8.4.1 Description

Repeat accumulate (RA) codes are a very simple but powerful type of error-correcting code. An RA codeword is created as follows: every bit in b is copied N _c/ N _b times, ^[1] resulting in a sequence d of length N _c. This sequence is passed to an interleaver, which essentially shuffles the bits around so that they look more random. We will denote the interleaver function by ?( ) and set e = ?( d). Finally, the coded sequence c is passed through an accumulator:

It is easily verified that we can express the relationship between b and c in a generator-matrix representation as

where G _rep is an N _c N _b repeater matrix with N _c/ N _b consecutive 1s per column and one 1 per row, the matrix G _? is an N _c N _c interleaver matrix (more commonly referred to as a permutation matrix), with exactly one 1 per row and per column, and G _acc is an N _c N _c accumulator matrix. An accumulator matrix is defined as follows: introduce 1 _k as a row-vector of k 1s, and 0 _k as a row-vector of k 0s, then the kth row (1 ? k ? N _c) is given by .