A method for Bose-Chaudhuri-Hocquenghem (BCH) soft error decoding includes receiving a codeword x, wherein the received codeword x has τ=t+r errors for some r≥1; computing a minimal monotone basis {λ.sub.i(x)}.sub.1≤i≤r+1.Math.F[x] of an affine space V={λ(x)∈F[x]:λ(x).Math.S(x)=λ′(x) (mod x.sup.2t), λ(0)=1, deg(λ(x)≤t+r}, wherein λ(x) is an error locator polynomial and S(x) is a syndrome; computing a matrix A≡(λ.sub.j(β.sub.i)).sub.i∈[w],j∈[r+1], wherein W={β.sub.1, . . . , β.sub.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.

A method for Bose-Chaudhuri-Hocquenghem (BCH) soft error decoding includes receiving a codeword x, wherein the received codeword x has τ=t+r errors for some r≥1; computing a minimal monotone basis {λ.sub.i(x)}.sub.1≤i≤r+1.Math.F[x] of an affine space V={λ(x)ϵF[x]: λ(x).Math.S(x)=λ′(x) (mod x.sup.2t), λ(0)=1, deg(λ(x)≤t+r}, wherein λ(x) is an error locator polynomial and S(x) is a syndrome; computing a matrix A≡(λ.sub.jβ.sub.i)).sub.iϵ[W],jϵ[r+1], wherein W={β.sub.i, . . . , β.sub.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.

Provided is an error correcting method including: executing, by an error-position detector, a determination process if a received word fails to satisfy a predetermined condition, the received word having a plurality of symbols, the determination process including determining if a determination-target symbol has an error or not, and detecting an error position, the error position being a position of the symbol having an error; changing, by a determination-target changing unit, the position of the determination-target symbol of the received word every time the determination process is executed; detecting, by an undetected-position detector, if the predetermined condition is satisfied, the error position of the symbol, for which the determination process is not executed, based on a relation between the error position and a variable generated from the received word; and correcting, by an error corrector, an error at the error position detected by the error-position detector and the undetected-position detector.

Disclosed is an encoding method for semantic channels. According to the encoding method, a semantic codebook and synonymous mapping relationships may be obtained at first. Then, a syntactic information sequence to be transmitted may be determined. Further, a syntactic codeword from the semantic codebook corresponding to the syntactic information sequence may be determined based on the synonymous mapping relationships. Finally, the syntactic codeword may be modulated to obtain a signal vector.

Disclosed is an encoding method for semantic channels. According to the encoding method, a semantic codebook and synonymous mapping relationships may be obtained at first. Then, a syntactic information sequence to be transmitted may be determined. Further, a syntactic codeword from the semantic codebook corresponding to the syntactic information sequence may be determined based on the synonymous mapping relationships. Finally, the syntactic codeword may be modulated to obtain a signal vector.