5.1 BCH Codes—Part I
69
to r , that is, C is an [n, n − 2(r − 2) − 1, d ≥ r ] q 2 code. Applying the Hermitian
construction to the code C, one can get an [[n, n − 4(r − 2) − 2, d ≥ r ]] q code, as
desired.
Corollary 5.1.3 Suppose q > 3 is a prime power and n > q
2 is an integer such that
gcd(q
2
, n) = 1. Assume also (q
2
− 1) | n and m = ord n (q
2
) = 2. Then there exist
quantum codes with parameters [[n, n − 4c − 2, d ≥ c + 2]] q , where 2 ≤ c < r − 2
and n = r (q
2
− 1).
Proof Let C be the BCH code generated by
M
(r )
(x)M
(r +1)
(x) · . . . · M
(r +c)
(x).
Proceeding similarly as in the proof of Theorem 5.1.8, the result follows.
Theorem 5.1.9 Let q ≥ 3 be a prime power, n > q
2 be a prime number and consider that m = ord n (q
2
) ≥ 2. Let C [s] be the q-coset containing s and s + 1.
Assume that Z = C [s] ∪ C [s+2] ∪ . . . ∪ C [s+r ] , where all the q-ary cosets C [s+i] ,
i = 0, 2, 3, . . . , r , are mutually disjoint, and suppose that Z ∩ Z
−q
= ∅. Then there
exists an [[n, n − 2mr, d ≥ r + 2]] q quantum code.
Proof We know that gcd(q, n) = 1 holds. Let C be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r )
(x).
Since Z ∩ Z
−q
= ∅ holds, it follows from Lemma 5.1.4 that C is Hermitian
dual-containing. From the BCH bound, the minimum distance of C is greater
than or equal to r + 2. It is easy to see that all the cosets C [s+i] , where i =
0, 2, 3, . . . , r , have m elements and they are mutually disjoint. Thus C has parameters [n, n − mr, d ≥ r + 2] q 2 . Applying the Hermitian construction one can get an
[[n, n − 2mr, d ≥ r + 2]] q quantum code.
We finish this subsection by showing how Lemma 5.1.2 works for constructing
quantum MDS-BCH codes.
Example 5.1.6 Let us consider q = 5 and n = 13. Since gcd(13, 24) = 1, the
linear congruence (q
2
− 1)x ≡ 1 mod n has a solution, so there exists at least
one q
2 -ary coset containing two consecutive integers, namely, the coset C [6] =
{6, 7}. Let C = =M
(6)
(x). Since C [4] and C [6] are disjoint, C is Hermitian dualcontaining and has parameters [13, 11, d ≥ 3] 5 . Applying the Hermitian construction, an [[13, 9, 3]] 5 quantum MDS-BCH code is constructed. Similarly, we can also
construct an [[17, 13, 3]] 4 and an [[17, 9, 5]] 4 quantum MDS-BCH code.
69
to r , that is, C is an [n, n − 2(r − 2) − 1, d ≥ r ] q 2 code. Applying the Hermitian
construction to the code C, one can get an [[n, n − 4(r − 2) − 2, d ≥ r ]] q code, as
desired.
Corollary 5.1.3 Suppose q > 3 is a prime power and n > q
2 is an integer such that
gcd(q
2
, n) = 1. Assume also (q
2
− 1) | n and m = ord n (q
2
) = 2. Then there exist
quantum codes with parameters [[n, n − 4c − 2, d ≥ c + 2]] q , where 2 ≤ c < r − 2
and n = r (q
2
− 1).
Proof Let C be the BCH code generated by
M
(r )
(x)M
(r +1)
(x) · . . . · M
(r +c)
(x).
Proceeding similarly as in the proof of Theorem 5.1.8, the result follows.
Theorem 5.1.9 Let q ≥ 3 be a prime power, n > q
2 be a prime number and consider that m = ord n (q
2
) ≥ 2. Let C [s] be the q-coset containing s and s + 1.
Assume that Z = C [s] ∪ C [s+2] ∪ . . . ∪ C [s+r ] , where all the q-ary cosets C [s+i] ,
i = 0, 2, 3, . . . , r , are mutually disjoint, and suppose that Z ∩ Z
−q
= ∅. Then there
exists an [[n, n − 2mr, d ≥ r + 2]] q quantum code.
Proof We know that gcd(q, n) = 1 holds. Let C be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r )
(x).
Since Z ∩ Z
−q
= ∅ holds, it follows from Lemma 5.1.4 that C is Hermitian
dual-containing. From the BCH bound, the minimum distance of C is greater
than or equal to r + 2. It is easy to see that all the cosets C [s+i] , where i =
0, 2, 3, . . . , r , have m elements and they are mutually disjoint. Thus C has parameters [n, n − mr, d ≥ r + 2] q 2 . Applying the Hermitian construction one can get an
[[n, n − 2mr, d ≥ r + 2]] q quantum code.
We finish this subsection by showing how Lemma 5.1.2 works for constructing
quantum MDS-BCH codes.
Example 5.1.6 Let us consider q = 5 and n = 13. Since gcd(13, 24) = 1, the
linear congruence (q
2
− 1)x ≡ 1 mod n has a solution, so there exists at least
one q
2 -ary coset containing two consecutive integers, namely, the coset C [6] =
{6, 7}. Let C = =M
(6)
(x). Since C [4] and C [6] are disjoint, C is Hermitian dualcontaining and has parameters [13, 11, d ≥ 3] 5 . Applying the Hermitian construction, an [[13, 9, 3]] 5 quantum MDS-BCH code is constructed. Similarly, we can also
construct an [[17, 13, 3]] 4 and an [[17, 9, 5]] 4 quantum MDS-BCH code.