144
6 Asymmetric Quantum Codes
Let us recall the following result on tensor product of cyclic codes.
Theorem 6.6.1 ([19, Chap. 10.2]) Let C 1 = [n 1 , k 1 , d 1 ] q and C 2 = [n 2 , k 2 , d 2 ] q be
cyclic codes with generator polynomial g 1 (x) and g 2 (y), respectively. Then the code
C 1 ⊗ C 2 is a bicyclic code generate by g 1 (x)g 2 (y). The codewords of C 1 ⊗ C 2 correspond to all polynomials of two variables of the form c(x, y) = l(x, y)g 1 (x)g 2 (y)
modulo de ideal generated by X
n 1 − 1 and Y
n 2 − 1, where l(x, y) ∈ F q [x, y]. The
Euclidean dual (C 1 ⊗ C 2 )
⊥ of C 1 ⊗ C 2 consists of all polynomials that are multiple
of h 1 (x) or h 2 (y), where h 1 (x) and h 2 (y) are the generator polynomials of C
⊥
1 and
C
⊥
2 , respectively.
The next result can be found in [57] (see Theorem 8).
Theorem 6.6.2 The product code C 1 ⊗ C 2 of two Reed–Solomon codes C 1 =
[q − 1, q − δ 1 , δ 1 ] q and C 2 = [q − 1, q − δ 2 , δ 2 ] q over F q is an [(q − 1)
2
, (q −
δ 1 )(q − δ 2 ), δ 1 δ 2 ] q code. The Euclidean dual (C 1 ⊗ C 2 )
⊥ of C 1 ⊗ C 2 has parameters
[(q − 1)
2
, k
⊥
, d
⊥
] q , where k
⊥
= q(δ 1 + δ 2 − 2) − δ 1 δ 2 + 1, and d
⊥
= min{q −
δ 1 , q − δ 2 }, where δ 1 and δ 2 are the minimum distances of C 1 and C 2 , respectively.
Lemmas 6.6.1 establishes conditions under which product codes are nested.
Lemma 6.6.1 Let C 1 , C 2 , C 3 and C 4 be four cyclic codes over F q of same length
such that the inclusions C 2 ⊂ C 1 and C 4 ⊂ C 3 hold. Then the product code C 1 ⊗ C 3
contains the product code C 2 ⊗ C 4 .
Proof Let C 1 , C 2 , C 3 and C 4 be cyclic code generated by the polynomials g 1 (x),
g 2 (x), g 3 (y) and g 4 (y), respectively. Since C 2 ⊂ C 1 , then there exists a polynomial
a(x) such that C 2 is generated by a(x)g 1 (x), i.e., g 2 (x) = a(x)g 1 (x). Analogously, as
C 4 ⊂ C 3 , t follows that g 4 (y) = b(y)g 3 (y). From Theorem 6.6.1, the code C 2 ⊗ C 4
is bicyclic code generated by the polynomial p(x, y) = g 2 (x)g 4 (y), i.e., p(x, y) =
a(x)g 1 (x)b(y)g 3 (y). Furthermore, applying again Theorem 6.6.1 we can construct a
bicyclic code C 1 ⊗ C 3 generated by the polynomial q(x, y) = g 1 (x)g 3 (x). Because
q(x, y)| p(x, y), we have C 2 ⊗ C 4 ⊂ C 1 ⊗ C 3 . The proof is complete.
As an immediate consequence of the previous lemma, we can conclude the following.
Corollary 6.6.1 Let C 1 and C 2 be cyclic codes over F q of same length such that
C 2 ⊂ C 1 . Then the inclusion C 2 ⊗ C 2 ⊂ C 1 ⊗ C 1 holds.
In Theorem 6.6.3 we construct asymmetric quantum product codes.
Theorem 6.6.3 There exists an
[[(q − 1)
2
, (q − d 1 )(q − d 3 ) − (q − d 2 )(q − d 4 ), d z /d x ]] q
asymmetric quantum code, where d 1 , d 2 , d 3 and d 4 are integers satisfying the inequalities 2 ≤ d 1 ≤ d 2 < q − 1, 2 ≤ d 3 ≤ d 4 < q − 1 and d z ≥ max{d 1 d 3 , min{q − d 2 ,
q − d 4 }}, and d x ≥ min{d 1 d 3 , min{q − d 2 , q − d 4 }}.
6 Asymmetric Quantum Codes
Let us recall the following result on tensor product of cyclic codes.
Theorem 6.6.1 ([19, Chap. 10.2]) Let C 1 = [n 1 , k 1 , d 1 ] q and C 2 = [n 2 , k 2 , d 2 ] q be
cyclic codes with generator polynomial g 1 (x) and g 2 (y), respectively. Then the code
C 1 ⊗ C 2 is a bicyclic code generate by g 1 (x)g 2 (y). The codewords of C 1 ⊗ C 2 correspond to all polynomials of two variables of the form c(x, y) = l(x, y)g 1 (x)g 2 (y)
modulo de ideal generated by X
n 1 − 1 and Y
n 2 − 1, where l(x, y) ∈ F q [x, y]. The
Euclidean dual (C 1 ⊗ C 2 )
⊥ of C 1 ⊗ C 2 consists of all polynomials that are multiple
of h 1 (x) or h 2 (y), where h 1 (x) and h 2 (y) are the generator polynomials of C
⊥
1 and
C
⊥
2 , respectively.
The next result can be found in [57] (see Theorem 8).
Theorem 6.6.2 The product code C 1 ⊗ C 2 of two Reed–Solomon codes C 1 =
[q − 1, q − δ 1 , δ 1 ] q and C 2 = [q − 1, q − δ 2 , δ 2 ] q over F q is an [(q − 1)
2
, (q −
δ 1 )(q − δ 2 ), δ 1 δ 2 ] q code. The Euclidean dual (C 1 ⊗ C 2 )
⊥ of C 1 ⊗ C 2 has parameters
[(q − 1)
2
, k
⊥
, d
⊥
] q , where k
⊥
= q(δ 1 + δ 2 − 2) − δ 1 δ 2 + 1, and d
⊥
= min{q −
δ 1 , q − δ 2 }, where δ 1 and δ 2 are the minimum distances of C 1 and C 2 , respectively.
Lemmas 6.6.1 establishes conditions under which product codes are nested.
Lemma 6.6.1 Let C 1 , C 2 , C 3 and C 4 be four cyclic codes over F q of same length
such that the inclusions C 2 ⊂ C 1 and C 4 ⊂ C 3 hold. Then the product code C 1 ⊗ C 3
contains the product code C 2 ⊗ C 4 .
Proof Let C 1 , C 2 , C 3 and C 4 be cyclic code generated by the polynomials g 1 (x),
g 2 (x), g 3 (y) and g 4 (y), respectively. Since C 2 ⊂ C 1 , then there exists a polynomial
a(x) such that C 2 is generated by a(x)g 1 (x), i.e., g 2 (x) = a(x)g 1 (x). Analogously, as
C 4 ⊂ C 3 , t follows that g 4 (y) = b(y)g 3 (y). From Theorem 6.6.1, the code C 2 ⊗ C 4
is bicyclic code generated by the polynomial p(x, y) = g 2 (x)g 4 (y), i.e., p(x, y) =
a(x)g 1 (x)b(y)g 3 (y). Furthermore, applying again Theorem 6.6.1 we can construct a
bicyclic code C 1 ⊗ C 3 generated by the polynomial q(x, y) = g 1 (x)g 3 (x). Because
q(x, y)| p(x, y), we have C 2 ⊗ C 4 ⊂ C 1 ⊗ C 3 . The proof is complete.
As an immediate consequence of the previous lemma, we can conclude the following.
Corollary 6.6.1 Let C 1 and C 2 be cyclic codes over F q of same length such that
C 2 ⊂ C 1 . Then the inclusion C 2 ⊗ C 2 ⊂ C 1 ⊗ C 1 holds.
In Theorem 6.6.3 we construct asymmetric quantum product codes.
Theorem 6.6.3 There exists an
[[(q − 1)
2
, (q − d 1 )(q − d 3 ) − (q − d 2 )(q − d 4 ), d z /d x ]] q
asymmetric quantum code, where d 1 , d 2 , d 3 and d 4 are integers satisfying the inequalities 2 ≤ d 1 ≤ d 2 < q − 1, 2 ≤ d 3 ≤ d 4 < q − 1 and d z ≥ max{d 1 d 3 , min{q − d 2 ,
q − d 4 }}, and d x ≥ min{d 1 d 3 , min{q − d 2 , q − d 4 }}.