78
5 Quantum Code Constructions
and q
m
− 1, it follows that sq
l
= (q
m
− 1)a + t, where 0 ≤ t < q
m
− 1 is even.
Since t and q
m
− 1 are even, then also is sq
l ; because q
l is odd, it follows that s is
even. The proof is complete.
From Proposition 5.2.1, we obtain two useful results.
Corollary 5.2.1 If q is an odd prime power, then there is no consecutive integers
belonging to the same q-ary coset modulo n = q
m
− 1.
Proof Immediate.
Remark 5.2.1 Note that, in the binary case, there exists at least one coset containing
two consecutive elements, namely, C 1 .
Corollary 5.2.2 Suppose that C x and C y are two q-ary cosets, where q is an odd
prime power. Assume also that a ∈ C x and b ∈ C y . If a ≡ b mod 2, then C x = C y .
Proof It follows directly from Proposition 5.2.1.
The next result shows that the minimum distance among elements belonging to
the same q-ary coset is greater than or equal to q − 1, with equality in some cases.
Theorem 5.2.1 Let q = 2 be a prime power and let C s be a q-ary coset (modulo q
m
− 1) with representative s. Define L s = min{| sq
j
− sq
l
|: 0 ≤ j, l ≤ m s −
1, j = l}, where | · | is the absolute value function, and sq
j
− sq
l is considered modulo q
m
− 1. Then it follows that L s ≥ q − 1, for all s, where s runs through the coset
representatives. Moreover, there exists at least one coset C s ∗ such that L s ∗ = q − 1.
Proof Assume without loss of generality (w. l. o. g.) that j > l are integers such that
1 ≤ j, l ≤ m s − 1, and let C s be an arbitrary coset. Applying the division algorithm
for sq
j
− sq
l and q
m
− 1 we have
sq
j
− sq
l
= (q
m
− 1)b + r, 0 ≤ r < q
m
− 1 =⇒
=⇒ sq
l
(q
j−l
− 1) − (q
m
− 1)b = r.
Since q − 1 divides q
j−l
− 1 and q
m
− 1, it follows that q − 1 divides r , i.e., L s ≥
q − 1.
To complete the proof, it suffices to consider the coset containing the element
q
m −1
2
+ 1 since this coset also contains the element
q
m −1
2
+ q.
Corollary 5.2.3 can be utilized to compute the upper bound for the designed
distance of certain classes of cyclic codes.
Corollary 5.2.3 Let q = 2 be a prime power. If C is a q-ary cyclic code whose
defining set Z contains i cosets, where 1 ≤ i < q − 2, then its designed distance δ
is at most i + 2. In particular, if Z consists of only one q-ary coset, then δ = 2.
5 Quantum Code Constructions
and q
m
− 1, it follows that sq
l
= (q
m
− 1)a + t, where 0 ≤ t < q
m
− 1 is even.
Since t and q
m
− 1 are even, then also is sq
l ; because q
l is odd, it follows that s is
even. The proof is complete.
From Proposition 5.2.1, we obtain two useful results.
Corollary 5.2.1 If q is an odd prime power, then there is no consecutive integers
belonging to the same q-ary coset modulo n = q
m
− 1.
Proof Immediate.
Remark 5.2.1 Note that, in the binary case, there exists at least one coset containing
two consecutive elements, namely, C 1 .
Corollary 5.2.2 Suppose that C x and C y are two q-ary cosets, where q is an odd
prime power. Assume also that a ∈ C x and b ∈ C y . If a ≡ b mod 2, then C x = C y .
Proof It follows directly from Proposition 5.2.1.
The next result shows that the minimum distance among elements belonging to
the same q-ary coset is greater than or equal to q − 1, with equality in some cases.
Theorem 5.2.1 Let q = 2 be a prime power and let C s be a q-ary coset (modulo q
m
− 1) with representative s. Define L s = min{| sq
j
− sq
l
|: 0 ≤ j, l ≤ m s −
1, j = l}, where | · | is the absolute value function, and sq
j
− sq
l is considered modulo q
m
− 1. Then it follows that L s ≥ q − 1, for all s, where s runs through the coset
representatives. Moreover, there exists at least one coset C s ∗ such that L s ∗ = q − 1.
Proof Assume without loss of generality (w. l. o. g.) that j > l are integers such that
1 ≤ j, l ≤ m s − 1, and let C s be an arbitrary coset. Applying the division algorithm
for sq
j
− sq
l and q
m
− 1 we have
sq
j
− sq
l
= (q
m
− 1)b + r, 0 ≤ r < q
m
− 1 =⇒
=⇒ sq
l
(q
j−l
− 1) − (q
m
− 1)b = r.
Since q − 1 divides q
j−l
− 1 and q
m
− 1, it follows that q − 1 divides r , i.e., L s ≥
q − 1.
To complete the proof, it suffices to consider the coset containing the element
q
m −1
2
+ 1 since this coset also contains the element
q
m −1
2
+ q.
Corollary 5.2.3 can be utilized to compute the upper bound for the designed
distance of certain classes of cyclic codes.
Corollary 5.2.3 Let q = 2 be a prime power. If C is a q-ary cyclic code whose
defining set Z contains i cosets, where 1 ≤ i < q − 2, then its designed distance δ
is at most i + 2. In particular, if Z consists of only one q-ary coset, then δ = 2.