170
7 Constructions of QCCs
2(i + 1), so the dimension k C 2 of C 2 equals k C 2 = n − deg(g 2 (x)) = n − 2(i + 1).
Moreover, the defining set of the code C 2 consists of the sequence {a − i, a − i +
1, . . . , a, a + 1, . . . , a + i + 1} of 2i + 2 consecutive integers, so, from the BCH
bound, the minimum distance d C 2 of C 2 satisfies d C 2 ≥ 2i + 3. Thus, C 2 is a MDS
code with parameters [n, n − 2i − 2, 2i + 3] q and, consequently, its (Euclidean)
dual code has dimension 2i + 2.
Let C 1 be the BCH code of length n over F q generated by g 1 (x) given by
g 1 (x) = M
(a−i+1)
(x)M
(a−i+2)
(x) · · · · · M
(a−1)
(x)M
(a)
(x).
We know that C 1 has a parity check matrix derived from the matrix
H 2i+1,a−i+1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
1 α
(a−i+2)
α
2(a−i+2)
· · · α
(n−1)(a−i+2)
. . .
. . .
. . .
. . .
. . .
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
1 α
a
· · ·
· · · α
(n−1)a
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) with respect to β
(already done, since H 2i+1,a−i+1 is a submatrix of H 2i+3,a−i ). After performing the
expansion for all entries, such new matrix is denoted by H C 1 (H C 1 is a submatrix of
H C 2 ). Applying Lemma 7.3.2 and proceeding in the same way as above, it follows
that C 1 is an [n, n − 2i, 2i + 1] q MDS code.
Let us now consider C as the BCH code of length n over F q generated by the
minimal polynomial M
(a−i)
(x); C is an [n, n − 2, d ≥ 2] q code. A parity check
matrix H C of C is given by expanding each entry of the matrix
H 2,a−i =
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
with respect to β (already done, since H 2,a−i is a submatrix of H 2i+3,a−i ). Because
C has dimension n − 2, H C has rank 2 (H C is also a submatrix of H C 2 ).
We here rearrange the rows of H C 2 in the form
H =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
a
· · ·
· · · α
(n−1)a
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
. . .
. . .
. . .
. . .
. . .
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
(to simplify the notation we write H in terms of powers of α, although it is clear
from the context that this matrix has entries in F q , which are derived by expanding
each entry with respect to β already performed).
We then split H into two disjoint submatrices H 0 and H 1 of the forms
7 Constructions of QCCs
2(i + 1), so the dimension k C 2 of C 2 equals k C 2 = n − deg(g 2 (x)) = n − 2(i + 1).
Moreover, the defining set of the code C 2 consists of the sequence {a − i, a − i +
1, . . . , a, a + 1, . . . , a + i + 1} of 2i + 2 consecutive integers, so, from the BCH
bound, the minimum distance d C 2 of C 2 satisfies d C 2 ≥ 2i + 3. Thus, C 2 is a MDS
code with parameters [n, n − 2i − 2, 2i + 3] q and, consequently, its (Euclidean)
dual code has dimension 2i + 2.
Let C 1 be the BCH code of length n over F q generated by g 1 (x) given by
g 1 (x) = M
(a−i+1)
(x)M
(a−i+2)
(x) · · · · · M
(a−1)
(x)M
(a)
(x).
We know that C 1 has a parity check matrix derived from the matrix
H 2i+1,a−i+1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
1 α
(a−i+2)
α
2(a−i+2)
· · · α
(n−1)(a−i+2)
. . .
. . .
. . .
. . .
. . .
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
1 α
a
· · ·
· · · α
(n−1)a
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) with respect to β
(already done, since H 2i+1,a−i+1 is a submatrix of H 2i+3,a−i ). After performing the
expansion for all entries, such new matrix is denoted by H C 1 (H C 1 is a submatrix of
H C 2 ). Applying Lemma 7.3.2 and proceeding in the same way as above, it follows
that C 1 is an [n, n − 2i, 2i + 1] q MDS code.
Let us now consider C as the BCH code of length n over F q generated by the
minimal polynomial M
(a−i)
(x); C is an [n, n − 2, d ≥ 2] q code. A parity check
matrix H C of C is given by expanding each entry of the matrix
H 2,a−i =
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
with respect to β (already done, since H 2,a−i is a submatrix of H 2i+3,a−i ). Because
C has dimension n − 2, H C has rank 2 (H C is also a submatrix of H C 2 ).
We here rearrange the rows of H C 2 in the form
H =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
a
· · ·
· · · α
(n−1)a
1 α
(a−1)
· · ·
· · · α
(n−1)(a−1)
. . .
. . .
. . .
. . .
. . .
1 α
(a−i+1)
α
2(a−i+1)
· · · α
(n−1)(a−i+1)
1 α
(a−i)
α
2(a−i)
· · · α
(n−1)(a−i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
(to simplify the notation we write H in terms of powers of α, although it is clear
from the context that this matrix has entries in F q , which are derived by expanding
each entry with respect to β already performed).
We then split H into two disjoint submatrices H 0 and H 1 of the forms