Systems and methods are disclosed for optically communicating data by, at a transmitter side, encoding a block of input bits by one or more outer encoders, and after interleaving the encoded bits, permuting the encoded bits according to a predetermined sequence or order, and further encoding the encoded bits by an inner encoder, and at a receiver side, decoding received bits with an inner decoder, and after the encoded bits are permuted, subsequently decoding by and outer decoder, and returning information bits at an outer decoder as an output. The soft-decision and hard-decision outputs from the outer BCH code help the inner LDPC decoder to have better estimation of the received bits and gain performance. The performance in higher-order modulation formats could be as large as 0.5 dB in one embodiment.

Examples are given for techniques associated with error correction for encoded data. In some examples, error correction code (ECC) information for the ECC encoded data may be received that indicates the ECC encoded data has bit errors that are not able to be corrected by the ECC used to encode the ECC encoded data. A soft decision decoding may be implemented that includes flipping a given number of bits of a selected portion of the ECC encoded data based on a combinatorial operation or method. One or more successful decodes may result from this selective flipping to enable the ECC to successfully decode the ECC encoded data.

According to the embodiments, a memory system includes a non-volatile memory, a control unit that reads a received word from the non-volatile memory, and a decoder that performs soft-decision decode to the received word. The decoder includes a test pattern generating unit that generates test patterns, a hard decision decoder that performs hard-decision decode by using the test pattern and the received word and outputs a decoded word, and a distance calculating unit that calculates Euclidean distance between the decoded word and the received word based on the decoded words of which the number is less than that of the test patterns of all the combinations in a case where the number of flips is of one to a predetermined value and selects a decoded word which is the decoding result from among the decoded words output from the hard decision decoder based on the Euclidean distance.

A method for soft decoding of generalized Reed-Solomon (RS) error correction codes, includes receiving a codeword through a digital electronic communication channel; verifying that an HD error-and-erasures decoding has failed; finding a Groebner basis that accounts for a fixed set of erasures; constructing a Chase and GMD decoding tree on the set of Chase coordinates and the set of GMD coordinates; traversing the decoding tree using polynomials of the Groebner basis as a basis that represents updated coefficient polynomials on the Chase and GMD decoding tree; updating polynomials on the decoding tree using root and derivate steps that flip an edge, or a root step for an erasure edge; calculating error locations by polynomial evaluation of candidate polynomials from the decoding tree, and calculating error values by using Forney's formula; and correcting the received codeword according to the calculated error locations and calculated error values.

A method for soft decoding of generalized Reed-Solomon (RS) error correction codes, includes receiving a codeword through a digital electronic communication channel; verifying that an HD error-and-erasures decoding has failed; finding a Groebner basis that accounts for a fixed set of erasures; constructing a Chase and GMD decoding tree on the set of Chase coordinates and the set of GMD coordinates; traversing the decoding tree using polynomials of the Groebner basis as a basis that represents updated coefficient polynomials on the Chase and GMD decoding tree; updating polynomials on the decoding tree using root and derivate steps that flip an edge, or a root step for an erasure edge; calculating error locations by polynomial evaluation of candidate polynomials from the decoding tree, and calculating error values by using Forney's formula; and correcting the received codeword according to the calculated error locations and calculated error values.

A least reliable bit (LRB) identification approach based on a cumulative distribution function (CDF) is disclosed. In some embodiments, a receiver includes a detector and a decoder. The detector is configured to receive a codeword and determine a list of reliability values for the bits included in the codeword. The decoder is configured to receive, from the detector, the codeword and the list of reliability values, compute a list of CDFs of the reliability values for the codeword, identify, from the CDF list, a group including a specific number of LRBs that have the reliability values within a threshold range, and determine a location of each LRB of the group in the codeword.

The present application relates to multi-candidate successive cancellation list (SCL) decoding of polar codes. In an example, an apparatus processes multiple candidate codewords using a same SCL decoder. The SCL decoder may be cyclic redundancy code (CRC)-aided. The multiple candidate codewords may have different code configurations and/or different inputs. In an example, the multiple candidate codewords are packaged so that at most one of the candidate codewords in the package is valid.