182
7 Constructions of QCCs
where μ + 1 ≤ i < q
2
− 1. Proceeding similarly as in the proof of Theorem 7.4.5,
the results follows.
Exercise 7.4.2 Complete the proof of Theorem 7.4.6.
Remark 7.4.1 Note that in Theorem 7.4.6, we can also consider that C is generated
by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+i)
(x) · M
(s−1)
(x) . . . M
(s− j)
(x),
for all 1 ≤ i, j ≤ q
2
− 1, because from Lemma 7.4.6, C is Hermitian self-orthogonal.
After this we choose suitable range for μ (greater than displayed in Theorem 7.4.6),
generating therefore more QCCs. Consequently, also in this case, the proposed construction method holds.
7.4.3 Construction III
In this subsection, we construct QCCs with respect to the Euclidean inner product.
Let us recall some results proved in [89].
Lemma 7.4.7 Suppose that n = q
m
− 1, where q ≥ 4 is a prime power and m =
ord n (q) ≥ 3. Let s =
m−1
i=0
q
i . Then the following hold:
(a) The q-coset C [s] has only one element;
(b) Each one of the q-ary cosets C [s+i] are mutually disjoints, where 1 ≤ i ≤ q − 1;
(c) Each one of the q-ary cosets C [s− j] are mutually disjoints, where 1 ≤ j ≤ q − 1;
(d) The q-cosets of the forms C [s+i] and C [s− j] are mutually disjoints, where 1 ≤
i, j ≤ q − 1;
(e) The cosets of the form C [s+i] , where 1 ≤ i ≤ q − 1, have m elements;
(f) The cosets of the form C [s− j] have m elements, where 1 ≤ j ≤ q − 1.
Proof See [89, Lemmas III.7, III.8, and III.9].
Lemma 7.4.8 Let q ≥ 4 be a prime power and n = q
m
− 1. Let m = ord n (q) ≥ 3
and s =
m−1
i=0
q
i . If C is the cyclic code generated by
M
(s)
(x)M
(s+1)
(x) . . . M
(s+ j)
(x) · M
(s−1)
(x) . . . M
(s− j)
(x),
where 1 ≤ j ≤ q − 1, then C is Euclidean self-orthogonal.
Proof See [89, Lemma III.10].