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7 Constructions of QCCs
γ 1 = μ(rk H μ + rk H
μ ) +
μ−1
i=1
(μ − i)[rk H
(μ−i) − rk H
(μ−i+1) ],
so V
⊥
1 has also degree γ 1 .
We know that V 2 ⊂ V 1 . The corresponding CSS-type code derived from V 1 and V 2
has frame size n, k = rk H 0 logical qudits per frame, degree γ = γ 1 + γ 2 , (d x ) f ≥
(d 1 ) f ≥ d
⊥ and (d z ) f ≥ (d 2 )
⊥
f , where min{d
0 + d
μ , d
∗
} ≤ (d 2 )
⊥
f ≤ d
∗ . Thus one
can get an [(n, rk H 0 , μ
∗
; γ 1 + γ 2 , (d z ) f /(d x ) f )] q AQCC. If H 1 (D) is a generator
matrix of the code V
⊥
1 then a stabilizer matrix of our AQCC is given by
G 2 (D) | 0
0 | H 1 (D)
.
Other variant of this construction can be obtained by considering a CSS-type code
derived from the pair of classical convolutional codes V
⊥
1 ⊂ V
⊥
2 . The proof is complete.
7.7.2 Construction II
In this subsection, we utilize Bose–Chaudhuri–Hocquenghem (BCH) codes to construct families of AQCCs. To proceed with Construction II, we fix some notation.
As always, q is a prime power and n a positive integer such that gcd(q, n) = 1. Let
α be a primitive nth root of unity in some extension field.
Recall that 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)},
i.e., 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. It is well known that
the minimum distance of a BCH code is greater than or equal to its designed distance
δ. 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)
⎤
⎥
⎥
⎥
⎦
,