Abstract: Devices, systems, and methods for managing a storage device configured to store a plurality of codewords, including: obtaining a two-dimensional (2D) generalized concatenated code (GCC) codeword from the storage device; providing the codeword to a sequential decoder; based on detecting a first failure by the sequential decoder, updating the codeword and transposing the updated codeword to obtain a transposed codeword; providing the transposed codeword to the sequential decoder; and obtaining information bits corresponding to the codeword based on a result obtained by the sequential decoder.

Abstract: Methods, devices, and systems for controlling a storage system, including: a storage device configured to store a plurality of code words; and at least one processor configured to: obtain information bits; encode the information bits using a first code to obtain a first plurality of code words; encode the first plurality of code words using a second code to generate a second plurality of code words; update the first plurality of code words based on the second plurality of code words to generate an adaptive generalized concatenated code (A-GCC) code word; and store the A-GCC code word in the storage device, wherein each code word of the first plurality of code words is encoded using a different error protection level.

Abstract: Devices, systems, and methods for managing a storage device configured to store a plurality of codewords, including: obtaining a two-dimensional (2D) generalized concatenated code (GCC) codeword from the storage device; providing the codeword to a sequential decoder; based on detecting a first failure by the sequential decoder, updating the codeword and transposing the updated codeword to obtain a transposed codeword; providing the transposed codeword to the sequential decoder; and obtaining information bits corresponding to the codeword based on a result obtained by the sequential decoder.

Abstract: Devices, systems, and methods for managing a storage device configured to store a plurality of codewords, including: obtaining a two-dimensional (2D) generalized concatenated code (GCC) codeword from the storage device; providing the codeword to a sequential decoder; based on detecting a first failure by the sequential decoder, updating the codeword and transposing the updated codeword to obtain a transposed codeword; providing the transposed codeword to the sequential decoder; and obtaining information bits corresponding to the codeword based on a result obtained by the sequential decoder.

Abstract: Methods, devices, and systems for controlling a storage system, including: a storage device configured to store a plurality of code words; and at least one processor configured to: obtain information bits; encode the information bits using a first code to obtain a first plurality of code words; encode the first plurality of code words using a second code to generate a second plurality of code words; update the first plurality of code words based on the second plurality of code words to generate an adaptive generalized concatenated code (A-GCC) code word; and store the A-GCC code word in the storage device, wherein each code word of the first plurality of code words is encoded using a different error protection level.

Abstract: Systems, devices, and methods for encoding information bits for storage, including obtaining information bits; encoding the information bits using an inner code to obtain a plurality of inner code words; encoding the plurality of inner code words using an outer code to generate an outer code word; and storing the outer code word in a storage device, wherein at least one of the inner code and the outer code includes a generalized concatenated code (GCC), and wherein the outer code word includes a hierarchical-GCC (H-GCC) code word.

Abstract: Systems, devices, and methods for encoding information bits for storage, including obtaining information bits; encoding the information bits using an inner code to obtain a plurality of inner code words; encoding the plurality of inner code words using an outer code to generate an outer code word; and storing the outer code word in a storage device, wherein at least one of the inner code and the outer code includes a generalized concatenated code (GCC), and wherein the outer code word includes a hierarchical-GCC (H-GCC) code word.

Abstract: A method of correcting data stored in a memory device includes: applying an iterative decoder to the data; determining a total number of rows in first data the decoder attempted to correct; estimating first visible error rows among the total number that continue to have an error after the attempt; estimating residual error rows among the total number that no longer have an error after the attempt; determining second visible error rows in second data of the decoder that continue to have an error by permuting indices of the residual error rows according to a permutation; and correcting the first data using the first visible error rows.

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 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 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: A method of correcting data stored in a memory device includes: applying an iterative decoder to the data; determining a total number of rows in first data the decoder attempted to correct; estimating first visible error rows among the total number that continue to have an error after the attempt; estimating residual error rows among the total number that no longer have an error after the attempt; determining second visible error rows in second data of the decoder that continue to have an error by permuting indices of the residual error rows according to a permutation; and correcting the first data using the first visible error rows.

Abstract: A method of correcting data stored in a memory device includes: applying an iterative decoder to the data; determining a total number of rows in first data the decoder attempted to correct; estimating first visible error rows among the total number that continue to have an error after the attempt; estimating residual error rows among the total number that no longer have an error after the attempt; determining second visible error rows in second data of the decoder that continue to have an error by permuting indices of the residual error rows according to a permutation; and correcting the first data using the first visible error rows.

Abstract: A method of correcting data stored in a memory device includes: applying an iterative decoder to the data; determining a total number of rows in first data the decoder attempted to correct; estimating first visible error rows among the total number that continue to have an error after the attempt; estimating residual error rows among the total number that no longer have an error after the attempt; determining second visible error rows in second data of the decoder that continue to have an error by permuting indices of the residual error rows according to a permutation; and correcting the first data using the first visible error rows.

Abstract: A method for Bose-Chaudhuri-Hocquenghem (BCH) soft error decoding includes receiving a codeword x, wherein the received codeword x has ?=t+r errors for some r?1; computing a minimal monotone basis {?i(x)}1?i?r+1?F[x] of an affine space V={?(x)?F[x]:?(x)·S(x)=??(x) (mod x2t), ?(0)=1, deg(?(x)?t+r}, wherein ?(x) is an error locator polynomial and S(x) is a syndrome; computing a matrix A?(?j(?i))i?[w],j?[r+1], wherein W={?1, . . . , ?w} is a set of weak bits in x; constructing a submatrix of r+1 rows from sub matrices of r+1 rows of the subsets of A such that the last column is a linear combination of the other columns; forming a candidate error locating polynomial using coefficients of the minimal monotone basis that result from the constructed submatrix; performing a fast Chien search to verify the candidate error locating polynomial; and flipping channel hard decision at error locations found in the candidate error locating polynomial.

Abstract: A method for Bose-Chaudhuri-Hocquenghem (BCH) soft error decoding includes receiving a codeword x, wherein the received codeword x has ?=t+r errors for some r?1; computing a minimal monotone basis {?i(x)}1?i?r+1?F[x] of an affine space V={?(x)?F[x]: ?(x)·S(x)=??(x) (mod x2t), ?(0)=1, deg(?(x)?t+r}, wherein ?(x) is an error locator polynomial and S(x) is a syndrome; computing a matrix A?(?j?i))i?[W],j?[r+1], wherein W={?i, . . . , ?W} is a set of weak bits in x; constructing a submatrix of r+1 rows from sub matrices of r+1 rows of the subsets of A such that the last column is a linear combination of the other columns; forming a candidate error locating polynomial using coefficients of the minimal monotone basis that result from the constructed submatrix; performing a fast Chien search to verify the candidate error locating polynomial; and flipping channel hard decision at error locations found in the candidate error locating polynomial.

Abstract: A method of performing division operations in an error correction code includes the steps of receiving an output ??F†{0} wherein F=GF(2r) is a Galois field of 2r elements, ?=?0?i?r?1?i×?i wherein ? is a fixed primitive element of F, and ?i?GF(2), wherein K=GF(2s) is a subfield of F, and {1, ?} is a basis of F in a linear subspace of K; choosing a primitive element ??K, wherein ?=?1+?×?2, ?1=?0?i?s?1 ?i×?i?K, ?2=?0?i?s?1 ?i+s×?i?K, and ?=[?0, . . . , ?r?1]T?GF(2)r; accessing a first table with ?1 to obtain ?3=?1?1, computing ?2×?3 in field K, accessing a second table with ?2=?3 to obtain (1+?×?2×?3)?1=?4+?×?5, wherein ??1=(?1×(1+?×?2×?3))?1=?3×(?4+?×?5)=?3×?4+?×?3×?5; and computing products ?3×?4 and ?3×?5 to obtain ??1=?0?i?s?1?i×?i+?·?i?i?s?1?i+s=?i where ?i?GF(2).

Abstract: A method of performing division operations in an error correction code includes the steps of receiving an output ??F†{0} wherein F=GF(2r) is a Galois field of 2r elements, ?=?0?i?r?1?i×?i wherein ? is a fixed primitive element of F, and ?i?GF(2), wherein K=GF(2s) is a subfield of F, and {1, ?} is a basis of F in a linear subspace of K; choosing a primitive element ??K, wherein ?=?1+?×?2, ?1=?0?i?s?1?i×?i?K, ?2=?0?i?s?1?i+s×?i?K, and ?=[?0, . . . , ?r?1]T?GF(2)r; accessing a first table with ?1 to obtain ?3=?1?1, computing ?2×?3 in field K, accessing a second table with ?2=?3 to obtain (1+?×?2×?3)?1=?4+?×?5, wherein ??1=(?1×(1+?×?2×?3))?1=?3×(?4+?×?5)=?3×?4+?×?3×?5; and computing products ?3×?4 and ?3×?5 to obtain ??1=?0?i?s?1?i×?i+?·?i?i?s?1?i+s=?i where ?i?GF(2).

Abstract: Systems and methods are described for low power error correction coding (ECC) for embedded universal flash storage (eUFS) are described. The systems and methods may include identifying a first element of an algebraic field; generating a plurality of lookup tables for multiplying the first element; multiplying the first element by a plurality of additional elements of the algebraic field, wherein the multiplication for each of the additional elements is performed using an element from each of the lookup tables; and encoding information according to an ECC scheme based on the multiplication.

Abstract: Systems and methods are described for low power error correction coding (ECC) for embedded universal flash storage (eUFS) are described. The systems and methods may include identifying a first element of an algebraic field; generating a plurality of lookup tables for multiplying the first element; multiplying the first element by a plurality of additional elements of the algebraic field, wherein the multiplication for each of the additional elements is performed using an element from each of the lookup tables; and encoding information according to an ECC scheme based on the multiplication.