On the Exponent of Boolean Primitive Circulant Matrices
649 K; S, n-periodic elements in Z
n
and there are no k; S, n-periodic elements in Z
n
for k K.
R
EMARK
2. If the minimum nonzero element of a basis S of Z
n
, say b, is not 1, then S
is not 0-periodic as any b
′
, with 0 b
′
b, is not 0; S, n-periodic. We finish this section with the following lemma.
L
EMMA
4.4. [2] Let S
∈ S
n
and m be a divisor of n. Suppose that S contains an
element of order m. Then
order S
≤ n
m + m
− 2.
By Remark 1, all bases for Z
n
, n
≥ 3, with cardinality 2 have order n − 1, and therefore, they satisfy Conjecture 1. In the next section we focus on bases with cardinality
3.
5. Order of bases for Z
n
with cardinality 3. By S
n,r
we denote the set of all bases for Z
n
with cardinality r.
For a given positive integer n, we define p
n
as follows: p
n
= ½
⌊n2⌋ + 1, if n is odd ⌊n2⌋,
if n is even. 5.1
L
EMMA
5.1. Let S =
{0, s
1
, s
2
} ∈ S
n,3
. If gcds
1
, n = 1 or gcds
2
, n = 1 or gcds
2
− s
1
, n = 1, then there exists b ∈ Z
n
such that b
≤ p
n
and S
∼ {0, 1, b}. Proof. Without loss of generality, suppose that either
gcds
1
, n = 1 or gcds
2
− s
1
, n = 1. In the first case, s
1
is a unit in Z
n
and S
∼ S
1
= s
−1 1
S = {0, 1, s
−1 1
s
2
}. If
s
−1 1
s
2
≤ ⌊n2⌋, the claim holds with b = s
−1 1
s
2
. If s
−1 1
s
2
⌊n2⌋, then S
∼ S
2
= 1 − S
1
= {0, 1, n + 1 − s
−1 1
s
2
} and the claim holds with
b + 1 − s
−1 1
s
2
. In the second case, that is, gcds
2
− s
1
, n = 1, let S
1
= −s
1
+ S = {0, s
2
− s
1
, n − s
1
}. Then
S ∼ S
2
= s
′
S
1
, where s
′
= s
2
− s
1 −1
. Now the argument used above applies to show the result.
We note that if gcds
1
, n 6= 1, gcds
2
, n 6= 1 and gcds
2
− s
1
, n 6= 1 and one of s
1
, s
2
, s
2
− s
1
, n − s
1
, n − s
2
, n − s
2
+ s
1
is a product of a divisor of n and a unit in Z
n
, then
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650
M.I. Bueno and S. Furtado
there exist a, b
∈ Z
n
such that a is a divisor of n, a
6= 1 and S = {0, s
1
, s
2
} ∼ {0, a, b}. To see this, assume, without loss of generality, that
s
1
s
2
. Note that S
∼ S
1
= −s
1
+ S = {0, n − s
1
, s
2
− s
1
} ∼ S
2
= −s
2
+ S = {0, n + s
1
− s
2
, n − s
2
}. Suppose that
s
1
is a product of a divisor of n and a unit. The proof is analogous in the other
mentioned cases, eventually by considering S
1
or S
2
instead of S. If s
1
|n, then the result is clear; otherwise
s
1
= d
1
t
1
, where d
1
|n and gcdn, t
1
= 1. Then S
∼ t
−1 1
S = {0, d
1
, t
−1 1
s
2
}, and the result follows.
We were not able to prove that every basis S
∈ S
n,3
that is not equivalent to a basis of the form
{0, 1, b} is equivalent to a basis {0, a, b} with a 6= 1 a divisor of n. However, numerical experiments show that if these bases exist, they are rare.
T
HEOREM
5.2. Let S =
{0, a, b} ∈ S
n,3
, where a is a divisor of n and a 6= 1. Then
a − 1 ≤ orderS ≤
n a
+ a − 2.
Proof. The inequality on the right follows from Lemma 4.4. Consider the quotient map
f : Z
n
→ Z
a
. Notice that T = f S =
{0, fb}. Since S is a basis and
a divides n, gcda, b = 1 and f b 6= 0. Therefore, T is a basis for Z
a
. Moreover, by Remark 1, order
T = a − 1. Since orderS ≥ orderT , we get orderS ≥ a − 1.
From now on, we consider bases S of S
n,3
of the form {0, 1, b}. For convenience, we
write expq; b, n instead of expq; S, n; we also say that S is 0-periodic instead of 0; S, n-
periodic. If
S = {0, 1, b} ∈ S
n,3
, then, considering the standard addition and multiplication in Z, for
k ≥ 1,
kS = [0, k] ∪ [b, b + k − 1] ∪ [2b, 2b + k − 2] ∪ · · · ∪ [k − 1b, k − 1b + 1] ∪ [kb].
For j = 0, 1, . . . , k, let
I
j,k
= [jb, jb + k − j].
5.2 T
HEOREM
5.3. Let S =
{0, 1, b} ∈ S
n,3
. Then j
n b
k ≤ orderS ≤
j n
b k
+ b − 2.
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On the Exponent of Boolean Primitive Circulant Matrices
651
Proof. Let n = mb + t, with m =
¥
n b
¦ and 0 ≤ t b. Considering the standard addition and multiplication in Z
, the largest element in kS is kb. Thus, if k = orderS, then kb
≥ n − 1 = mb + t − 1, which implies the inequality on the left. We have
I
,b −1
∪ I
1 ,b
−1
= [0, 2b − 2] ⊂ b − 1S. Moreover, {0, b, . . . , m − 1b} ⊂
m − 1S. Thus,
b − 1 + m − 1S = b − 1S + m − 1S = Z
n
. which implies the inequality on the right.
We now focus on 0-periodic bases of S
n,3
. C
OROLLARY
5.4. Let S =
{0, 1, b} ∈ S
n,3
. If S is 0-periodic, then order
S = j
n b
k + b
− 2.
Proof. Suppose kS = Z
n
. Let j =
¥
n b
¦ − 1. Note that k ≥ j . Since S is 0-periodic,
necessarily [j
b, j b + b
− 1] ⊂ I
j ,k
, which implies that ³j
n b
k − 1
´ b + k
− ³j
n b
k − 1
´ ≥
j n
b k
b − 1,
that is, k
≥ ¥
n b
¦ + b − 2. Then orderS ≥ ¥
n b
¦ + b − 2. Since, by Theorem 5.3, orderS ≤ ¥
n b
¦ + b − 2, the result follows. We next characterize the
0-periodic bases in S
n,3
. Note that, by Remark 2, we may assume that these bases are of the form
{0, 1, b}. First, we add a technical lemma. It is clear that if
b and w are positive integers, with b ≥ 2, then the minimum x + y
among all the solutions of x + by = w, with x, y
≥ 0, is obtained when y is y =
¥
w b
¦. Since x = w
− by, the minimum value of x + y is x + y
= w −
¥
w b
¦ b − 1. Note also that x
= w − by
b. We then have the following lemma. L
EMMA
5.5. Let b, q
∈ Z
n
, with b ≥ 2 and q 6= 0. Then
expq; b, n = min ½
q + kn − b − 1
¹ q + kn b
º , k
≥ 0 ¾
.
Note that expq; b, n
≤ q − b − 1 j
q b
k ≤ q,
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652
M.I. Bueno and S. Furtado
where the first inequality follows from Lemma 5.5. L
EMMA
5.6. Let S =
{0, 1, b} ∈ S
n,3
. Then S is 0-periodic if and only if either b divides
n or b
− 1 ³j
n b
k + 1
´ ≤ n.
5.3 Proof. Suppose that
S is 0-periodic and b is not a divisor of n. Let q = ¥
n b
¦ b − 1. By Lemma 5.5 and Definition 4.3,
q − b − 1
j q
b k
≤ q + n − b − 1 ¹ n + q
b º
. 5.4
As n + q = 2
¥
n b
¦ b + t − 1, for some 1 ≤ t b, it follows that ¥
n+q b
¦ = 2 ¥
n b
¦ . Also, ¥
q b
¦ = ¥
n b
¦ − 1. Thus, 5.4 is equivalent to 5.3. To prove the converse note that, from Lemma 5.5, if
b divides n, S is 0-periodic, as, for any
k 0, kn
− b − 1 kn
b 0.
Now suppose that 5.3 holds. According to Lemma 5.5, we need to show that, for any k 0
and any q
∈ Z
n
\{0}, q
− b − 1 j
q b
k ≤ q + kn − b − 1
¹ q + kn b
º ,
or equivalently, b
− 1 µ¹ q + kn
b º
− j
q b
k ¶
≤ kn. Because of 5.3, it is enough to show that
¹ q + kn b
º −
j q
b k
≤ k ³j
n b
k + 1
´ .
For n =
¥
n b
¦ b + t, 0 ≤ t b, kn + q = k
j n
b k
b + kt + q = ³
k j
n b
k +
j q
b k
+ k ´
b + kt − b +
³ q
− j
q b
k b
´ .
As, kt
− b + q − ¥
q b
¦ b b, then ¹ q + kn
b º
≤ k j
n b
k +
j q
b k
+ k, completing the proof.
T
HEOREM
5.7. Let S =
{0, 1, b} ∈ S
n,3
. Then S is 0-periodic if and only if one of the following conditions is satisfied:
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On the Exponent of Boolean Primitive Circulant Matrices
653 i
b is a divisor of n; ii
b = j
n j
k + 1, for some nonnegative integer j. In this case, j is unique and is given
by ⌊nb⌋ + 1.
Proof. If b is a divisor of n, then Lemma 5.6 implies that S is 0-periodic. Now suppose
that b is not a divisor of n. Let i =
⌊nb⌋ and n = bi + t, with 0 t b. If t i then b =
⌊
n i
⌋, otherwise b ⌊
n i
⌋. Since n = i + 1b + t − b and t − b 0, it follows that ⌊
n i+1
⌋ b. Therefore, ¹
n i + 1
º b
≤ j
n i
k .
Suppose that b =
j
n i+1
k + 1. Since
b − 1 =
¹ n
i + 1 º
≤ n
i + 1 ,
we have n
≥ b − 1i + 1. Therefore, by Lemma 5.6,
S is 0-periodic. Now suppose that
¥
n i
¦ ≥ b ≥ j
n i+1
k + 2. Then
n i + 1
b − 1
and, by Lemma 5.6, S is not 0-periodic.
6. The conjecture for bases in
