U. Glaeser

FIGURE 34.63 Interleaving m times of code C.
FIGURE 34.64 Product code C 1 × C 2.
parameter m is called the depth of interleaving. If each of the individual codes can correct up to s errors,
the interleaved scheme can correct up to s bursts of length up to m bytes each, or (m − 1)b + 1 bits each.
This occurs because a burst of length up to m bytes is distributed among m different codewords.
Intuitively, interleaving “randomizes” a burst.
The drawback of interleaving is delay. Notice that we need to read most of the information bytes before
we are able to calculate and write the redundant bytes. Thus, we need enough buffer space to accomplish
this.
Interleaving of RS codes has been widely used in magnetic recording. For instance, in a disk, the data
are written in concentric tracks, and each track contains a number of information sectors. Typically, a
sector consists of 512 information 8-bit bytes (although the latest trends tend to larger sectors). A typical
embodiment would consist in dividing the 512 bytes into four codewords, each one containing 128
information bytes and six redundant bytes (i.e., each interleaved shortened RS codeword can correct up
to three bytes). Therefore, this scheme can correct up to three bursts of length up to 25 bits each.
A natural generalization of the interleaved scheme described above is product codes. In effect, we may
consider that both rows and columns are encoded into error-correcting codes. The product of an [n 1, k 1]
code C 1 with an [n 2,k 2] code C 2, denoted C 1 × C 2, is illustrated in Fig. 34.64. If C 1 has minimum distance
d 1 and C 2 has minimum distance d 2, it is easy to see that C 1 × C 2 has minimum distance d 1d 2.
In general, the symbols are read out in row order (although other readouts, like diagonal readouts,
are also possible). For encoding, first the column redundant symbols are obtained, and then the row
redundant symbols. For obtaining the checks on checks c i,j, k 1 ≤ i ≤ n 1 − 1, k 2 ≤ j ≤ n 2 − 1, it is easy to
see that it is irrelevant if we encode on columns or on rows first. If the symbols are read in row order,
normally C 1 is called the outer code and C 2 the inner code. For decoding, many possible procedures are
used. The idea is to correct long bursts together with random errors. The inner code C 2 corrects first. In
that case, two events may happen when its error-correcting capability is exceeded: either the code will
detect the error event or it will miscorrect. If the code detects an error event (that may well have been
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