52
4 Linear Block Codes
that is, the code has a sequence of δ − 1 consecutive powers of α as zeros. Then the
minimum distance of C is greater than or equal to δ.
In the sequence, we present the definition of a BCH code.
Definition 4.4.4 Let q be a prime power and let n be a positive integer such that
gcd(q, n) = 1. Assume that α is a primitive nth root of unity. A cyclic code C of
length n over F q is a BCH code with designed distance δ if, for some integer b ≥ 0,
we have
g(x) = l. c. m.{M
(b)
(x), M
(b+1)
(x), . . . , M
(b+δ−2)
(x)},
that is, g(x) is the monic polynomial of smallest degree over F q having α
b
, α
b+1
,
. . . , α
b+δ−2 as zeros.
Therefore, c ∈ C if and only if c(α
b
) = c(α
b+1
) = . . . = c(α
b+δ−2
) = 0. Thus
the code has a string of δ − 1 consecutive powers of α as zeros, Hence, from the
BCH bound, its minimum distance is at least δ. If n = q
l
− 1, then the BCH code is
called primitive and if b = 1 it is called narrow-sense.
Definition 4.4.5 The q-ary cyclotomic coset (or q-ary coset or q-coset) modulo n
containing an element s is defined by {s, sq, sq
2
, sq
3
, . . . , sq
m s −1
}, where m s is the
smallest positive integer such that sq
m s ≡ s mod n. If s is the smallest number in
coset, this coset is denoted by C s .
In terms of cyclotomic cosets, the generator polynomial of a BCH code is of the
form
g(x) =
z∈Z
(x − α
z
),
where Z = C b ∪ C b+1 ∪ · · · ∪ C b+δ−2 is the defining set of the code.
A parity check matrix for C is given by
H δ,b =
⎡
⎢
⎢
⎢
⎣
1 α
b
α
2b
· · · α
(n−1)b
1 α
(b+1)
α
2(b+1)
· · · α
(n−1)(b+1)
. . .
. . .
. . .
. . .
. . .
1 α
(b+δ−2)
· · · · · · α
(n−1)(b+δ−2)
⎤
⎥
⎥
⎥
⎦
,
where each entry is replaced by the corresrponding column of l = ord n (q) elements
from F q , then removing any linearly dependent rows. The rows of the resulting matrix
over F q are the parity checks satisfied by C.