5.2 BCH Codes—Part II
81
Lemma 5.2.1 ([4, Lemma 1]) Assume that gcd(q, n) = 1. A cyclic code of length n
over F q with defining set Z contains its Euclidean dual code if and only if Z ∩ Z
−1
=
∅, where Z
−1
= {−z mod n | z ∈ Z }.
The next result characterizes Euclidean dual-containing cyclic codes in terms of
complementary cosets.
Proposition 5.2.3 Assume that gcd(q, n) = 1. A cyclic codes of length n over F q
with defining set Z = C r 1 ∪ C r 2 ∪ . . . ∪ C r n contains its Euclidean dual code if and
only if the union of the complementary cosets C i , where i = r 1 , r 2 , . . . , r n , and Z
does not have common elements.
Proof Since the coset C −i is the complementary coset of C i , for all i = r 1 , r 2 , . . . , r n ,
the result follows.
Remark 5.2.2 It is interesting to note that analogous result can be shown in the
Hermitian case.
Let us consider the following result shown in [4].
Lemma 5.2.2 ([4, Lemmas 8 and 9]) Let n ≥ 1 be an integer and q be a power of
a prime such that gcd(n, q) = 1 and q
m/2
< n ≤ q
m
− 1, where m = or d n (q).
(a) The cyclotomic coset C x = {xq
j mod n | 0 ≤ j < m} has cardinality m for all
x in the range 1 ≤ x ≤ nq
m/2
/(q
m
− 1).
(b) If x and y are distinct integers in the range 1 ≤ x, y ≤ min{{nq
m/2
/(q
m
−
1) − 1, n − 1} such that x, y ≡ 0, then the q-ary cosets of x and y modulo n
are disjoint.
We next improve the upper bound for the number of disjoint q-ary cosets modulo
q
m
− 1.
Theorem 5.2.2 Let n = q
m
− 1, where q is a prime power and m is even. If x and
y are distinct integers in the range 1 ≤ x, y ≤ 2q
m/2 , such that x ≡ 0 mod q and
y ≡ 0 mod q, then the q-ary cosets of x and y modulo n are disjoint.
Proof Let us consider the following result.
Theorem 5.2.3 ([165, Theorem 2.3]) Let n = q
m
− 1, where q is a prime power
and m is even. Denote s
∗
= min{t : t ∈ C s } be the minimum coset representative.
If 0 ≤ s ≤ T , where T := 2q
m/2 , and q s then s = s
∗ , and T is the greatest value
having this property.
Applying Theorem 5.2.3, the result follows directly. To see this, note that if the
inequalities 0 ≤ s ≤ T := 2q
m/2 hold, it follows that s = s
∗ . Hence, there exist at
least 2q
m/2 q-cosets in the range 0 ≤ s ≤ T . Because the minimum coset representatives belongs to disjoint cosets, there are exactly 2q
m/2 disjoint coset in the range
0 ≤ s ≤ T := 2q
m/2 . The proof is complete.
Theorem 5.2.2 will be used in the construction of new families of quantum codes,
(see Sect. 5.2.2).
81
Lemma 5.2.1 ([4, Lemma 1]) Assume that gcd(q, n) = 1. A cyclic code of length n
over F q with defining set Z contains its Euclidean dual code if and only if Z ∩ Z
−1
=
∅, where Z
−1
= {−z mod n | z ∈ Z }.
The next result characterizes Euclidean dual-containing cyclic codes in terms of
complementary cosets.
Proposition 5.2.3 Assume that gcd(q, n) = 1. A cyclic codes of length n over F q
with defining set Z = C r 1 ∪ C r 2 ∪ . . . ∪ C r n contains its Euclidean dual code if and
only if the union of the complementary cosets C i , where i = r 1 , r 2 , . . . , r n , and Z
does not have common elements.
Proof Since the coset C −i is the complementary coset of C i , for all i = r 1 , r 2 , . . . , r n ,
the result follows.
Remark 5.2.2 It is interesting to note that analogous result can be shown in the
Hermitian case.
Let us consider the following result shown in [4].
Lemma 5.2.2 ([4, Lemmas 8 and 9]) Let n ≥ 1 be an integer and q be a power of
a prime such that gcd(n, q) = 1 and q
m/2
< n ≤ q
m
− 1, where m = or d n (q).
(a) The cyclotomic coset C x = {xq
j mod n | 0 ≤ j < m} has cardinality m for all
x in the range 1 ≤ x ≤ nq
m/2
/(q
m
− 1).
(b) If x and y are distinct integers in the range 1 ≤ x, y ≤ min{{nq
m/2
/(q
m
−
1) − 1, n − 1} such that x, y ≡ 0, then the q-ary cosets of x and y modulo n
are disjoint.
We next improve the upper bound for the number of disjoint q-ary cosets modulo
q
m
− 1.
Theorem 5.2.2 Let n = q
m
− 1, where q is a prime power and m is even. If x and
y are distinct integers in the range 1 ≤ x, y ≤ 2q
m/2 , such that x ≡ 0 mod q and
y ≡ 0 mod q, then the q-ary cosets of x and y modulo n are disjoint.
Proof Let us consider the following result.
Theorem 5.2.3 ([165, Theorem 2.3]) Let n = q
m
− 1, where q is a prime power
and m is even. Denote s
∗
= min{t : t ∈ C s } be the minimum coset representative.
If 0 ≤ s ≤ T , where T := 2q
m/2 , and q s then s = s
∗ , and T is the greatest value
having this property.
Applying Theorem 5.2.3, the result follows directly. To see this, note that if the
inequalities 0 ≤ s ≤ T := 2q
m/2 hold, it follows that s = s
∗ . Hence, there exist at
least 2q
m/2 q-cosets in the range 0 ≤ s ≤ T . Because the minimum coset representatives belongs to disjoint cosets, there are exactly 2q
m/2 disjoint coset in the range
0 ≤ s ≤ T := 2q
m/2 . The proof is complete.
Theorem 5.2.2 will be used in the construction of new families of quantum codes,
(see Sect. 5.2.2).