Page 4 of 16
Zhang and Zhang
SpringerPlus (2016) 5:1845
Lemma 4 [see Golub and Loan (1996)] Let A, B, C, D be suitably sized matrices. en
Let a quasigroup
(Q, ∗)
of order p
kd
be given by the multiplication scheme in
Table 1, where
q
(j)
i
∈
Q
,
(i
,
j
= 0, 1, ...,
p
kd
− 1
)
. For given i and
∀j �= j
′
, we have
q
(j)
i
�= q
(j
′
)
i
; for given j and
∀i �= i
′
, we have
q
(j)
i
�= q
(j)
i
′
. One can choose two bijections
κ : Q →{
0, 1, ...,
p −
1
}
dk
and
ι :{
0, 1, ...,
p −
1
}
k
→{
0, 1, ...,
p
k
−
1
}
. Collect the ele-
ments of Table1 into a vector
and convert every element of the vector into a kd-ary sequence over GF(p) according
to the bijection
κ
. en, divide every kd-ary binary sequence into d groups from left to
right, where every group is a k-ary sequence, and represent every group by a unique ele
-
ment in
{
0, 1,
...
,
p
k
−
1
}
according to the bijection
ι
. In this way, we obtain a
p
2kd
× d
matrix
[b
1
, ..., b
d
]
, where every
b
s
(s = 1, ..., d)
is a
p
2kd
dimensional column vector
over finite field
GF(p
k
)
.
According to Lemma 1, whether a given quasigroup is an MQQ over
GF
(
p
k
)
mainly
lies in whether there is 2d-ary quadratic Boolean function set
{f
1
, f
2
, ..., f
d
}
satisfying
Table1. Note that, any
f
s
(x
1
, ..., x
d
, x
d+1
, ..., x
2d
)
can be written in the form
where
A
s
is a matrix of order
2d + 1
over finite field
GF(p
k
)
. By (2) and Table1, when
(x
1
, ..., x
d
)
and
(x
d+1
, ..., x
2d
)
are respectively assigned d-ary sequences in the order
of
{
0, 1, ...,
p
kd
− 1
}
in which every element is written by d-ary sequence over
GF
(
p
k
)
,
namely,
(1, x
1
, ..., x
d
, x
d+1
, ..., x
2d
)
in
f
s
are assigned all row vectors of the following
p
2kd
×
(2
d +
1
)
matrix of the form
(A + B) ⊗ (C + D) = A ⊗ C + A ⊗ D + B ⊗ C + B ⊗ D.
(3)
q
(0)
0
, q
(0)
1
, ..., q
(0)
p
kd
−1
, q
(1)
0
, q
(1)
1
, ..., q
(1)
p
kd
−1
, ..., q
(p
kd
−1)
0
, q
(p
kd
−1)
1
, ..., q
(p
kd
−1)
p
kd
−1
T
,
(4)
f
s
= (1, x
1
, ..., x
d
, x
d+1
, ..., x
2d
)A
s
1
x
1
.
.
.
x
d
x
d+1
.
.
.
x
2d
, (s = 1, 2, ..., d)
,
Table 1 A quasigroup
(Q,
∗
)
oforder p
kd
* 0 1 2
...
p
kd
−
1
0
q
(0)
0
q
(0)
1
q
(0)
2
···
q
(0)
p
kd
−
1
1
q
(1)
0
q
(1)
1
q
(1)
2
···
q
(1)
p
kd
−
1
2
q
(2)
0
q
(2)
1
q
(2)
2
···
q
(2)
p
kd
−
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
p
kd
−
1
q
(p
kd
−1
)
0
q
(p
kd
−1
)
1
q
(p
kd
−1
)
2
···
q
(p
kd
−1
)
p
kd
−1
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