192
7 Constructions of QCCs
7.6.1 Convolutional AG Code
In this subsection, we present the constructions of convolutional codes from AG
codes.
Theorem 7.6.1 Assume that F/F q is an algebraic function field of genus g. Consider the AG code C (D, G) shown in Definition 4.5.13. Then there exists an
(n, k − r, r ; 1, d f ≥ d) q unit-memory CC, where r ≤ k/2, k = l(G) − l(G − D)
and d ≥ n − deg(G).
Proof Let C (D, G) be the AG code with parity check matrix
H =
⎡
⎢
⎣
x 1 (P 1 ) x 1 (P 2 ) . . . x 1 (P n )
. . .
. . .
. . .
x k (P 1 ) x k (P 2 ) . . . x k (P n )
⎤
⎥
⎦ ,
where {x 1 , . . . , x k } is a basis of L(G). Let C L (D, G) be the Euclidean dual of
C (D, G). The matrix H is a generator matrix for C L (D, G). We know that
C L (D, G) is an [n, k = l(G) − l(G − D), d ≥ n − deg(G)] q AG code, where n =
deg(D). We define a convolutional code with generator matrix M(D) = H 0 + ˜
H 1 D,
where H 0 is the submatrix of H consisting of the k − r first rows and ˜
H 1 is the matrix
consisting of the last r rows of H by adding zero-rows at the bottom such that the
matrix ˜
H 1 has k − r rows in total. From hypothesis, it follows that rk H 0 ≥ rk ˜
H 1 .
From Theorem 7.1.1, M(D) is reduced and basic matrix. Again, from Theorem 7.1.1,
it follows that the CC generated by M(D) is a unit-memory code with dimension
k − r , degree r and free distance d f ≥ d, and the code follows.
Applying Proposition 4.5.1 in Theorem 7.6.1, we derive more families of CCs:
Corollary 7.6.1 Assume all the hypotheses of Theorem 7.6.1 hold. If 2g − 2 <
deg(G) < n, then there exists an (n, k − r, r ; 1, d f ≥ d) q , where n = deg(D), k =
deg(G) + 1 − g, r ≤ k/2 and d ≥ n − deg(G).
Corollary 7.6.2 Assume all the hypotheses of Theorem 7.6.1 hold. If 2g − 2 <
deg(G) < n, then there exists an (n, k − 1, 1; 1, d f ≥ d) q , where k = deg(G) +
1 − g and d ≥ n − deg(G).
Remark 7.6.1 Applying the generalized Singleton bound to the CCs constructed in
Corollary 7.6.2, it follows the free distance of such codes satisfy d f ≤ n − k + 3,
where n and k are the parameters of C L (D, G). Further, d f ≥ n − k + 1 − g, i.e.,
n − k + 1 − g ≤ d f ≤ n − k + 3. In particular, if the genus of the function fields is
zero, these convolutional codes are almost near MDS or near MDS or MDS, because
the Singleton defect is at most two.
Corollary 7.6.3 Assume that F = F q (z) is a rational function field. For β ∈ F q ,
let P β be the zero of z − β and P ∞ be the pole of z in F q (z). Then there exists an
(q, t, 1; 1, d f ≥ q − t) q , where 1 ≤ t ≤ q − 1.
7 Constructions of QCCs
7.6.1 Convolutional AG Code
In this subsection, we present the constructions of convolutional codes from AG
codes.
Theorem 7.6.1 Assume that F/F q is an algebraic function field of genus g. Consider the AG code C (D, G) shown in Definition 4.5.13. Then there exists an
(n, k − r, r ; 1, d f ≥ d) q unit-memory CC, where r ≤ k/2, k = l(G) − l(G − D)
and d ≥ n − deg(G).
Proof Let C (D, G) be the AG code with parity check matrix
H =
⎡
⎢
⎣
x 1 (P 1 ) x 1 (P 2 ) . . . x 1 (P n )
. . .
. . .
. . .
x k (P 1 ) x k (P 2 ) . . . x k (P n )
⎤
⎥
⎦ ,
where {x 1 , . . . , x k } is a basis of L(G). Let C L (D, G) be the Euclidean dual of
C (D, G). The matrix H is a generator matrix for C L (D, G). We know that
C L (D, G) is an [n, k = l(G) − l(G − D), d ≥ n − deg(G)] q AG code, where n =
deg(D). We define a convolutional code with generator matrix M(D) = H 0 + ˜
H 1 D,
where H 0 is the submatrix of H consisting of the k − r first rows and ˜
H 1 is the matrix
consisting of the last r rows of H by adding zero-rows at the bottom such that the
matrix ˜
H 1 has k − r rows in total. From hypothesis, it follows that rk H 0 ≥ rk ˜
H 1 .
From Theorem 7.1.1, M(D) is reduced and basic matrix. Again, from Theorem 7.1.1,
it follows that the CC generated by M(D) is a unit-memory code with dimension
k − r , degree r and free distance d f ≥ d, and the code follows.
Applying Proposition 4.5.1 in Theorem 7.6.1, we derive more families of CCs:
Corollary 7.6.1 Assume all the hypotheses of Theorem 7.6.1 hold. If 2g − 2 <
deg(G) < n, then there exists an (n, k − r, r ; 1, d f ≥ d) q , where n = deg(D), k =
deg(G) + 1 − g, r ≤ k/2 and d ≥ n − deg(G).
Corollary 7.6.2 Assume all the hypotheses of Theorem 7.6.1 hold. If 2g − 2 <
deg(G) < n, then there exists an (n, k − 1, 1; 1, d f ≥ d) q , where k = deg(G) +
1 − g and d ≥ n − deg(G).
Remark 7.6.1 Applying the generalized Singleton bound to the CCs constructed in
Corollary 7.6.2, it follows the free distance of such codes satisfy d f ≤ n − k + 3,
where n and k are the parameters of C L (D, G). Further, d f ≥ n − k + 1 − g, i.e.,
n − k + 1 − g ≤ d f ≤ n − k + 3. In particular, if the genus of the function fields is
zero, these convolutional codes are almost near MDS or near MDS or MDS, because
the Singleton defect is at most two.
Corollary 7.6.3 Assume that F = F q (z) is a rational function field. For β ∈ F q ,
let P β be the zero of z − β and P ∞ be the pole of z in F q (z). Then there exists an
(q, t, 1; 1, d f ≥ q − t) q , where 1 ≤ t ≤ q − 1.