# A Simple Cocyclic Jacket Matrices

Authors: Moon Ho Lee; Gui-Liang Feng; Zhu Chen
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

We present a new class of cocyclic Jacket matrices over complex number field with any size. We also construct cocyclic Jacket matrices over the finite field. Such kind of matrices has close relation with unitary matrices which are a first hand tool in solving many problems in mathematical and theoretical physics. Based on the analysis of the relation between cocyclic Jacket matrices and unitary matrices, the common method for factorizing these two kinds of matrices is presented.


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## Body

## 1. Introduction

The  Walsh-Hadamard matrix is
widely  used for the Walsh representation of the data sequence in image coding
and for Hadamard transform orthogonal code design for spread spectrum
communications and quantum computation [1–4]. Their basic functions are sampled
Walsh functions which can be expressed in terms of the Hadamard [H]N matrices.
Using the orthogonality of Hadamard matrices, more general  matrices have
been developed [5]. These matrices are called as Jacket matrices and denoted by Jk.
From [6], we have the following definition of Jacket matrix (http://en.wikipedia.org/wiki/Category:Matrices;
http://en.wikipedia.org/wiki/user:Jacket Matrix).Definition 1.1.
If a matrix of sizem×m
has nonzero elements, and an inverse form which is only from the element-wise
inverse and then transpose, such as(1.1)[J]m×m=[j0,0j0,1⋯j0,m−1j1,0j1,1⋯j1,m−1⋮⋮⋱⋮jm−1,0jm−1,1⋯jm−1,m−1], and its inverse is(1.2)[J]m×m−1=Δ1C[j0,0−1j0,1−1⋯j0,m−1−1j1,0−1j1,1−1⋯j1,m−1−1⋮⋮⋱⋮jm−1,0−1jm−1,1−1⋯jm−1,m−1−1]T, where C is
the normalized value for this matrix, and T is the transpose, then this matrix is called as Jacket matrix.
Many
interesting matrices, such as Hadamard, DFT, and Haar, belong to the Jacket
family [6, 7]. In many applications, cocyclic matrices are very useful. The
definition cocyclic matrix is as follows [8–10].Definition 1.2.
IfG is a finite group of order v with operation “∘,” and C is a finite abelian group of order w,
a “two-dimensional“ cocycle is a mapping φ:G×G→C,
satisfying
(1.3)φ(g,h)φ(g∘h,k)=φ(g,h∘k)φ(h,k), where g,h,k∈G.
A square matrix Mφ, whose rows and columns are indexed by the
elements of G,
with entry φ(g,h) in the position (g,h),
that is, Mφ=[φ(g,h)] where g,h∈G and φ(g,h)∈C,
can be called as a cocyclic matrix.In [11], it is demonstrated that many well-known
binary, quaternary, and q-ary codes are cocyclic Hadamard codes, that is, derived
from a cocyclic generalized Hadamard matrix or its equivalents. In [9, 12, 13], Lee
et al. proved that many Jacket matrices derived in [12, 14–16] are all cocyclic
matrices and they are called cocyclic Jacket matrices. Hence, the Jacket matrices
have many applications [9, 10, 17]. However, the derived Jacket matrices have
only the sizes N=2l,2lp,
where p is an odd prime. In this
paper, we present an explicit construction of cocyclic Jacket matrices over
complex field and finite field with any sizes. As a byproduct, a factorization
of unitary matrices is given, which can be useful in many domains of mathematical
and theoretical physics [18].This paper is organized as follows: in Section2, we
present a class of cocyclic Jacket matrices over complex number field. The
known Jacket matrices belong to this class of matrices. A class of cocyclic Jacket
matrices over finite field is presented in Section 3. In Section 4, factorization
of cocyclic Jacket matrices and unitary matrices is presented. Finally,
conclusions are drawn in Section 5.


## 2. Cocyclic Jacket Matrices over Complex Number Field

In
this section, we present a class of cocyclic Jacket matrices over complex
number field.


## 2.1. Basic Notations and Results

Letp be an odd prime integer and α=e−1(2π/p).
Thus, we have αp=1, and Fp={0,1,2,…,p−1} with the operations for 〈a⋅b〉 are the finite field, where(2.1)〈a〉=Δamod⁡p. Let a∈Fp,
we define a function(2.2)fa(x)=Δ〈a×x〉. Let V→=(αv0,αv1,…,αvp−1) be a vector, where vi∈Fp for i=0,1,…,p−1. We define a vector(2.3)V→a=(αfa(v0),αfa(v1),…,αfa(vp−1)). We have the following lemma.Lemma 2.1.
LetV→=(1,α1,α2,…,αp−1),
then(2.4)V0→×V0→T=p,Va→×Vb→T=p,for〈a+b〉=0,Va→×Vb→T=0,for〈a+b〉≠0.Proof.
The first equation can be easily
proved becauseV0→=(1,1,…,1).
For the second equation, since 〈a+b〉=0,
we have
(2.5)(fa(vi)+fb(vi))mod⁡p=(a+b)vimod⁡p=0.
Thus the second equation is
also true. Now we consider the last equation since p is an odd prime, we know that for any 0<c<p,(2.6){0,1,2,…,p−1}={0,〈c〉,〈2c〉,…,〈(p−1)c〉}. Furthermore, for 〈a+b〉≠0,
we have
(2.7){0,1,2,…,p−1}={0,〈a+b〉,〈2(a+b)〉,…,〈(p−1)(a+b)〉}, that is,(2.8)V→a×V→b=∑i=0p−1αi. On the other hand, from αp=1,
we have
(2.9)0=αp−1=(α−1)∑i=0p−1αi. Since α−1≠0, ∑i=0p−1αi should be zero.
Thus the last equation is also true. The proof is completed.Example 2.2.
Let
us considerp=5 and α=e−1(2π/5).
We have α5=1 and
(2.10)F5={0,1,2,3,4}.Let V→={1,α,α2,α3,α4},
then we have(2.11)V0→=(11111),V1→=(1αα2α3α4),V2→=(1α2α4αα3),V3→=(1α3αα4α2),V4→=(1α4α3α2α). It can be seen that(2.12)V0→×V0→T=5,Va→×Vb→T=5,for〈a+b〉=0,Va→×Vb→T=0,for〈a+b〉≠0.


## 2.2. Cocyclic Jacket Matrix with Sizep

Now we are going to constructp×p cocyclic Jacket matrix over complex number
field. For a given odd prime p,
let α=e−1(2π/p),
and(2.13)V→={1,α,α2,…,αp−2,αp−1}.Definition 2.3.
One has the following equation:(2.14)[J]p=Δ[V0→V1→⋮V→p−1]. The  inverse  of [J]p is denoted by [J]p−1. From Lemma 2.1, it can be easily checked that if [6](2.15)[J]p−1=1p[V0→TV1→T⋯V→p−1T]=1p[J]pT, then
(2.16)[J]p×[J]p−1=[I]p=[J]p−1×[J]p,[J]p×[J]pT=p[I]p.
According  to the Definition 1.1, from
(2.13)–(2.15),
[J]p
is a Jacket matrix over complex number of field. The following lemma shows that
[J]p
is acocyclic Jacket matrix [10].Lemma 2.4.
LetG=Fp with the operation i∘j=Δ〈i+j〉, C={1,α,…,αp−1} with traditional multiplication, the rows and
columns are indexed by the elements of Fp under the increasing order (i.e., 0,1,…,(p−1)), and the entry of (q,h) is φ(g,h).
Then, the Jacket matrix [J]p is a symmetric normalized cocyclic matrix.Proof.
Letg,h,j,i∈G=Fp.
Based on the above increasing order and from (1.3), we have
(2.17)φ(g,0)=φ(0,g)=φ(0,0)=1,φ(g,h)=α〈gh〉,φ(g,h∘k)=α〈g(h+k)〉,φ(g,h)φ(i,k)=α〈gh+ik〉.
Therefore, for any g,h,k∈G,
we have
(2.18)φ(g,h)φ(g∘h,k)=α〈g,h〉×α〈(g+h)k〉=α〈gh+(g+h)k〉,φ(g,h∘k)φ(h,k)=α〈g(h+k)〉×α〈hk〉=α〈g(h+k)+hk〉. Since
(2.19)〈gh+(g+h)k〉=〈g(h+k)+hk〉, we have
(2.20)φ(g,h)φ(g∘h,k)=φ(g,h∘k)φ(h,k). Therefore, [J]p is a cocyclic matrix.Hence, we have the following theorem.Theorem 2.5.
The matrix[J]p is a cocyclic Jacket matrix with size p over complex number field.Example 2.6.
Let
us considerp=5.
From Example 2.2, we have
(2.21)V0→=(11111),V1→=(1αα2α3α4),V2→=(1α2α4αα3),V3→=(1α3αα4α2),V4→=(1α4α3α2α),[J]5=[111111αα2α3α41α2α4αα31α3αα4α21α4α3α2α],([J]5)−1=15[111111α4α3α2α1α3αα4α21α2α4αα31αα2α3α4]=15[j−1]5T. Moreover, the Jacket matrix [J]5 can be mapped as shown in Table 1. It can be
verified that [J]5 is a cocyclic matrix.Table 1g∖h0123401111111αα2α3α421α2α4αα331α3αα4α241α4α3α2αExample 2.7.
Let us considerp=2,
this p is not an odd prime, but it is a prime. Let α=e−1(π/2),
we have α2=−1.
We have V→=(1,α2) and
(2.22)V0→=(1,1),V1→=(1,α2)=(1,−1).
Thus, we have
(2.23)[J]2=[111α2]=[111−1]=[H]2,[J]2−1=12[111α2]=12[111−1]=[H]2−1=12[J]2T,
where [H]2 is Walsh-Hadamard matrix.


## 2.3. Cocyclic Jacket Matrix
with Sizep1e1p2e2⋯pses

First we introduce some lemmas which
are useful to derive the construction of the cocyclic Jacket matrix with sizep1e1p2e2⋯pses.Lemma 2.8.
One has the following equation:(2.24)(Ai×j⊗Bh×k)(Cj×s⊗Dk×t)=(Ai×j×Cj×s)⊗(Bh×k×Dk×t), where ⊗ denotes the Kronecker product [1–3, 5, 6].Lemma 2.9.
One has the following equation:(2.25)(Ah⊗Bk)−1=Ah−1⊗Bk−1,(Ah⊗Bk)T=AhT⊗BkT. Now we are going to prove the
following theorem.Theorem 2.10.
IfAu×u and Bv×v are cocyclic Jacket matrices, then Au×u⊗Bv×v is also a cocyclic Jacket matrix with size uv.Proof.
SinceAu=[ai,j]u×u and Bv=[bs,t]v×v are cocyclic Jacket matrices, according to the
property of Jacket matrix, we have(2.26)Au×u−1=1Ca[ai,j−1]u×uT,Bv×v−1=1Cb[bs,t−1]v×vT. Let(2.27)Au×u⊗Bv×v=[miu+s,ju+t]uv×uv, where miu+s,ju+t=ai,jbs,t.
On the other hand, from (2.25) and (2.26), we have(2.28)(Au×u⊗Bv×v)−1=1CaCb[(ai,jbs,t)−1]uv×uvT=1CaCb[(miu+s,ju+t)−1]uv×uvT. From (2.27), (2.28), and
Definition 1.1, Au×u⊗Bv×v is a Jacket matrix. Next, we will prove that Au×u⊗Bv×v is also a cocyclic matrix.
Assume thatAu×u and Bv×v are cocyclic under the following row and column index
orders:(2.29)gs1(A)≺gs2(A)≺⋯≺gsu(A),forgsj(A)∈GA,gs1(B)≺gs2(B)≺⋯≺gsv(B),forgsk(B)∈GB, where s=r or c, grj(A) and gcj(A) denote the jth row index and jth column index of matrix A. Similarly, grk(B) and gck(B) denote the kth row index and kth column index of matrix B. Then, for matrix Au⊗Bv, the row and
column index orders are defined as follows:(2.30)gsj(A)gsk(B)≺gsi(A)gsh(B),(2.31)φAB(gri(A)grh(B),gcj(A)gck(B))=ΔφA(gri(A),gcj(A))φB(grh(B),gck(B)). In order to understand (2.29),
(2.30), and (2.31) better, we interpret matrices Au×u,Bv×v, and Au⊗Bv as the following three forms shown in Table 2. Since Au×u and Bv×v are cocyclic matrices, thus their elements φA(gri(A),gcj(A)) and φB(grh(B),gck(B)) should satisfy (1.3). From (2.31), and the above
fact, it can be verified that φAB(gri(A)grh(B),gcj(A)gck(B))=ΔφA(gri(A),gcj(A))φB(grh(B),gck(B)) is also satisfied (1.3) under the index orders
(2.30). Hence, Au×u⊗Bv×v is a cocyclic matrix.Table 2(a)g∖h⋯gcj(A)⋯⋮⋱⋮⋱gri(A)⋯φA(gri(A),gcj(A))⋯⋮⋱⋮⋱(b)g∖h⋯gck(B)⋯⋮⋱⋮⋱grh(B)⋯φB(grh(B),gck(B))⋯⋮⋱⋮⋱(c)g∖h⋯gcj(A)gck(B)⋯⋮⋱⋮⋱gri(A)grh(B)⋯φAB(gri(A)grh(B),gcj(A)gck(B))⋯⋮⋱⋮⋱Example 2.11.
Let us consider[J]3⊗[J]5,
let β=e−1(2π/5), and let μ=e−1(2π/3).
Then we have
(2.32)[J]3⊗[J]5=[111111ββ2β3β41β2β4ββ31β3ββ4β21β4β3β2β111111ββ2β3β41β2β4ββ31β3ββ4β21β4β3β2β111111ββ2β3β41β2β4ββ31β3ββ4β21β4β3β2β111111ββ2β3β41β2β4ββ31β3ββ4β21β4β3β2βμμμμμμβμβ2μβ3μβ4μμβ2μβ4μβμβ3μμβ3μβμβ4μβ2μμβ4μβ3μβ2μβμμ2μ2μ2μ2μ2μ2βμ2β2μ2β3μ2β4μ2μ2β2μ2β4μ2βμ2β3μ2μ2β3μ2βμ2β4μ2β2μ2μ2β4μ2β3μ2β2μ2βμ2111111ββ2β3β41β2β4ββ31β3ββ4β21β4β3β2βμ2μ2μ2μ2μ2μ2βμ2β2μ2β3μ2β4μ2μ2β2μ2β4μ2βμ2β3μ2μ2β3μ2βμ2β4μ2β2μ2μ2β4μ2β3μ2β2μ2βμ2μμμμμμβμβ2μβ3μβ4μμβ2μβ4μβμβ3μμβ3μβμβ4μβ2μμβ4μβ3μβ2μβμ].
It can be
easily verified that [J]3⊗[J]5 is a Jacket matrix. We also present its index
order matrix
as shown in Table 3, where
the row and column index orders are
(2.33)00≺01≺02≺03≺04≺10≺11≺12≺13≺14≺20≺21≺22≺23≺24,ij∘hk=Δ〈i+h〉3〈j+k〉5. For
example, 23∘14=02.
It can be easily verified that [J]5⊗[J]3 is a cocyclic matrix.
Next, we are going to construct a
cocyclic Jacket matrix using the complex number field with sizep1e1p2e2⋯pses,
where pi,
for i=1,2,…,s, are primes.Table 3g∖h00010203041011121314202122232400111111111111111011ββ2β3β41ββ2β3β41ββ2β3β4021β2β4ββ31β2β4ββ31β2β4ββ3031β3ββ4β21β3ββ4β21β3ββ4β2041β4β3β2β1β4β3β2β1β4β3β2β1011111μμμμμμ2μ2μ2μ2μ2111ββ2β3β4μβμβ2μβ3μβ4μμ2βμ2β2μ2β3μ2β4μ2121β2β4ββ3μβ2μβ4μβμβ3μμ2β2μ2β4μ2βμ2β3μ2131β3ββ4β2μβ3μβμβ4μβ2μμ2β3μ2βμ2β4μ2β2μ2141β4β3β2βμβ4μβ3μβ2μβμμ2β4μ2β3μ2β2μ2βμ22011111μ2μ2μ2μ2μ2μμμμμ211ββ2β3β4μ2βμ2β2μ2β3μ2β4μ2μβμβ2μβ3μβ4μ221β2β4ββ3μ2β2μ2β4μ2βμ2β3μ2μβ2μβ4μβμβ3μ231β3ββ4β2μ2β3μ2βμ2β4μ2β2μ2μβ3μβμβ4μβ2μ241β4β3β2βμ2β4μ2β3μ2β2μ2βμ2μβ4μβ3μβ2μβμDefinition 2.12.
One has
the following equation:(2.34)[J]p1e1p2e2⋯pses=Δ[J]p1e1⊗[J]p2e2⊗⋯⊗[J]pses, where
(2.35)[J]piei=Δ[J]pi⊗[J]pi⊗⋯⊗[J]pifori=1,2,…,s.
From
Lemma2.8 and Theorem 2.10, we have the following theorem.Theorem 2.13.
The
matrix from Definition2.12 is a cocyclic Jacket matrix over the complex number
field.Example 2.14.
Let us considerp1=3,
p2=2, and e1=e2=1. Thus,
N=2×3=6. Let β=e−1(π/3) and α=e−1(2π/3), that is, α=β2.
We have
(2.36)[J]6=[J]3⊗[J]2=[1111αα21α2α]⊗[111−1]=[1111111−11−11−111ααα2α21−1−ααα2−α211α2α2αα1−1α2−α2α−α]=[1111111−11−11−111β2β2β4β41−1β2β5β4β11β4β4β2β21−1β4ββ2β5].
From [19],
we know that (2.37)[JM]6=[1111111ββ2β5β4−11β2β4β4β211β5β4ββ2−11β4β2β2β411−11−11−1].
It can be seen that
(2.38)[100000000001001000000100000010010000]×[JM]6×[100000000001000010010000001000000100]=[J]6,
where [JM]6 is the generalized Jacket matrix of order 6.From Lemma2.9 and the definition
of Jp1e1p2e2⋯pses, it can be verified that Jp1e1p2e2⋯pses is an orthogonal matrix and its inverse matrix
can be determined as
(2.39)Jp1e1p2e2⋯pses−1=(Jp1e1⊗Jp2e2⊗⋯⊗Jpses)−1p1e1p2e2⋯pses=1p1e1p2e2⋯pses(Jp1e1−1⊗Jp2e2−2⊗⋯⊗Jpses−1),
where Jpiei−1=Jpi−1⊗Jpi−1⊗⋯⊗Jpi−1︸ei.Example 2.15.
Let us considerJ4[J]4=[J]2⊗[J]2=[H]2⊗[H]2.
Thus, we have
(2.40)[J]4=[11111−11−111−1−11−1−11].
The
Jacket matrix J4 can be mapped as shown in Table 4. Then J4 is also a cocyclic matrix.Table 4g∖h01230111111−11−1211−1−131−1−11


## 3. Cocyclic Jacket Matrices over Finite Field

In this
section, we will construct the cocyclic Jacket matrices overGF(2m).
Let α be a primitive element of GF(2m).
Then,(3.1)GF(2m)={0,1,α,α2,…,α2m−2}, and we have
the following lemma.Lemma 3.1.
One has the following
equation:(3.2)∑i=02m−2αri={1,forr=0,0,for1≤r≤2m−2.Proof.
It is evident that∑i=02m−2αri
contains 2m−1 terms, that is, odd terms. If r=0,
then
∑i=02m−2αri
is a sum of odd 1’s and should be 1. Thus, the first equation is
proved.
We now
consider the case of1≤r≤2m−2. Since αr(2m−2)=1 =1, we have(3.3)0=αr(2m−1)+1=(αr+1)(∑i=02m−2ari). Since 1≤r≤2m−2,
that is, αr+1≠0, we have ∑i=02m−2ari=0. The proof is completed.Let[JF]2m−1=[mi,j](2m−1)(2m−1), where
(3.4)mij=αijfor0≤i,j≤2m−2, then, we have
the following theorem.Theorem 3.2.
[JF]2m−1 is a cocyclic Jacket matrix.Proof.
Let(3.5)[JF]2m−1−1=[mi,j−1](2m−1)×(2m−1)T,where mi,j=αij. From the definition of [JF]2m−1 and Lemma 3.1, we have(3.6)[JF]2m−1×[JF]2m−1−1=[JF]2m−1−1×[JF]2m−1=I(2m−1). Hence, [JF]2m−1 is a Jacket matrix. Next, we will prove that [JF]2m−1 is also a cocyclic matrix. Let φ(i,j) be the entry of row i and column j, where the
order of rows and columns is from 0 to 2m−2.
From (3.4), we have
(3.7)φ(i,0)=φ(0,i)=φ(0,0)=α0=1,φ(i,j)=αij,φ(i,j∘h)=αi(j+k),φ(i,j)φ(h,k)=αij+hk.
Therefore,
for  any g,h,k∈Z2m−1, we have
(3.8)φ(g,h)φ(g∘h,k)=αgh×α(g+h)k=αgh+(g+h)k,φ(g,h∘k)φ(h,k)=αg(h+k)×αhk=αg(h+k)+hk.
Since 〈gh+(g+h)k〉=〈g(h+k)+hk〉,
we have(3.9)φ(g,h)φ(g∘h,k)=φ(g,h∘k)φ(h,k).
In terms of
(1.3), [JF]2m−1 is a cocyclic matrix. The proof is completed.Example 3.3.
Let us consider[JF]7=[JF]23−1.
Let α and x3+x+1=0 be the primitive element and primitive
polynomial of GF(23), respectively. Thus, GF(23)={α,α2,α3,α4,α5,α6}  and α7=1.
On the other hand, any element β∈GF(23) can be represented as a binary vector (b0,b1,b2), where bi∈{0,1} for i=0,1,2 such that
(3.10)β=b0+b1α+b2α2, as
shown in Table 5.Table 5
Binary representation ofGF(23).
ElementsBinary representation0(000)1(100)α(010)α2(001)α3(110)α4(011)α5(111)α6(101)Using
Table5, it can be easily checked that (3.9) is true for GF(23). Thus, we have
(3.11)[JF]7=[11111111αα2α3α4α5α61α2α4α6αα3α51α3α6α2α5αα41α4αα5α2α6α31α5α3αα6α4α21α6α5α4α3α2α],[JF]7−1=[11111111α6α5α4α3α2α1α5α3αα6α4α21α4αα5α2α6α31α3α6α2α5αα41α2α4α6αα3α51αα2α3α4α5α6], and
index
mapping of order-8 Cocyclic Jacket matrix (see Table 6).Table 6
Index mapping of order-8 Cocyclic Jacket matrix.
g∖h01234560111111111αα2α3α4α5α621α2α4α6αα3α531α3α6α2α5αα441α4αα5α2α6α351α5α3αα6α4α261α6α5α4α3α2αIt
can be verified that[JF]7 is a cocyclic Jacket matrix over GF(23).Example 3.4.
Let us consider[JF]32−1=[JF]8 over GF(32). Let α and x2+x+2=0 be the primitive and primitive polynomial of GF(32), respectively. Thus, GF(32)={0,1,α,α2,α3,α4,α5,α6,α7} and α8=1.
Conversely, any element β∈GF(32) can be represented as a vector over  GF(3):β=b0+b1α, where b0,b1∈{0,1,2} (see Table 7).Table 7
Binary representation ofGF(32).
ElementsBinary representation0(00)1(10)α(00)α2(21)α3(22)α4(02)α5(20)α6(12)α7(11)Using this table, it is easy to
deduce that (3.2) is true for GF(32) (change 2m to 3m). Thus, we have
(3.12)[JF]8=[111111111αα2α3α4α5α6α71α2α4α61α2α4α61α3α6αα4α7α2α51α41α41α41α41α5α2α7α4αα6α31α6α4α21α6α4α21α7α6α5α4α3α2α],[JF]8−1=[111111111α7α6α5α4α3α2α1α6α4α21α6α4α21α5α2α7α4αα6α31α41α41α41α41α3α6αα4α7α2α51α2α4α61α2α4α61αα2α3α4α5α6α7],
and index
mapping of order-9 Cocyclic Jacket matrix (see Table 8).Table 8
Index mapping of order-9 Cocyclic Jacket matrix.
g∖h0123456701111111111αα2α3α4α5α6α721α2α4α61α2α4α631α3α6αα4α7α2α541α41α41α41α451α5α2α7α4αα6α361α6α4α21α6α4α271α7α6α5α4α3α2αIt is easy
to verify that[JF]8 is a cocyclic Jacket matrix over GF(32).Remark 3.5.
We can also construct cocyclic Jacket matrices
based on additive characters of the finite fieldFq and first-order q-ary Reed-Muller codes RMp(1,n) [20], where Fq={α1=0,α2,…,αq} is a finite field of q elements, q=pm, and p is a prime number. The way of construction is described by the following
lemma.Lemma 3.6.
The cocyclic Jacket matrix with
orderN=pn is Jpn=[ωi→∘j→], where ω=exp⁡(2π−1/p),
and one
defines
(3.13)i→·j→=(i0,i1,…,in−1)·(j0,j1,…,jn−1)=〈i0×j0〉p+〈i1×j1〉p+⋯+〈in−1×jn−1〉p for  0≤ik, jk≤p−1
(0≤k≤n−1).Example 3.7.
Letp=3, n=2, and m=1, the finite field of q=p1=3 elements F3={0,1,2},RM3(1,2) is as follows:
(3.14)RM3(1,2)=(000012021000012021000012021000012021111120102222201210000012021222201210111120102). The entries of J32 are shown in
Table 9.Table 9
The correspondence between the indices and the
entries ofJ32.
i/j→j/j→012345678000102101112202122000ω0ω0ω0ω0ω0ω0ω0ω0ω0101ω0ω1ω2ω0ω1ω2ω0ω1ω2202ω0ω2ω1ω0ω2ω1ω0ω2ω1310ω0ω0ω0ω1ω1ω1ω2ω2ω2411ω0ω1ω2ω1ω2ω0ω2ω0ω1512ω0ω2ω1ω1ω0ω2ω2ω1ω0620ω0ω0ω0ω2ω2ω2ω1ω1ω1721ω0ω1ω2ω2ω0ω1ω1ω2ω0822ω0ω2ω1ω2ω1ω0ω1ω0ω2From Table9, we can see
when i=2, j=4,
we have
(3.15)ω(02)∘(11)=ω〈0×1〉3+〈2×1〉3=ω2. The other entries can be
obtained using the same
fashion, perfectly.


## 4. The Factorization of Cocyclic Jacket Matrices and Unitary Matrices

Definition 4.1.
A square matrixU is a unitary matrix if U−1=UH, where UH denote the conjugate transpose and U−1 is the matrix inverse.Proposition 4.2.
The matrixUn=(1/c)Jn is a unitary matrix where Jn is the cocyclic Jacket matrix, c is the normalized value for Jn.Proof.
From
the definition of Jacket matrix, we haveJn=[j]n−1=(1/c)[j−1]nT,
and the entries in cocyclic Jacket matrices
also satisfy ∥j∥=1,
we have j−1=j¯, then [j]n−1=(1/c)[j¯]nT,[j¯]nT=[j]nH=c[j]n−1.
Certainly,(4.1)Un⋅UnH=1c[j]n⋅1c[j]nH=1c[j]n⋅[j]nH=1c[j]n⋅c[j]n−1=In.Example 4.3.
Based
on Example2.14, we have(4.2)[J]6=[J]3⊗[J]2=[1111111−11−11−111ααα2α21−1−ααα2−α211α2α2αα1−1α2−α2α−α], where α=e−1(2π/3)=eiθ,
i=1, θ=2π/3,
then(4.3)[U]6⋅[U]6H=16[J]6⋅[J]6H=16[1111111−11−11−111eiθeiθei2θei2θ1−1−eiθeiθei2θ−ei2θ11ei2θei2θeiθeiθ1−1ei2θ−ei2θeiθ−eiθ][1111111−11−11−111e−iθ−e−iθe−2iθe−2iθ1−1e−iθe−iθe−2iθ−e−2iθ11e−2iθe−2iθe−iθe−iθ1−1e−2iθ−e−2iθe−iθ−e−iθ]=[I]6. A special  feature of cocyclic
Jacket matrices has been introduced in [21]. If the cocyclic Jacket matrices with order N=p1p2⋯pn, pi is the prime number, then (4.4)JN=Jp1⊗Jp2⊗⋯⊗Jpn=Ap11Ap22⋯Apnn,where(4.5)Apmm=Ip1⊗Ip2⊗⋯Ipm−1︸m−1⊗Jpm⊗Ipm+1⊗Ipm+2⊗⋯⊗Ipn︸n−m, and based on this characteristic
of cocyclic Jacket matrices, we can easy decompose the unitary matrices with
sparse matrices(4.6)UN=1NJN=1NAp11Ap22⋯Apnn. From (4.6), the U6 can be decomposed as
(4.7)U6=16(I3⊗J2)(J3⊗I2)=16(I3⊗2U2)(3U3⊗I2)=(I3⊗U2)(U3⊗I2)=16[1100001−10000001100001−10000001100001−1][10101001010110eiθ0ei2θ0010eiθ0ei2θ10ei2θ0eiθ0010ei2θ0eiθ].
Clearly,
(4.7) is the new factorization matrix.


## 5. Conclusions

In
this paper, we present a new class of cocyclic
Jacket matrices over complex number field and finite field. Using this way, we
can get such kind of matrix with orderpk directly, for the other orders N=p1k1p2k2⋯pnkn, they can be obtained from
the Kronecker product with some matrices whose orders are piki. The cocyclic Jacket matrices also have a close
relation with unitary matrices. In particular, the factorizations of unitary
matrices have the similar patterns with that of cocyclic Jacket matrices. Therefore,
the door for using cocyclic Jacket matrices in signal processing [7], cryptography
[9], mobile communication [4, 6], Jacket transform coding [13, 20],
and quantum processing [17, 22] is opened.



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