7.4 More QCCs from BCH Codes
175
Proof We know the equalities gcd(q, n) = 1 and or d n (q) = 2 hold. Assume first
that C is the BCH code of length n = q
4
− 1 over F q 2 , generated by the product of
the minimal polynomials
C = =M
(q
2 +1)
(x)M
(q
2 +2)
(x) · · · · · · · · M
(q
2 +i−1)
(x)M
(q
2 +i)
(x),
where 3 ≤ i ≤ q
2
− 1. A parity check matrix of C is obtained from the matrix
H i+1,q 2 +1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(q
2 +1)
α
2(q
2 +1)
· · · α
(n−1)(q
2 +1)
1 α
(q
2 +2)
α
2(q
2 +2)
· · · α
(n−1)(q
2 +2)
. . .
. . .
. . .
. . .
. . .
1 α
(q
2 +i−1)
· · · · · · α
(n−1)(q
2 +i−1)
1 α
(q
2 +i)
· · · · · · α
(n−1)(q
2 +i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (in this case, containing 2 rows) over some
F q 2 -basis β of F q 4 and then removing any linearly dependent rows. We denote this
new matrix by H . From Lemma 7.4.3, C has parameters [n, n − 2(i − 1) − 1, d ≥
i + 1] q 2 . Moreover, since C has dimension n − 2(i − 1) − 1, H has 2(i − 1) + 1
linearly independent rows.
We next consider that C 0 is the BCH code of length n = q
4
− 1, over F q 2 , generated by the product of the minimal polynomials
C 0 = =M
(q
2 +1)
(x)M
(q
2 +2)
(x) · · · · · · · · M
(q
2 +i−2)
(x)M
(q
2 +i−1)
(x).
Analogously, C 0 has a parity check matrix derived from the matrix
H i,q 2 +1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(q
2 +1)
α
2(q
2 +1)
· · · α
(n−1)(q
2 +1)
1 α
(q
2 +2)
α
2(q
2 +2)
· · · α
(n−1)(q
2 +2)
. . .
. . .
. . .
. . .
. . .
1 α
(q
2 +i−2)
· · · · · · α
(n−1)(q
2 +i−2)
1 α
(q
2 +i−1)
· · · · · · α
(n−1)(q
2 +i−1)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) over some F q 2 -basis
β of F q 4 , then removing any linearly dependent rows. After these operations the new
matrix is denoted by H 0 . Applying again Lemma 7.4.3, the code C 0 has parameters
[n, n − 2(i − 2) − 1, d 0 ≥ i] q 2 . Since C 0 has dimension n − 2(i − 2) − 1, H 0 has
2(i − 2) + 1 linearly independent rows.
Let C 1 be the BCH code of length n = q
4
− 1 over F q 2 , generated by M
(q
2 +i)
(x);
C 1 has parameters [n, n − 2, d 1 ≥ 2] q 2 and a parity check matrix of C 1 is given by
expanding each entry of the matrix
175
Proof We know the equalities gcd(q, n) = 1 and or d n (q) = 2 hold. Assume first
that C is the BCH code of length n = q
4
− 1 over F q 2 , generated by the product of
the minimal polynomials
C = =M
(q
2 +1)
(x)M
(q
2 +2)
(x) · · · · · · · · M
(q
2 +i−1)
(x)M
(q
2 +i)
(x),
where 3 ≤ i ≤ q
2
− 1. A parity check matrix of C is obtained from the matrix
H i+1,q 2 +1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(q
2 +1)
α
2(q
2 +1)
· · · α
(n−1)(q
2 +1)
1 α
(q
2 +2)
α
2(q
2 +2)
· · · α
(n−1)(q
2 +2)
. . .
. . .
. . .
. . .
. . .
1 α
(q
2 +i−1)
· · · · · · α
(n−1)(q
2 +i−1)
1 α
(q
2 +i)
· · · · · · α
(n−1)(q
2 +i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (in this case, containing 2 rows) over some
F q 2 -basis β of F q 4 and then removing any linearly dependent rows. We denote this
new matrix by H . From Lemma 7.4.3, C has parameters [n, n − 2(i − 1) − 1, d ≥
i + 1] q 2 . Moreover, since C has dimension n − 2(i − 1) − 1, H has 2(i − 1) + 1
linearly independent rows.
We next consider that C 0 is the BCH code of length n = q
4
− 1, over F q 2 , generated by the product of the minimal polynomials
C 0 = =M
(q
2 +1)
(x)M
(q
2 +2)
(x) · · · · · · · · M
(q
2 +i−2)
(x)M
(q
2 +i−1)
(x).
Analogously, C 0 has a parity check matrix derived from the matrix
H i,q 2 +1 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1 α
(q
2 +1)
α
2(q
2 +1)
· · · α
(n−1)(q
2 +1)
1 α
(q
2 +2)
α
2(q
2 +2)
· · · α
(n−1)(q
2 +2)
. . .
. . .
. . .
. . .
. . .
1 α
(q
2 +i−2)
· · · · · · α
(n−1)(q
2 +i−2)
1 α
(q
2 +i−1)
· · · · · · α
(n−1)(q
2 +i−1)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
by expanding each entry as a column vector (containing 2 rows) over some F q 2 -basis
β of F q 4 , then removing any linearly dependent rows. After these operations the new
matrix is denoted by H 0 . Applying again Lemma 7.4.3, the code C 0 has parameters
[n, n − 2(i − 2) − 1, d 0 ≥ i] q 2 . Since C 0 has dimension n − 2(i − 2) − 1, H 0 has
2(i − 2) + 1 linearly independent rows.
Let C 1 be the BCH code of length n = q
4
− 1 over F q 2 , generated by M
(q
2 +i)
(x);
C 1 has parameters [n, n − 2, d 1 ≥ 2] q 2 and a parity check matrix of C 1 is given by
expanding each entry of the matrix