5.1 BCH Codes—Part I
63
Theorem 5.1.3 Let q ≥ 3 be a prime power, n > q be a prime number and consider m = ord n (q) ≥ 2. Suppose also that the q-cosets C [s] and C [−s] are disjoint,
where C [s] is a q-coset containing two consecutive integers. Then there exists an
[[n, n − 2m, d ≥ 3]] q quantum code.
Proof Note that gcd(q, n) = 1. Let C 1 be the code generated by M
(s)
(x) and C 2
generated by
i
M
(i)
(x), where i = −s and i runs through the coset representatives
mod n. It is easy to see that the cosets C [s] and C [−s] contain m elements. Proceeding
similarly as in the proof of Theorem 5.1.1, the result follows.
Theorem 5.1.4 Let q ≥ 3 be a prime power, n > q be a prime and consider that m =
ord n (q) ≥ 2. Let C [s] be the q-coset containing s and s + 1. Suppose also that all
the q-ary cosets C [s] , C [s+2] , . . . , C [s+r ] , C [−s] , C [−s−2] , . . . , C [−s−r ] , are mutually
disjoint. Then there exists a quantum code with parameters [[n, n − 2mr, d ≥ r +
2]] q .
Proof We know that gcd(q, n) = 1 and the coset C [−s] also contains two consecutive
integers, namely, −s − 1 and −s. Let C 1 be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r )
(x),
and let C 2 be the cyclic code generated by the polynomial g 2 (x), that is the product
of the minimal polynomials
g 2 (x) =
j
M
( j)
(x),
where j /
∈ {−s − r, . . . , −s − 2, −s} and j runs through the coset representatives
mod n.
From the BCH bound, the minimum distance of C 1 is greater than or equal to r + 2
because its defining set contains the sequence of r + 1 consecutive integers given
by s, s + 1, s + 2, . . . , s + r . Similarly, the defining set of the code C generated by
the polynomial h 2 (x) = (x
n
− 1)/g 2 (x), contains a sequence of r + 1 consecutive
integers given by −s − r, . . . , −s − 2, −s − 1, −s. Again, from the BCH bound,
C has minimum distance greater than or equal to r + 2. Since C is equivalent to
C
⊥
2 , it follows that C
⊥
2 also has minimum distance greater than or equal to r + 2.
Therefore, the resulting CSS code have minimum distance greater than or equal to
r + 2. If s ∈ [1, n − 1] satisfies gcd(s, n) = 1 then the coset C s has cardinality m.
In fact, if |C s | = c < m it follows that n|s(q
c
− 1), so n|(q
c
− 1), a contradiction.
Thus, since n is prime, each of the cosets C s , where s ∈ [1, n − 1], has cardinality m.
Additionally, from the hypotheses, all the q-ary cosets C [s] , C [s+2] , . . . , C [s+r ] , are
mutually disjoint. Thus C 1 has dimension k 1 = n − mr and C 2 has dimension k 2 =
mr, since there exist r disjoint q-cosets not contained in the defining set of C 2 , where
each of them has cardinality m. Therefore, the corresponding CSS code has dimension
K = n − 2mr. Since the cosets C [s] , C [s+2] , . . . , C [s+r ] , C [−s] , C [−s−2] , . . . , C [−s−r ] ,
63
Theorem 5.1.3 Let q ≥ 3 be a prime power, n > q be a prime number and consider m = ord n (q) ≥ 2. Suppose also that the q-cosets C [s] and C [−s] are disjoint,
where C [s] is a q-coset containing two consecutive integers. Then there exists an
[[n, n − 2m, d ≥ 3]] q quantum code.
Proof Note that gcd(q, n) = 1. Let C 1 be the code generated by M
(s)
(x) and C 2
generated by
i
M
(i)
(x), where i = −s and i runs through the coset representatives
mod n. It is easy to see that the cosets C [s] and C [−s] contain m elements. Proceeding
similarly as in the proof of Theorem 5.1.1, the result follows.
Theorem 5.1.4 Let q ≥ 3 be a prime power, n > q be a prime and consider that m =
ord n (q) ≥ 2. Let C [s] be the q-coset containing s and s + 1. Suppose also that all
the q-ary cosets C [s] , C [s+2] , . . . , C [s+r ] , C [−s] , C [−s−2] , . . . , C [−s−r ] , are mutually
disjoint. Then there exists a quantum code with parameters [[n, n − 2mr, d ≥ r +
2]] q .
Proof We know that gcd(q, n) = 1 and the coset C [−s] also contains two consecutive
integers, namely, −s − 1 and −s. Let C 1 be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r )
(x),
and let C 2 be the cyclic code generated by the polynomial g 2 (x), that is the product
of the minimal polynomials
g 2 (x) =
j
M
( j)
(x),
where j /
∈ {−s − r, . . . , −s − 2, −s} and j runs through the coset representatives
mod n.
From the BCH bound, the minimum distance of C 1 is greater than or equal to r + 2
because its defining set contains the sequence of r + 1 consecutive integers given
by s, s + 1, s + 2, . . . , s + r . Similarly, the defining set of the code C generated by
the polynomial h 2 (x) = (x
n
− 1)/g 2 (x), contains a sequence of r + 1 consecutive
integers given by −s − r, . . . , −s − 2, −s − 1, −s. Again, from the BCH bound,
C has minimum distance greater than or equal to r + 2. Since C is equivalent to
C
⊥
2 , it follows that C
⊥
2 also has minimum distance greater than or equal to r + 2.
Therefore, the resulting CSS code have minimum distance greater than or equal to
r + 2. If s ∈ [1, n − 1] satisfies gcd(s, n) = 1 then the coset C s has cardinality m.
In fact, if |C s | = c < m it follows that n|s(q
c
− 1), so n|(q
c
− 1), a contradiction.
Thus, since n is prime, each of the cosets C s , where s ∈ [1, n − 1], has cardinality m.
Additionally, from the hypotheses, all the q-ary cosets C [s] , C [s+2] , . . . , C [s+r ] , are
mutually disjoint. Thus C 1 has dimension k 1 = n − mr and C 2 has dimension k 2 =
mr, since there exist r disjoint q-cosets not contained in the defining set of C 2 , where
each of them has cardinality m. Therefore, the corresponding CSS code has dimension
K = n − 2mr. Since the cosets C [s] , C [s+2] , . . . , C [s+r ] , C [−s] , C [−s−2] , . . . , C [−s−r ] ,