186
7 Constructions of QCCs
Let us consider that m = or d 2n (q
2
) and let β be a primitive 2nth root of unity in
F q 2m ; hence, α = β
2
∈ F q 2m is a primitive nth root of unity. The roots of x
n
+ 1 are
given by β
2i+1 , 0 ≤ i ≤ n − 1. Put O 2n = {1, 3, . . . , 2n − 1}.
Definition 7.5.1 The defining set of a negacyclic code C of length n generated by
g(x) is the set
Z = {i ∈ O 2n |β
i is root of g(x)}.
The defining set is a union of q
2 -ary cosets defined by
C i = {i, iq
2
, . . . , iq
2(m i −1)
},
where m i is the smallest positive integer such that iq
2(m i )
≡ i mod 2n.
Definition 7.5.2 The minimal polynomial M
( j)
(x), over F q 2 , of β
j
∈ F q 2m is given
by
M
( j)
(x) =
j∈C i
(x − β
j
).
The dimension of C is equal to n − |Z|.
The BCH bound for constacyclic codes (see for example [14, 87]) asserts, that
is, C is a q
2 -ary negacyclic code of length n with generator polynomial g(x), and if
g(x) has the elements {β
2i+1
|0 ≤ i ≤ d − 2} as roots, where β is a primitive 2nth
root of unity, then C has minimum distance d C lower bounded by d, that is, d C ≥ d.
A definition of a negacyclic BCH code is given below.
Definition 7.5.3 (Negacyclic BCH codes) Let q be a power of an odd prime with
gcd(n, q) = 1. Let β be a primitive 2nth root of unity in F q m . A negacyclic code C
of length n over F q is a BCH code with designed distance δ if, for some odd integer
b ≥ 1, we have
g(x) = lcm{M
(b)
(x), M
(b+2)
(x), . . . , M
[b+2(δ−2)]
(x)},
that is, g(x) is the monic polynomial of smallest degree over F q having α
b
, α
b+2
, . . . ,
α
[b+2(δ−2)] as zeros.
Therefore, c ∈ C if and only if c(α
b
) = c(α
(b+2)
) = · · · = c(α
[b+2(δ−2)]
) = 0. Thus
the code has a string of δ − 1 consecutive odd powers of β as zeros.
7.5.2 Convolutional MDS Codes
In this subsection, families of convolutional codes are constructed. First, we recall
some results shown in the literature.
7 Constructions of QCCs
Let us consider that m = or d 2n (q
2
) and let β be a primitive 2nth root of unity in
F q 2m ; hence, α = β
2
∈ F q 2m is a primitive nth root of unity. The roots of x
n
+ 1 are
given by β
2i+1 , 0 ≤ i ≤ n − 1. Put O 2n = {1, 3, . . . , 2n − 1}.
Definition 7.5.1 The defining set of a negacyclic code C of length n generated by
g(x) is the set
Z = {i ∈ O 2n |β
i is root of g(x)}.
The defining set is a union of q
2 -ary cosets defined by
C i = {i, iq
2
, . . . , iq
2(m i −1)
},
where m i is the smallest positive integer such that iq
2(m i )
≡ i mod 2n.
Definition 7.5.2 The minimal polynomial M
( j)
(x), over F q 2 , of β
j
∈ F q 2m is given
by
M
( j)
(x) =
j∈C i
(x − β
j
).
The dimension of C is equal to n − |Z|.
The BCH bound for constacyclic codes (see for example [14, 87]) asserts, that
is, C is a q
2 -ary negacyclic code of length n with generator polynomial g(x), and if
g(x) has the elements {β
2i+1
|0 ≤ i ≤ d − 2} as roots, where β is a primitive 2nth
root of unity, then C has minimum distance d C lower bounded by d, that is, d C ≥ d.
A definition of a negacyclic BCH code is given below.
Definition 7.5.3 (Negacyclic BCH codes) Let q be a power of an odd prime with
gcd(n, q) = 1. Let β be a primitive 2nth root of unity in F q m . A negacyclic code C
of length n over F q is a BCH code with designed distance δ if, for some odd integer
b ≥ 1, we have
g(x) = lcm{M
(b)
(x), M
(b+2)
(x), . . . , M
[b+2(δ−2)]
(x)},
that is, g(x) is the monic polynomial of smallest degree over F q having α
b
, α
b+2
, . . . ,
α
[b+2(δ−2)] as zeros.
Therefore, c ∈ C if and only if c(α
b
) = c(α
(b+2)
) = · · · = c(α
[b+2(δ−2)]
) = 0. Thus
the code has a string of δ − 1 consecutive odd powers of β as zeros.
7.5.2 Convolutional MDS Codes
In this subsection, families of convolutional codes are constructed. First, we recall
some results shown in the literature.