ALGEBRAIC CODING THEORY 93
Besides the factor −Y
j
, this expression is identical to the expression derived above for the
error evaluator polynomial. By relating these expressions, Forney’s formula
Y
j
=−
ω(X
−1
j
)
λ
(X
−1
j
)
results for calculating the error values Y
j
.
In summary, we obtain the algebraic decoding algorithm for narrow-sense BCH codes
in Figure 2.61 (Neubauer, 2006b). This algebraic decoding algorithm can be illustrated by
the block diagram in Figure 2.62 showing the individual steps of the algorithm (Lee, 2003).
In this block diagram the main parameters and polynomials determined during the course
of the algorithm are shown. Here, Chien search refers to the sequential search for the zeros
of the error locator polynomial (Berlekamp, 1984). The arithmetics are carried out in the
extension field F
q
l
. For a Reed– Solomon code as a special case of primitive BCH codes
with code word length n = q −1 the calculations are executed in the finite field F
q
. Further
details of Reed–Solomon decoders, including data about the implementation complexity
which is measured by the number of gates needed to realise the corresponding integrated
circuit module, can be found elsewhere (Lee, 2003, 2005).
Erasure Correction
Erasures are defined as errors with known error positions i
j
but unknown error values Y
j
.
For the correction of these erasures, the algebraic decoding algorithm can be simplified.
Since the error positions i
j
as well as the number of errors w are known, the error locators
X
j
= α
i
j
and the error locator polynomial λ(z) can be directly formulated. Figure 2.63
illustrates the algebraic decoding algorithm for the correction of erasures.
2.4 Summary
In this chapter we have introduced the basic concepts of algebraic coding theory. Lin-
ear block codes have been discussed which can be defined by their respective generator
and parity-check matrices. Several code construction techniques have been presented. As
important examples of linear block codes we have treated the repetition code, parity-check
code, Hamming code, simplex code and Reed–Muller code. So-called low density parity-
check or LDPC codes, which are currently under research, will be presented in Section 4.1.
Further information about other important linear and non-linear codes can be found else-
where (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004;
MacWilliams and Sloane, 1998; McEliece, 2002; van Lint, 1999).
By introducing further algebraic structures, cyclic codes were presented as an important
subclass of linear block codes for which efficient algebraic decoding algorithms exist. Cyclic
codes can be defined with the help of suitable generator or parity-check polynomials. Owing
to their specific properties, efficient encoding and decoding architectures are available, based
on linear feedback shift registers. With the help of the zeros of the respective generator
polynomial, BCH codes and Reed–Solomon codes were defined. Further details about
cyclic codes can be found elsewhere (Berlekamp, 1984; Bossert, 1999; Lin and Costello,
2004; Ling and Xing, 2004).