6.3 Asymmetric BCH Codes
129
families are quite different from the ones shown in [105]. Additionally, the lower
bounds for the minimum distances d z and d x of codes displayed in [105] are the
same, whereas we here construct AQECCs where the lower bound for d z is greater
than the lower bound for d x . In other words, in the proposed codes one has d z more
large than d x . This fact allows us to generate quantum codes capable of correcting
quantum errors with great asymmetry.
To compare the parameters of the asymmetric quantum BCH codes constructed
here with the ones shown in the literature, we utilize the usual criterion: for fixed
values of the code length n, and for fixed values of the lower bounds for d z and d x ,
our codes achieve greater values of the number of qudits than the ones available in
the literature.
The asymmetric quantum BCH codes have parameters given by
(i) [[n, n − m(2c − l − 4) − 2, d z ≥ c/d x ≥ (c − l)]] q , 2 ≤ c ≤ q and 0 ≤ l ≤
c − 2;
(ii) [[n, n − m(2c − l − 6) − 2, d z ≥ c/d x ≥ (c − l)]] q , q + 2 < c ≤ 2q and 0 ≤
l ≤ c − q − 3;
(iii) [[n, n − m(4q − l − 5) − 1, d z ≥ (2q + 1)/d x ≥ (2q − l)]] q , 0 ≤ l ≤ q − 2;
(iv) [[n, n − m(4q − l − 5) − 2, d z ≥ (2q + 2)/d x ≥ (2q − l)]] q , 0 ≤ l ≤ q − 2,
where q is an odd prime power and n = q
m
− 1.
Notation. We always assume that q is an odd prime power, n = q
m
− 1 is the code
length, F q denotes the finite field with q elements, α denotes a primitive element
of F q m , M
( j)
(x) denotes the minimal polynomial of α
j
∈ F q m , the congruence ≡
is considered modulo n (mod n), CSS(C 1 , C 2 ) denotes the asymmetric CSS code
derived from two distinct classical linear codes C 1 and C 2 , C
⊥ denotes the Euclidean
dual code of a code C and C [a] denotes the cyclotomic coset containing a, where a
is not necessarily the smallest number in the coset C [a] .
The following lemma will be applied in the proposed construction:
Lemma 6.3.1 ([4, Lemmas 8 and 9]) Let n ≥ 1 be an integer and q be a power
of a prime such that gcd(n, q) = 1 and q
m/2
< n ≤ q
m
− 1, where m = or d n (q)
denotes the multiplicative order of q modulo n. Then the coset C x = {xq
j mod n |
0 ≤ j < m} has cardinality m for all x in the range 1 ≤ x ≤ nq
m/2
/(q
m
− 1).
Moreover, if x and y are distinct integers in the range 1 ≤ x, y ≤ min{{nq
m/2
/
(q
m
− 1) − 1, n − 1} such that the congruence x, y ≡ 0 mod q does not hold, then
the q-ary cyclotomic cosets of x and y modulo n are distinct.
6.3.1 Code Constructions
The main results presented in this subsection are Theorems 6.3.1 and 6.3.2 and Corollary 6.3.1. They provide several families of asymmetric quantum codes derived from
BCH codes. In order to obtain good codes, the main idea subjacent Theorem 6.3.1 is
129
families are quite different from the ones shown in [105]. Additionally, the lower
bounds for the minimum distances d z and d x of codes displayed in [105] are the
same, whereas we here construct AQECCs where the lower bound for d z is greater
than the lower bound for d x . In other words, in the proposed codes one has d z more
large than d x . This fact allows us to generate quantum codes capable of correcting
quantum errors with great asymmetry.
To compare the parameters of the asymmetric quantum BCH codes constructed
here with the ones shown in the literature, we utilize the usual criterion: for fixed
values of the code length n, and for fixed values of the lower bounds for d z and d x ,
our codes achieve greater values of the number of qudits than the ones available in
the literature.
The asymmetric quantum BCH codes have parameters given by
(i) [[n, n − m(2c − l − 4) − 2, d z ≥ c/d x ≥ (c − l)]] q , 2 ≤ c ≤ q and 0 ≤ l ≤
c − 2;
(ii) [[n, n − m(2c − l − 6) − 2, d z ≥ c/d x ≥ (c − l)]] q , q + 2 < c ≤ 2q and 0 ≤
l ≤ c − q − 3;
(iii) [[n, n − m(4q − l − 5) − 1, d z ≥ (2q + 1)/d x ≥ (2q − l)]] q , 0 ≤ l ≤ q − 2;
(iv) [[n, n − m(4q − l − 5) − 2, d z ≥ (2q + 2)/d x ≥ (2q − l)]] q , 0 ≤ l ≤ q − 2,
where q is an odd prime power and n = q
m
− 1.
Notation. We always assume that q is an odd prime power, n = q
m
− 1 is the code
length, F q denotes the finite field with q elements, α denotes a primitive element
of F q m , M
( j)
(x) denotes the minimal polynomial of α
j
∈ F q m , the congruence ≡
is considered modulo n (mod n), CSS(C 1 , C 2 ) denotes the asymmetric CSS code
derived from two distinct classical linear codes C 1 and C 2 , C
⊥ denotes the Euclidean
dual code of a code C and C [a] denotes the cyclotomic coset containing a, where a
is not necessarily the smallest number in the coset C [a] .
The following lemma will be applied in the proposed construction:
Lemma 6.3.1 ([4, Lemmas 8 and 9]) Let n ≥ 1 be an integer and q be a power
of a prime such that gcd(n, q) = 1 and q
m/2
< n ≤ q
m
− 1, where m = or d n (q)
denotes the multiplicative order of q modulo n. Then the coset C x = {xq
j mod n |
0 ≤ j < m} has cardinality m for all x in the range 1 ≤ x ≤ nq
m/2
/(q
m
− 1).
Moreover, if x and y are distinct integers in the range 1 ≤ x, y ≤ min{{nq
m/2
/
(q
m
− 1) − 1, n − 1} such that the congruence x, y ≡ 0 mod q does not hold, then
the q-ary cyclotomic cosets of x and y modulo n are distinct.
6.3.1 Code Constructions
The main results presented in this subsection are Theorems 6.3.1 and 6.3.2 and Corollary 6.3.1. They provide several families of asymmetric quantum codes derived from
BCH codes. In order to obtain good codes, the main idea subjacent Theorem 6.3.1 is