of each sub-data stream segment, second equalization processing on each sub-data stream segment, performing second FEC decoding on the data stream, and outputting the data stream obtained according to the second FEC decoding in response to a preset iteration termination condition being met, or performing, in response to the preset iteration termination condition not being met, according to the equalization termination state of each sub-data stream segment obtained according to the first equalization, the second equalization processing on each sub-data stream segment obtained according to the second FEC decoding.

Abstract: 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.

Abstract: According to one embodiment, a memory system includes a nonvolatile memory, and a memory controller. Each of first storage regions of each of the nonvolatile memory includes a plurality of second storage regions. Each of pieces of first data includes pieces of second data as storage target data. Third data includes pieces of the second data that are selected one by one from each of the pieces of first data. The memory controller executes first decoding of decoding each of the pieces of first data on the basis of a first error correcting code generated by using the first data, and executes second decoding of decoding the third data including a bit of which reliability, which relates to each bit in each of the second storage regions that fail in the first decoding, is less than reliability of other bits on the basis of a second error correcting code.

Abstract: A method of transmitting a data signal in sync with a clock signal in a DTV receiver. The method includes performing demodulation processing and error correction processing on an input carrier wave and outputting signals resulting from these types of processing; acquiring a transport stream (TS) packet included in the signals; acquiring a variable-length packet included in the signals; and selecting either the TS packet or the variable-length packet and outputting the selected packet as the data signal, where to variable-length packet is either a type length value (TLV) packet or an Internet protocol (IP) packet.

Abstract: A method is provided for multiplying two polynomials. In the method, first and second polynomials are evaluated at 2t inputs, where t is greater than or equal to one, and where each input is a fixed power of two 2l/(2t) multiplied with a different power of a primitive root of unity, thereby creating 2 times 2t integers, where l is an integer such that 2l is at least as large as the largest coefficient of the resulting product multiplying the first and second polynomials. The 2 times 2t integers are then multiplied pairwise, and a modular reduction is performed to get 2t integers. A linear combination of the 2t integers multiplied with primitive roots of unity is computed to get 2t integers whose limbs in the base 2l-bit representation correspond to coefficients of the product of the first and second polynomials. The method can be implemented on a processor designed for performing RSA and/or ECC type cryptographic operations.

Abstract: Disclosed are devices, systems and methods for improving a bit-flipping algorithm for an irregular LDPC code in a non-volatile memory device. An example method includes receiving a noisy codeword, the codeword having been generated from an irregular low-density parity-check code, performing a first iteration of a bit-flipping algorithm on the noisy codeword, computing a first syndrome based on an output codeword of the first iteration, determining that the first syndrome comprises a non-zero vector and no bits of the noisy codeword were flipped during the first iteration of the bit-flipping algorithm, flipping, based on the determining, at least one bit of the output codeword, the at least one bit corresponding to a variable node of the plurality of variable nodes with a smallest column weight connected to one or more unsatisfied check nodes of the plurality of check nodes, and computing, subsequent to the flipping, a second syndrome.

Abstract: A transmission apparatus includes a signal processing circuit configured to obtain information data bits to be transmitted; add known information data bits to the information data bits to generate first data blocks; perform error-correction coding on the first data blocks to generate first coded data blocks including parity data blocks such that the first coded data blocks satisfy a first code rate; remove the known information data bits from the first coded data blocks to generate second coded data blocks, the second coded data blocks satisfying a second code rate different from the first code rate; and modulate the second coded data blocks using a modulation scheme to generate a modulated signal, which is then transmitted. A number of the known information data bits depends on a number of the information data bits such that the first code rate is fixed regardless of the number of the information data bits.

Abstract: A forward-backward Kalman filter for estimating phase noise present in a received signal. Both the forward and backward Kalman filters use hard-decision measurements of the received symbols. The phase noise estimate from the forward Kalman filter is used as a coarse phase noise estimate for the backward Kalman filter and vice versa. The final phase noise estimate is an optimal combination of the forward phase noise estimate and backward phase noise estimate.

Abstract: The disclosure may provide for a communication method and system. A transmitter of the communication system may include an interleaver and a first encoder for determining parity bits. The transmitter also may include a multiplexer for joining the parity bits with the data. A second encoder may be positioned after the multiplexer for implementing an error correcting code. A receiver of the communication system may include a decoder followed by an interleaver. When errors are detected in received data at the decoder, one or more processors of the receiver may be configured to correct portions of the received data and combine the corrected portions with the received data.

Abstract: Disclosed are techniques for generating a parity check matrix representing an error correcting code (ECC) for protecting a plurality of bits of a message. In an aspect, a method includes initializing a matrix M to store selected three-bit codes, selecting a first three-bit code from a set L of three-bit combinations of a number of bits n of the ECC that minimizes a sum of squared row weights of each row of the matrix M, comparing the first three-bit code with each of a plurality of error syndromes, and calculating, based on no comparison of the first three-bit code with each of the plurality of error syndromes, new error syndromes and storing the new error syndromes, wherein the new error syndromes are calculated by comparing the first three-bit code with each three-bit code in the matrix M, and storing the first three-bit code in the matrix M.

Abstract: Devices, systems, and methods that reduce the latency of detecting that a codeword is uncorrectable are disclosed and described. Such devices, systems, and methods allow the determination that a codeword is uncorrectable prior to determining error locations in the codeword, thus eliminating the need for such an error location search.

Abstract: A dual-mode Reed-Solomon decoder is configured to perform error correction for two different encoding schemes. The decoder includes a syndrome calculator block, a key equation solver block, a polynomial evaluation block, and an error correction block. The syndrome calculator block receives encoded input data and calculates syndromes, with the number of calculated syndromes based on the selected decoding mode. The key equation solver block calculates an error locator polynomial and an error evaluator polynomial for the encoded input data, with the degree of the polynomials based on the selected decoding mode. The polynomial evaluation block identifies error locations and magnitudes in the encoded data, with an array of constants input to the block based on the selected decoding mode. The error correction block decodes the encoded input data based on the identified error locations and error magnitudes.

Abstract: According to one embodiment, a memory system encodes a plurality of data frames written in the same block in an inter-frame direction and generates first parity data, encodes the first parity data in the inter-frame direction and generates second parity data, generates a plurality of pieces of first frame data by concatenating at least a part of the first or second parity data with each of the plurality of data frames, encodes each of the plurality of pieces of first frame data in an intra-frame direction and generates a plurality of third parity data, and writes a plurality of pieces of second frame data obtained by concatenating each of the plurality of pieces of first frame data and each of the plurality of pieces of third parity data in a plurality of pages in the same block in the non-volatile memory one by one.

Abstract: A circuit for error correction comprises a first RS syndrome generator to generate a first RS syndrome for a RS(n, k) code according to a received symbol stream to be decoded, wherein k and n are respective the number of data symbols and the total number of code symbols in the received symbol stream to be decoded; a first decision unit communicatively coupled to the first RS syndrome generator and configured to determine whether there are at least N symbols in the first RS syndrome that equal 0, wherein N is related to a code distance of the RS(n, k) code; and a first adder communicatively coupled to the first decision unit and configured to output a corrected decoded codeword by adding the first RS syndrome to the received symbol stream to be decoded if there are at least N symbols in the first RS syndrome that equal 0.

Abstract: Techniques are described for performing a bit-flipping decoding scheme on a G-LDPC codeword. In an example, a decoding system uses two syndrome tables. The first syndrome table identifies a predefined syndrome for a component codeword that protects a bit of the G-LDPC codeword. This predefined syndrome is identified based on a location of the bit and is used to update a current syndrome of the component codeword. The second syndrome table identifies one or more bit error locations for the component codeword. The bit error locations are identified from the second syndrome table based on the current syndrome of the component codeword, as updated. In an example, the error locations are used to update a reliability of the bit if its location corresponds to one of the error locations. A bit flipping decision is made for the bit based on its reliability.

Abstract: A wireless receiver includes a wireless communication component and a controller. The wireless communication component receives from a wireless transmitter a wireless signal that includes content data and encoded data having first error correction information and second error correction information of a different type from that of the first error correction information, for correcting errors in the content data. The controller determines which of the first error correction information and the second error correction information is to be given priority based on the signal quality of the wireless signal.

Abstract: A method of operation of a decoder includes receiving first data at the decoder. The method further includes generating second data at the decoder based on the first data. The second data is generated by adjusting an error locator polynomial based on an error parity of the first data.

Abstract: Examples disclosed herein relate to very large-scale integration (VLSI) circuit implementations of list decode circuits. In accordance with some examples disclosed herein, a list decode circuit may include a syndrome calculation circuit, a symbol erasure circuit, an erasure syndrome calculation circuit and a Berlekamp-Massey algorithm circuit (BMA), and an error locator polynomial (ELP) evaluation circuit. The syndrome calculation circuit may calculate a baseline syndrome and erasure syndrome calculation circuit may calculate erasure syndromes from error locator polynomials calculated by the symbol erasure circuit. The BMA circuit may use the calculated syndromes to generate a series of ELPs, which may be used by the ELP evaluation circuit to identify error locations in a codeword.

Abstract: The present invention discloses a heterogeneous computation framework, of Association. Rule Mining (ARM) using Micron's Autotmata Processor (AP). This framework is based on the Apriori algorithm. Two Automaton designs are proposed to match and count the individual itemset. Several performance improvement strategies are proposed including minimizing the number of reporting vectors and reduce reconfiguration delays. The experiment results show up to 94× speed ups of the proposed AP-accelerated Apriori on six synthetic and real-world datasets, when compared with the Apriori single-core CPU implementation. The proposed AP-accelerated Apriori solution also outperforms the state-of-the-art multicore and GPU implementations of Equivalence Class Transformation (Eclat) algorithm on big datasets.

Abstract: Error correction methods for arrays of resistive change elements are disclosed. An array of resistive change elements is organized into a plurality of subsections. Each subsection includes at least one flag bit and a plurality of data bits. At the start of a write operation, all bits in a subsection are initialized. If any data bits fail to initialize, the pattern of errors is compared to the input data pattern. The flag cells are then activated to indicate the appropriate encoding pattern to apply to the input data to match the errors. The input data is then encoded according to this encoding pattern before being written to the array. A second error correction algorithm can be used to correct remaining errors. During a read operation, the encoding pattern indicated by the flag bits is used to decode the read data and retrieve the original input data.

Abstract: An error correction encoder (10) includes an interleaver circuit (31), encoding circuits (321, 322) and a deinterleaver circuit (33). The interleaver circuit (31) generates, in a standard speed mode, a single series of yet-to-be-coded bit sequences (IL1) on the basis of the bits in plural columns that are arranged at an interval of C columns in a single series of transmission frames, and generates, in a two-times speed mode, two series of yet-to-be-coded bit sequences (IL1, IL2) on the basis of the bits in plural columns that are arranged at an interval of C/2 columns in each of two series of transmission frames. The encoding circuits (321, 322) apply error-correction coding to either the single series of yet-to-be-coded bit sequences (IL1) or the two series of yet-to-be-coded bit sequences (IL1, IL2).

Abstract: A decoding method in a mobile communication system using a non-binary LDPC code according to various embodiments of the present disclosure includes: selecting a message value having the highest reliability from each column and each row of an input vector message; generating a configuration set using the message value selected for each column and a GF element corresponding to the message value; and generating a check node output message using the generated configuration set and an extra output message value. According to various embodiments of the present disclosure, a decoding time period is reduced.

Abstract: The disclosure relates to noiseless coding and, in particular, to the use of twisted polar code in data encoding and decoding. The disclosure increases the speed of encoding and/or decoding through the reduction in the number of iterations to be performed. The object is attained in method for encoding data, comprising the steps of: pre-coding, by a pre-coding module, data presented in the form of k-dimensional binary vector x, the pre-coding consists in computing u(0)=xW, where W is a matrix of dimension ?×2m, and m is a code parameter; performing, by the pre-coding module, m-layer twisted polarization transformation of vector u(0), wherein i-th transformation layer consists in partitioning vector u(i?1) into 2m?1 subvectors of length 2, multiplying the subvectors by matrix ( 1 0 1 1 ) , merging the resulting subvectors into one vector of dimension 2m.

Abstract: Decoder circuitry for an input channel having a data rate, where a codeword on the input channel includes a plurality of symbols, includes options to provide a first output channel having that data rate, and a plurality of second output channels having slower data rates. The decoder circuitry includes syndrome calculation circuitry, polynomial calculation circuitry, and search-and-correct circuitry. The syndrome calculation circuitry includes finite-field multipliers for multiplying each symbol by a power of a root of the field. Each multiplier other than a first multiplier multiplies a symbol by a higher power of the root than an adjacent multiplier. First-level adders add outputs of a number of groups of multipliers. A second-level adder adds outputs of the first-level adders to be accumulated as syndromes of the first output channel. Another plurality of accumulators accumulates outputs of the first-level adders, which after scaling, are syndromes of the second output channels.

Abstract: Implementations of Advanced Encryption Standard (AES) encryption and decryption processes are disclosed. In one embodiment of S-box processing, a block of 16 byte values is converted, each byte value being converted from a polynomial representation in GF(256) to a polynomial representation in GF((22)4). Multiplicative inverse polynomial representations in GF((22)4) are computed for each of the corresponding polynomial representations in GF((22)4). Finally corresponding multiplicative inverse polynomial representations in GF((22)4) are converted and an affine transformation is applied to generate corresponding polynomial representations in GF(256). In an alternative embodiment of S-box processing, powers of the polynomial representations are computed and multiplied together in GF(256) to generate multiplicative inverse polynomial representations in GF(256).

Abstract: In an illustrative example, a method includes receiving data that includes a set of data symbols. The method further includes generating a set of parity symbols based on the set of data symbols using an erasure correcting code. The set of parity symbols includes at least a first parity symbol that is generated based on a first proper subset of the set of data symbols. The first parity symbol enables recovery of a data symbol of the first proper subset independently of a second proper subset of the set of data symbols.

Abstract: A CRC code is generated from an original data, a BCH code is generated with respect to the original data and the CRC code, and the original data, the CRC code, and the BCH code are recorded in pages selected from different planes of a plurality of memory chips. An RS code is generated from the original data across pages, a CRC code is generated with respect to the RS code, a BCH code is generated with respect to the RS code and the CRC code, and the RS code, the CRC code, the BCH code are recorded in a memory chip different from a memory chip including the original data. When reading data, error correction is performed on the original data by using the BCH code, and then CRC is calculated. If the number of errors is the number of errors that is correctable by erasure correction using the RS code, the original data is corrected by the erasure correction.