60
5 Quantum Code Constructions
C [r +2] = {r + 2, r + 2q}, . . . , C [2r −2] = {2r − 2, r + (r − 2)q}, are mutually disjoint and, with exception of the cosets C [0] = {0} and C [r ] = {r }, each of them has
two elements.
The cosets C [0] and C [r ] have only one element. Let us show that each of the other
cosets has two elements. Since (r − 2)q < n, then the congruence l ≡ lq mod n
implies that l = lq, where 1 ≤ l ≤ r − 2, which is a contradiction. If r + s ≡ (r +
s)q mod n, where 1 ≤ s ≤ r − 2, then r + s = r + sq, which is a contradiction.
From now on, we show that all these cosets given above and C [0] and C [r ] are
mutually disjoint. We only consider the case C [r +l] = C [r −s] , where 1 ≤ l, s ≤ r − 2,
since the other cases are similar to this one. Seeking a contradiction, we assume
that C [r +l] = C [r −s] , where 1 ≤ l, s ≤ r − 2. If the congruence (r + l) ≡ (r − s)
mod n holds, we obtain
(r + l) ≡ (r − s) mod n =⇒ n | (l + s).
If l + s = 0, one has n ≤ l + s, which is a contradiction. If l + s = 0, this implies
l = −s, which is a contradiction.
On the other hand, if (r + l)q ≡ r − s mod n holds, we have
(r + l)q ≡ r − s =⇒ lq ≡ −s mod n
=⇒ n | (lq + s).
Since l, s ≤ r − 2 and r < q are true, if lq + s = 0 holds, it follows that lq + s < n,
which is a contradiction. If lq + s = 0 then lq = −s, which is a contradiction. Thus
all the q-ary cosets C [0] , C [1] , . . . , C [r −2] , are disjoint from each of the q-ary cosets
C [r ] , C [r +1] , . . . , C [2r −2] . Additionally, all the q-ary cosets C [0] , C [1] , . . . , C [r −2] , are
mutually disjoint and all the q-ary cosets C [r ] , C [r +1] , . . . , C [2r −2] are also mutually
disjoint.
Let C 1 be the cyclic code generated by the product of the minimal polynomials
M
(0)
(x)M
(1)
(x) · . . . · M
(r −2)
(x),
and C 2 be the cyclic code generated by g 2 (x), that is the product of the minimal
polynomials
g 2 (x) =
i
M
(i)
(x),
where i /
∈ {r, r + 1, . . . , 2r − 2} and i runs through the coset representatives
mod n. From construction it follows that C 2 C 1 . From the BCH bound, the minimum distance of C 1 is greater than or equal to r , because its defining set contains
the sequence 0, 1, . . . , r − 2, of r − 1 consecutive integers. Similarly, the defining
set of the code C generated by the polynomial h(x) =
x
n −1
g 2 (x)
contains the sequence
r, r + 1, . . . , 2r − 2, of r − 1 consecutive integers and so, from the BCH bound, C
5 Quantum Code Constructions
C [r +2] = {r + 2, r + 2q}, . . . , C [2r −2] = {2r − 2, r + (r − 2)q}, are mutually disjoint and, with exception of the cosets C [0] = {0} and C [r ] = {r }, each of them has
two elements.
The cosets C [0] and C [r ] have only one element. Let us show that each of the other
cosets has two elements. Since (r − 2)q < n, then the congruence l ≡ lq mod n
implies that l = lq, where 1 ≤ l ≤ r − 2, which is a contradiction. If r + s ≡ (r +
s)q mod n, where 1 ≤ s ≤ r − 2, then r + s = r + sq, which is a contradiction.
From now on, we show that all these cosets given above and C [0] and C [r ] are
mutually disjoint. We only consider the case C [r +l] = C [r −s] , where 1 ≤ l, s ≤ r − 2,
since the other cases are similar to this one. Seeking a contradiction, we assume
that C [r +l] = C [r −s] , where 1 ≤ l, s ≤ r − 2. If the congruence (r + l) ≡ (r − s)
mod n holds, we obtain
(r + l) ≡ (r − s) mod n =⇒ n | (l + s).
If l + s = 0, one has n ≤ l + s, which is a contradiction. If l + s = 0, this implies
l = −s, which is a contradiction.
On the other hand, if (r + l)q ≡ r − s mod n holds, we have
(r + l)q ≡ r − s =⇒ lq ≡ −s mod n
=⇒ n | (lq + s).
Since l, s ≤ r − 2 and r < q are true, if lq + s = 0 holds, it follows that lq + s < n,
which is a contradiction. If lq + s = 0 then lq = −s, which is a contradiction. Thus
all the q-ary cosets C [0] , C [1] , . . . , C [r −2] , are disjoint from each of the q-ary cosets
C [r ] , C [r +1] , . . . , C [2r −2] . Additionally, all the q-ary cosets C [0] , C [1] , . . . , C [r −2] , are
mutually disjoint and all the q-ary cosets C [r ] , C [r +1] , . . . , C [2r −2] are also mutually
disjoint.
Let C 1 be the cyclic code generated by the product of the minimal polynomials
M
(0)
(x)M
(1)
(x) · . . . · M
(r −2)
(x),
and C 2 be the cyclic code generated by g 2 (x), that is the product of the minimal
polynomials
g 2 (x) =
i
M
(i)
(x),
where i /
∈ {r, r + 1, . . . , 2r − 2} and i runs through the coset representatives
mod n. From construction it follows that C 2 C 1 . From the BCH bound, the minimum distance of C 1 is greater than or equal to r , because its defining set contains
the sequence 0, 1, . . . , r − 2, of r − 1 consecutive integers. Similarly, the defining
set of the code C generated by the polynomial h(x) =
x
n −1
g 2 (x)
contains the sequence
r, r + 1, . . . , 2r − 2, of r − 1 consecutive integers and so, from the BCH bound, C