62
5 Quantum Code Constructions
Example 5.1.1 As an example, let us consider that q = 9 and n = 40; then
gcd(9, 40) = 1, 8 | 40 and or d 40 (9) = 2. In this case we have r = 5. Theorem 5.1.1
asserts the existence of a quantum code with parameters [[40, 26, d ≥ 5]] 9 . Consider next q = 11 and n = 30. Let C 1 be the cyclic code generated by the product of the minimal polynomials M
(0)
(x)M
(1)
(x) . . . M
(6)
(x) and C 2 be the cyclic
code generated by the product of the minimal polynomials
i
M
(i)
(x), where
i /
∈ {7, 10, 15, 16, 18, 19, 21} and i runs through the coset representatives mod 30.
Proceeding similarly as in the proof of Theorem 5.1.1, an [[30, 8, d ≥ 8]] 11 quantum
code can be constructed.
5.1.2 Construction II
Here the attention is focused on cyclic codes of prime length. Among the contributions exhibited in this subsection, we prove there exists at least one q-ary coset
containing two consecutive integers (see Lemma 5.1.2). In order to proceed further,
let us recall a well-known result from number theory.
Theorem 5.1.2 A linear congruence ax ≡ b (mod m), where a = 0, admits an integer solution if and only if d = gcd(a, m) divides b.
Applying Theorem 5.1.2 we prove Lemma 5.1.2.
Lemma 5.1.2 Assume that q ≥ 3 is a prime power, n > q is a prime number and
consider that m = ord n (q) ≥ 2. Then there exists at least one q-ary coset containing
two consecutive integers.
Proof Note first that gcd(q, n) = 1. In order to prove this lemma, it suffices to
show that the congruence xq ≡ x + 1( mod n) has at least one solution for some
0 ≤ x ≤ n − 1 or, equivalently, the congruence (q − 1)x ≡ 1 (mod n) has at least
one solution. We know that gcd(q − 1, n) = 1, because n > q and n is prime. Since
q − 1 = 0, it follows from Theorem 5.1.2 that (q − 1)x ≡ 1 (mod n) has an integer
solution x 0 . Applying the division algorithm for x 0 and n we have x 0 = ns 0 + r 0 ,
where r 0 and s 0 are integers and 0 ≤ r 0 ≤ n − 1. Since (q − 1)x 0 ≡ 1 (mod n)
holds then the congruence (q − 1)r 0 ≡ 1 (mod n) also holds. Therefore, the result
follows.
Remark 5.1.1 Note that in Lemma 5.1.2 it is not necessary to assume that n
is a prime number. In fact, we need only to suppose that gcd(q − 1, n) = 1 and
gcd(q, n) = 1 hold. However, since the corresponding q-ary cosets of BCH codes
of prime length have nice properties, we have assumed that n is prime. Nevertheless,
if one assumes that gcd(q − 1, n) = 1 and gcd(q, n) = 1 hold, more good quantum
codes can be constructed.
5 Quantum Code Constructions
Example 5.1.1 As an example, let us consider that q = 9 and n = 40; then
gcd(9, 40) = 1, 8 | 40 and or d 40 (9) = 2. In this case we have r = 5. Theorem 5.1.1
asserts the existence of a quantum code with parameters [[40, 26, d ≥ 5]] 9 . Consider next q = 11 and n = 30. Let C 1 be the cyclic code generated by the product of the minimal polynomials M
(0)
(x)M
(1)
(x) . . . M
(6)
(x) and C 2 be the cyclic
code generated by the product of the minimal polynomials
i
M
(i)
(x), where
i /
∈ {7, 10, 15, 16, 18, 19, 21} and i runs through the coset representatives mod 30.
Proceeding similarly as in the proof of Theorem 5.1.1, an [[30, 8, d ≥ 8]] 11 quantum
code can be constructed.
5.1.2 Construction II
Here the attention is focused on cyclic codes of prime length. Among the contributions exhibited in this subsection, we prove there exists at least one q-ary coset
containing two consecutive integers (see Lemma 5.1.2). In order to proceed further,
let us recall a well-known result from number theory.
Theorem 5.1.2 A linear congruence ax ≡ b (mod m), where a = 0, admits an integer solution if and only if d = gcd(a, m) divides b.
Applying Theorem 5.1.2 we prove Lemma 5.1.2.
Lemma 5.1.2 Assume that q ≥ 3 is a prime power, n > q is a prime number and
consider that m = ord n (q) ≥ 2. Then there exists at least one q-ary coset containing
two consecutive integers.
Proof Note first that gcd(q, n) = 1. In order to prove this lemma, it suffices to
show that the congruence xq ≡ x + 1( mod n) has at least one solution for some
0 ≤ x ≤ n − 1 or, equivalently, the congruence (q − 1)x ≡ 1 (mod n) has at least
one solution. We know that gcd(q − 1, n) = 1, because n > q and n is prime. Since
q − 1 = 0, it follows from Theorem 5.1.2 that (q − 1)x ≡ 1 (mod n) has an integer
solution x 0 . Applying the division algorithm for x 0 and n we have x 0 = ns 0 + r 0 ,
where r 0 and s 0 are integers and 0 ≤ r 0 ≤ n − 1. Since (q − 1)x 0 ≡ 1 (mod n)
holds then the congruence (q − 1)r 0 ≡ 1 (mod n) also holds. Therefore, the result
follows.
Remark 5.1.1 Note that in Lemma 5.1.2 it is not necessary to assume that n
is a prime number. In fact, we need only to suppose that gcd(q − 1, n) = 1 and
gcd(q, n) = 1 hold. However, since the corresponding q-ary cosets of BCH codes
of prime length have nice properties, we have assumed that n is prime. Nevertheless,
if one assumes that gcd(q − 1, n) = 1 and gcd(q, n) = 1 hold, more good quantum
codes can be constructed.