208
7 Constructions of QCCs
us consider the set of polynomials of degree less than k, in F q [x], denoted by P k .
Assuming all these facts, we can define a GRS code.
Definition 7.7.2 The GRS code is defined as
GRS k (ζ, v) = {(v 0 f (ζ 0 ), v 1 f (ζ 1 ), . . . , v n−1 f (ζ n−1 ))| f ∈ P k }.
The generalized RS code GRS k (ζ, v) is a MDS code with parameters [n, k, n − k +
1] q . It is not difficult to see that the Euclidean dual GRS
⊥
k (ζ, v) of the GRS code
GRS k (ζ, v) is also a GRS code and GRS
⊥
k (ζ, w) = GRS n−k (ζ, v) for some n-tuple
w = (w 0 , . . . , w n−1 ) of nonzero elements of F q .
Exercise 7.7.2 Show that GRS k (ζ, v) is a maximum-distance-separable code.
A generator matrix of GRS k (ζ, v) is given as
G =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
v 0
v 1 · · · v n−1
v 0 ζ 0 v 1 ζ 1 · · · v n−1 ζ n−1
v 0 ζ
2
0
v 1 ζ
2
1 · · · v n−1 ζ
2
n−1
. . .
. . .
. . .
. . .
v 0 ζ
k−1
0
v 1 ζ
k−1
1
· · · v n−1 ζ
k−1
n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
A parity check matrix for this code is the matrix
H =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
w 0
w 1
· · ·
w n−1
w 0 ζ 0
w 1 ζ 1 · · · w n−1 ζ n−1
w 0 ζ
2
0
w 1 ζ
2
1
· · · w n−1 ζ
2
n−1
. . .
. . .
. . .
. . .
w 0 ζ
n−k−1
0
w 1 ζ
n−k−1
1
· · · w n−1 ζ
n−k−1
n−1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Exercise 7.7.3 Show that G and H are, in fact, a generator and a parity check matrix
for GRS k (ζ, v), respectively.
In the next result, we construct new AQCCs derived from GRS codes.
Theorem 7.7.7 Let q ≥ 5 be a prime power. Assume that k ≥ 1 and n ≥ 5 are
integers such that n ≤ q and k ≤ n − 4. Choose an n-tuple ζ = (ζ 0 , . . . , ζ n−1 ) of
distinct elements of F q and an n-tuple v = (v 0 , . . . , v n−1 ) of nonzero elements of F q .
Then there exists an [(n, n − t − k − 2, μ
∗
; 3, [d z ] f /[d x ] f )] q AQCC, where (d z ) f ≥
t + 2 and (d x ) f ≥ k + 1, 1 ≤ t ≤ n − k − 2.