5.1 BCH Codes—Part I
65
and has parameters [31, 25, d ≥ 4] 5 . From Lemma 5.1.3, it is easy to check that C is
Euclidean dual-containing. Furthermore, C can be enlarged to a code C
with parameters [31, 28, d ≥ 3] 5 , whose generator polynomial is M
(8)
(x). Applying Corollary 5.1.2 to C and C
, we obtain an [[31, 22, d ≥ 4]] 5 quantum code.
Theorem 5.1.5 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-ary coset containing s and s + 1 and let Z =
C [s] ∪ C [s+2] , where C [s] = C [s+2] . Assume also that Z ∩ Z
−1
= ∅ holds. Then there
exists an [[n, n − 3m, d ≥ 4]] q code.
Proof We know that gcd(q, n) = 1. Let C be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x). By hypothesis and from Lemma 5.1.3, we know that C is
Euclidean dual-containing; C has parameters [n, n − 2m, d ≥ 4] q . Let C
be the
cyclic code generated by M
(s)
(x). We know that C
is an enlargement of C and it has
parameters [n, n − m, d ≥ 3] q . Since m ≥ 2, it follows that k
− k = m ≥ 2, where
k
is the dimension of C
and k is the dimension of C. Applying the Steane code
construction to C and C
, since
q+1
q
> 1, we get an [[n, n − 3m, d ≥ 4]] q quantum
code.
Theorem 5.1.5 can be generalized in the following way.
Theorem 5.1.6 Assume that q ≥ 3 is a prime power, n > q is a prime number
and consider that m = ord n (q) ≥ 2. Let C [s] be the coset containing s and s +
1. Assume that Z = C [s] ∪ C [s+2] ∪ . . . ∪ C [s+r ] , where all the q-cosets C [s+i] , i =
0, 2, 3, . . . , r , are mutually disjoint. Assume also that Z ∩ Z
−1
= ∅. Then there exists
an [[n, n − m(2r − 1), d ≥ r + 2]] q quantum code.
Proof We know that gcd(q, n) = 1. Let C be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r )
(x).
Since Z ∩ Z
−1
= ∅, it follows from Lemma 5.1.3 that C is Euclidean dual-containing.
From the hypotheses, all the q-ary cosets C [s] , C [s+2] , . . . , C [s+r ] are mutually disjoint; hence, C has dimension k = n − mr and its minimum distance is lower
bounded by d ≥ r + 2, i.e., C is an [n, n − mr, d ≥ r + 2] q code. Let C
be the
cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r −1)
(x).
We know that C
is an enlargement of C and it has parameters [n, n − m(r − 1),
d ≥ r + 1] q . Since m ≥ 2, we have k
− k = m ≥ 2, where k
is the dimension of
C
and k is the dimension of C. Applying the Steane’s construction to C and C
we
obtain an [[n, n − m(2r − 1), d ≥ r + 2]] q code, as required.
Example 5.1.4 In this example we construct an [[31, 16, d ≥ 5]] 5 quantum code.
For this purpose we take n = 31 and q = 5; then m = ord n (q) = 3. Let C be the
65
and has parameters [31, 25, d ≥ 4] 5 . From Lemma 5.1.3, it is easy to check that C is
Euclidean dual-containing. Furthermore, C can be enlarged to a code C
with parameters [31, 28, d ≥ 3] 5 , whose generator polynomial is M
(8)
(x). Applying Corollary 5.1.2 to C and C
, we obtain an [[31, 22, d ≥ 4]] 5 quantum code.
Theorem 5.1.5 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-ary coset containing s and s + 1 and let Z =
C [s] ∪ C [s+2] , where C [s] = C [s+2] . Assume also that Z ∩ Z
−1
= ∅ holds. Then there
exists an [[n, n − 3m, d ≥ 4]] q code.
Proof We know that gcd(q, n) = 1. Let C be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x). By hypothesis and from Lemma 5.1.3, we know that C is
Euclidean dual-containing; C has parameters [n, n − 2m, d ≥ 4] q . Let C
be the
cyclic code generated by M
(s)
(x). We know that C
is an enlargement of C and it has
parameters [n, n − m, d ≥ 3] q . Since m ≥ 2, it follows that k
− k = m ≥ 2, where
k
is the dimension of C
and k is the dimension of C. Applying the Steane code
construction to C and C
, since
q+1
q
> 1, we get an [[n, n − 3m, d ≥ 4]] q quantum
code.
Theorem 5.1.5 can be generalized in the following way.
Theorem 5.1.6 Assume that q ≥ 3 is a prime power, n > q is a prime number
and consider that m = ord n (q) ≥ 2. Let C [s] be the coset containing s and s +
1. Assume that Z = C [s] ∪ C [s+2] ∪ . . . ∪ C [s+r ] , where all the q-cosets C [s+i] , i =
0, 2, 3, . . . , r , are mutually disjoint. Assume also that Z ∩ Z
−1
= ∅. Then there exists
an [[n, n − m(2r − 1), d ≥ r + 2]] q quantum code.
Proof We know that gcd(q, n) = 1. Let C be the cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r )
(x).
Since Z ∩ Z
−1
= ∅, it follows from Lemma 5.1.3 that C is Euclidean dual-containing.
From the hypotheses, all the q-ary cosets C [s] , C [s+2] , . . . , C [s+r ] are mutually disjoint; hence, C has dimension k = n − mr and its minimum distance is lower
bounded by d ≥ r + 2, i.e., C is an [n, n − mr, d ≥ r + 2] q code. Let C
be the
cyclic code generated by
M
(s)
(x)M
(s+2)
(x) · . . . · M
(s+r −1)
(x).
We know that C
is an enlargement of C and it has parameters [n, n − m(r − 1),
d ≥ r + 1] q . Since m ≥ 2, we have k
− k = m ≥ 2, where k
is the dimension of
C
and k is the dimension of C. Applying the Steane’s construction to C and C
we
obtain an [[n, n − m(2r − 1), d ≥ r + 2]] q code, as required.
Example 5.1.4 In this example we construct an [[31, 16, d ≥ 5]] 5 quantum code.
For this purpose we take n = 31 and q = 5; then m = ord n (q) = 3. Let C be the