Abstract: An encoding apparatus includes a processor a non-transitory computer-readable storage medium storing a program for encoding data x of dimension k into a codeword c of length n. The program includes instructions to encode the data x using a C(n, k, d) code. The code C(n,k,d) has a length n and a minimum distance d, n=2m1+ . . . +2ms, mh, h=1, . . . , s, is an integer, on the basis of c=uA, where ui=xji, 0?ji<k?1, if i?F, F is a set of n?k frozen bit indices of the code C(n,k,d), and ui=?s=0i?1?i,sus, if i?F, is a constraint matrix given by a solution of T=0, ?i is an index of a row of the matrix , having a last non-zero element in column i, and is a precoding matrix and A is a matrix defined by A m h = ( 1 0 1 1 ) ? m h , and Q?m denotes the m-times Kronecker product of a matrix Q with itself.