7.6 QCCs from AG Codes
195
Lemma 7.6.1 ([6, Props. 1 and 2]) Let C be an (n, (n − k)/2, γ; m) q 2 convolutional code satisfying C ⊆ C
⊥ h (respectively, C ⊆ C
⊥ , where C is an (n, (n −
k)/2, γ; m) q ). There there exists an [(n, k, m; γ, d f )] q quantum convolutional code,
where d f = wt(C
⊥ h \ C) (respectively, d f = wt(C
⊥
\ C)). If does not exist errors
of weight less than d f in the stabilizer of C, then C is said to be pure.
Theorem 7.6.8 Let C L (D, G) be an one-point self-orthogonal code (as given in
Definition 4.5.11) with parameters [n, k, d] q and generator matrix M. If d 1 is the minimum distance of the code with parity check matrix ˜
M 1 and C L (D, G
∗
) is an AG code
generated by the divisor G
∗ , where dim(L(G
∗
)) = k − l, then there exists an [(n, n −
2(k − l), 1; l, d f )] QCC, where d f ≥ min{d(C L (D, G
∗
)) + d 1 , d(C (D, G))}. In
particular, if 2g − 2 ≤ deg(G) ≤ n/2 + g, then one has d f ≥ deg(G) − (2g − 2).
Proof Let us consider S(Q) = {0 = ρ 1 < ρ 2 < · · · } be the Weierstrass semigroup
of Q. We construct the Weierstrass basis of the Riemann–Roch space L(r Q), where
r ≥ 0, with dimension l(r Q) = k as follows. The first element of the basis of L(r Q)
is a vector x 1 ∈ F/F q such that ν Q (x 1 ) = ρ 1 = 0. The choice for the second vector
follows the same idea: we take a vector x 2 ∈ F/F q with ν Q (x 2 ) = ρ 2 , and so on. From
construction, it follows that each of such vectors has different valuation in the place
Q; hence, the set {x 1 , x 2 , . . . , x k } contains k LI vectors. Additionally, these vectors
belong to the space L(r Q). In other words, {x 1 , x 2 , . . . , x k } is a basis of L(r Q) (called
a Weierstrass basis of L(r Q)). Hence, the set {x 1 , x 2 , . . . , x k−1 } is also a basis of a
Riemann–Roch space L(r
∗ Q), where L(r
∗ Q) ⊂ L(r Q) and l(r
∗ Q) = k − 1.
Let C L (D, G) be an [n, k, d] q one-point self-orthogonal code (as in Definition 4.5.11) with generator matrix M with lines {m 1 , . . . , m k }. Let {x 1 , . . . , x k } be
the basis of L(G) constructed above. Applying Theorem 7.1.1, the corresponding
convolutional code is also self-orthogonal and it has parameters (n, k − l, l; 1, d f ≥
d(C L (D, G))) q . Applying Lemma 7.6.1 for this code and since the set of vectors
{x 1 , . . . , x k−l } generates another Riemann–Roch space associated with the divisor
G
∗
= ρ k−l Q, where ρ k−l is the (k − l)th element of S(Q), we obtain an [(n, n −
2(k − l), 1; l, d f )] QCC, where d f ≥ min{d(C L (D, G
∗
)) + d 1 , d(C (D, G))}.
Remark 7.6.2 It is interesting to observe that our construction can be utilized over
any curve that satisfies the conditions of Definition 4.5.11.
Remark 7.6.3 Note that, in Theorem 7.6.8, by applying the generalized quantum
Singleton bound, it follows that the free distance of the QCCs previously constructed
are bounded by d f ≤ deg(G) − g + 2 (where 2g − 2 ≤ deg(G) ≤ n/2 + g). Further, d f ≥ deg(G) − 2g + 2; then the free distance d f is bounded by deg(G) −
2g + 2 ≤ d f ≤ deg(G) − g + 2. In particular, for function fields F/F q with g = 0,
the new QCCs are MDS. In the other cases, the generalized quantum Singleton
defect of the QCCs is at most equal to the genus of the curve utilized to construct
the corresponding AG code.
Let us see how to obtain optimal QCCs.