194
7 Constructions of QCCs
Then there exists an (n − 1, k − r, r ; 1, d f ) q CC, where d f ≥ d, k = deg(G) + 1 −
g and d ≥ n − deg(G).
Proof Let C L (D, G) be the AG with parameters [n, k, d] q considered in Theorem 7.6.1, where D = P 1 + · · · + P n . Assume that D
∗
= D − P j , where j ∈
{1, 2, . . . , n}. Define the puncture code C L (D
∗
, G) derived from C L (D, G) which
is also an AG code (see [126]). Since the supports of D
∗ and G are disjoint, the
definition of the code C L (D
∗
, G) makes sense. We know that C L (D
∗
, G) is an
[n − 1, k, d] q . Proceeding similarly as in the proof of Theorem 7.6.1, one has an
(n − 1, k − r, r ; 1, d f ) q CC. The proof is complete.
Theorem 7.6.5 Assume the same notation of Theorem 7.6.1. Then there exists an
(n + 1, k − r, r ; 1, d f ≥ d
e
) q CC, where d
e
= d or d
e
= d + 1, k = deg(G) + 1 −
g, r ≤ k/2 and d ≥ n − deg(G).
Proof Let C L (D, G) be the AG with parameters [n, k, d] q considered in Theorem 7.6.1. We construct a new code C
e
L (D, G) by extending C L (D, G); this new
code is an [n, k, d
e
] q , where d
e is specified in the hypotheses. Proceeding similarly
as in the proof of Theorem 7.6.1, we have the code.
Applying code expansion we have more families of CCs.
Theorem 7.6.6 Assume the same notation of Theorem 7.6.1, where C L (D, G) is
an AG code over F q r . Then there exists an (rn, rk − t, t; 1, d f ≥ d) q CC, where
k = deg(G) + 1 − g, t ≤ k/2 and d ≥ n − deg(G).
Proof The proof is left to the reader.
Theorem 7.6.7 Assume the same notation of Theorem 7.6.1. Then there exists an
(n
2
, k
2
− r, r ; 1, d f ≥ d
2
) q CC, where k = deg(G) + 1 − g, r ≤ k/2 and d ≥ n −
deg(G).
Proof Left to the reader.
7.6.2 Quantum Convolutional AG Codes
Fixing the Notation. As usual p denotes a prime number, q is a prime power, F q is
the finite field with q elements and F/F q denotes an algebraic function field over F q
of genus g.
In this subsection, we construct several families of quantum convolutional codes
derived from AG codes. Additionally, the family of QCCs constructed in Theorems 7.6.9 consists of maximum-distance-separable (MDS) codes.
In our construction, we need to utilize the following result from [6].
7 Constructions of QCCs
Then there exists an (n − 1, k − r, r ; 1, d f ) q CC, where d f ≥ d, k = deg(G) + 1 −
g and d ≥ n − deg(G).
Proof Let C L (D, G) be the AG with parameters [n, k, d] q considered in Theorem 7.6.1, where D = P 1 + · · · + P n . Assume that D
∗
= D − P j , where j ∈
{1, 2, . . . , n}. Define the puncture code C L (D
∗
, G) derived from C L (D, G) which
is also an AG code (see [126]). Since the supports of D
∗ and G are disjoint, the
definition of the code C L (D
∗
, G) makes sense. We know that C L (D
∗
, G) is an
[n − 1, k, d] q . Proceeding similarly as in the proof of Theorem 7.6.1, one has an
(n − 1, k − r, r ; 1, d f ) q CC. The proof is complete.
Theorem 7.6.5 Assume the same notation of Theorem 7.6.1. Then there exists an
(n + 1, k − r, r ; 1, d f ≥ d
e
) q CC, where d
e
= d or d
e
= d + 1, k = deg(G) + 1 −
g, r ≤ k/2 and d ≥ n − deg(G).
Proof Let C L (D, G) be the AG with parameters [n, k, d] q considered in Theorem 7.6.1. We construct a new code C
e
L (D, G) by extending C L (D, G); this new
code is an [n, k, d
e
] q , where d
e is specified in the hypotheses. Proceeding similarly
as in the proof of Theorem 7.6.1, we have the code.
Applying code expansion we have more families of CCs.
Theorem 7.6.6 Assume the same notation of Theorem 7.6.1, where C L (D, G) is
an AG code over F q r . Then there exists an (rn, rk − t, t; 1, d f ≥ d) q CC, where
k = deg(G) + 1 − g, t ≤ k/2 and d ≥ n − deg(G).
Proof The proof is left to the reader.
Theorem 7.6.7 Assume the same notation of Theorem 7.6.1. Then there exists an
(n
2
, k
2
− r, r ; 1, d f ≥ d
2
) q CC, where k = deg(G) + 1 − g, r ≤ k/2 and d ≥ n −
deg(G).
Proof Left to the reader.
7.6.2 Quantum Convolutional AG Codes
Fixing the Notation. As usual p denotes a prime number, q is a prime power, F q is
the finite field with q elements and F/F q denotes an algebraic function field over F q
of genus g.
In this subsection, we construct several families of quantum convolutional codes
derived from AG codes. Additionally, the family of QCCs constructed in Theorems 7.6.9 consists of maximum-distance-separable (MDS) codes.
In our construction, we need to utilize the following result from [6].