A Simple Cocyclic Jacket Matrices

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.

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 𝐽_𝑘. 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 size 𝑚×𝑚 has nonzero elements, and an inverse form which is only from the element-wise inverse and then transpose, such as[𝐽]_𝑚×𝑚=𝑗_0,0𝑗_0,1⋯𝑗_0,𝑚−1𝑗_1,0𝑗_1,1⋯𝑗_1,𝑚−1𝑗⋮⋮⋱⋮_𝑚−1,0𝑗_𝑚−1,1⋯𝑗_{𝑚−1,𝑚−1}, (1.1) and its inverse is[𝐽]^Δ−1_𝑚×𝑚=1𝐶𝑗⁻¹_0,0𝑗⁻¹_0,1⋯𝑗⁻¹_0,𝑚−1𝑗⁻¹_1,0𝑗⁻¹_1,1⋯𝑗⁻¹_1,𝑚−1𝑗⋮⋮⋱⋮_{−1𝑚−1,0}𝑗⁻¹_𝑚−1,1⋯𝑗⁻¹_{𝑚−1,𝑚−1}^𝑇, (1.2) 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. If 𝐺 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 𝜑(𝑔,ℎ)𝜑(𝑔∘ℎ,𝑘)=𝜑(𝑔,ℎ∘𝑘)𝜑(ℎ,𝑘), (1.3) where 𝑔,ℎ,𝑘∈𝐺. A square matrix 𝑀_𝜑, whose rows and columns are indexed by the elements of 𝐺, with entry 𝜑(𝑔,ℎ) in the position (𝑔,ℎ), that is, 𝑀_𝜑=[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=2^𝑙,2^𝑙𝑝, 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 Section 2, 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

Let 𝑝 be an odd prime integer and 𝛼=𝑒^{√−1(2𝜋/𝑝)}. Thus, we have 𝛼^𝑝=1, and 𝐹_𝑝={0,1,2,…,𝑝−1} with the operations for ⟨𝑎⋅𝑏⟩ are the finite field, where⟨𝑎⟩=^Δ𝑎mod𝑝. (2.1) Let 𝑎∈𝐹_𝑝, we define a function𝑓_𝑎(𝑥)=^Δ⟨𝑎×𝑥⟩. (2.2) Let 𝑉^→=(𝛼^𝑣0,𝛼^𝑣1,…,𝛼^𝑣𝑝−1) be a vector, where 𝑣_𝑖∈𝐹_𝑝 for 𝑖=0,1,…,𝑝−1. We define a vector𝑉^→_𝑎=(𝛼^{𝑓𝑎(𝑣0)},𝛼^{𝑓𝑎(𝑣1)},…,𝛼^{𝑓𝑎(𝑣𝑝−1)}). (2.3) We have the following lemma.

Lemma 2.1. Let 𝑉^→=(1,𝛼¹,𝛼²,…,𝛼^𝑝−1), then𝑉^→₀×𝑉^→₀^𝑇=𝑝,_→𝑉_𝑎×𝑉^→_𝑏^𝑇=𝑝,for⟨𝑎+𝑏⟩=0,_→𝑉_𝑎×𝑉^→_𝑏^𝑇=0,for⟨𝑎+𝑏⟩≠0. (2.4)

Proof. The first equation can be easily proved because 𝑉^→₀=(1,1,…,1). For the second equation, since ⟨𝑎+𝑏⟩=0, we have (𝑓_𝑎(𝑣_𝑖)+𝑓_𝑏(𝑣_𝑖))mod𝑝=(𝑎+𝑏)𝑣_𝑖mod𝑝=0. (2.5) 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,{0,1,2,…,𝑝−1}={0,⟨𝑐⟩,⟨2𝑐⟩,…,⟨(𝑝−1)𝑐⟩}. (2.6) Furthermore, for ⟨𝑎+𝑏⟩≠0, we have {0,1,2,…,𝑝−1}={0,⟨𝑎+𝑏⟩,⟨2(𝑎+𝑏)⟩,…,⟨(𝑝−1)(𝑎+𝑏)⟩}, (2.7) that is,𝑉^→_𝑎×𝑉^→_𝑏=^𝑝−1_𝑖=0𝛼^𝑖. (2.8) On the other hand, from 𝛼^𝑝=1, we have 0=𝛼^𝑝−1=(𝛼−1)^𝑝−1_𝑖=0𝛼^𝑖. (2.9) Since 𝛼−1≠0, ∑^𝑝−1_𝑖=0𝛼^𝑖 should be zero. Thus the last equation is also true. The proof is completed.

Example 2.2. Let us consider 𝑝=5 and 𝛼=𝑒^{√−1(2𝜋/5)}. We have 𝛼⁵=1 and 𝐹₅={0,1,2,3,4}. (2.10)Let 𝑉^→={1,𝛼,𝛼²,𝛼³,𝛼⁴}, then we have𝑉^→₀=11111_→,𝑉₁=1𝛼𝛼²𝛼³𝛼⁴,_→𝑉₂=1𝛼²𝛼⁴𝛼𝛼³,_→𝑉₃=1𝛼³𝛼𝛼⁴𝛼²,_→𝑉₄=1𝛼⁴𝛼³𝛼²𝛼. (2.11) It can be seen that𝑉^→₀×𝑉^→₀^𝑇=5,_→𝑉_𝑎×𝑉^→_𝑏^𝑇=5,for⟨𝑎+𝑏⟩=0,_→𝑉_𝑎×𝑉^→_𝑏^𝑇=0,for⟨𝑎+𝑏⟩≠0. (2.12)

2.2. Cocyclic Jacket Matrix with Size 𝑝

Now we are going to construct 𝑝×𝑝 cocyclic Jacket matrix over complex number field. For a given odd prime 𝑝, let 𝛼=𝑒^{√−1(2𝜋/𝑝)}, and𝑉^→={1,𝛼,𝛼²,…,𝛼^𝑝−2,𝛼^𝑝−1}. (2.13)

Definition 2.3. One has the following equation: [𝐽]=_𝑝^Δ→𝑉_0→𝑉_1→⋮𝑉_𝑝−1. (2.14) The inverse of [𝐽]_𝑝 is denoted by [𝐽]_𝑝⁻¹. From Lemma 2.1, it can be easily checked that if [6][𝐽]_𝑝⁻¹=1𝑝^→𝑉𝑉₀^𝑇→₁^𝑇⋯𝑉^→𝑇_𝑝−1=1𝑝[𝐽]^𝑇_𝑝, (2.15) then [𝐽]_𝑝×[𝐽]_𝑝⁻¹=[𝐼]_𝑝=[𝐽]_𝑝⁻¹×[𝐽]_𝑝,[𝐽]_𝑝×[𝐽]^𝑇_𝑝=𝑝[𝐼]_𝑝. (2.16) 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. Let 𝐺=𝐹_𝑝 with the operation 𝑖∘𝑗=^Δ⟨𝑖+𝑗⟩, 𝐶={1,𝛼,…,𝛼^𝑝−1} with traditional multiplication, the rows and columns are indexed by the elements of 𝐹_𝑝 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. Let 𝑔,ℎ,𝑗,𝑖∈𝐺=𝐹_𝑝. Based on the above increasing order and from (1.3), we have 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⟩}. (2.17) Therefore, for any 𝑔,ℎ,𝑘∈𝐺, we have 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⟩}. (2.18) Since ⟨𝑔ℎ+(𝑔+ℎ)𝑘⟩=⟨𝑔(ℎ+𝑘)+ℎ𝑘⟩, (2.19) we have 𝜑(𝑔,ℎ)𝜑(𝑔∘ℎ,𝑘)=𝜑(𝑔,ℎ∘𝑘)𝜑(ℎ,𝑘). (2.20) 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 consider 𝑝=5. From Example 2.2, we have 𝑉^→₀=11111_→,𝑉₁=1𝛼𝛼²𝛼³𝛼⁴,_→𝑉₂=1𝛼²𝛼⁴𝛼𝛼³,_→𝑉₃=1𝛼³𝛼𝛼⁴𝛼²,_→𝑉₄=1𝛼⁴𝛼³𝛼²𝛼,[𝐽]₅=111111𝛼𝛼²𝛼³𝛼⁴1𝛼²𝛼⁴𝛼𝛼³1𝛼³𝛼𝛼⁴𝛼²1𝛼⁴𝛼³𝛼²𝛼,([𝐽]₅)⁻¹=15111111𝛼⁴𝛼³𝛼²𝛼1𝛼³𝛼𝛼⁴𝛼²1𝛼²𝛼⁴𝛼𝛼³1𝛼𝛼²𝛼³𝛼⁴=15[𝑗⁻¹]^𝑇₅. (2.21) 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 1

Example 2.7. Let us consider 𝑝=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 𝑉^→₀=(1,1),_→𝑉₁=(1,𝛼²)=(1,−1). (2.22) Thus, we have [𝐽]₂=111𝛼²=111−1=[𝐻]₂,[𝐽]₂⁻¹=12111𝛼²=12111−1=[𝐻]₂⁻¹=12[𝐽]^𝑇₂, (2.23) where [𝐻]₂ is Walsh-Hadamard matrix.

2.3. Cocyclic Jacket Matrix with Size 𝑝^𝑒1₁𝑝^𝑒2₂⋯𝑝^𝑒𝑠_𝑠

First we introduce some lemmas which are useful to derive the construction of the cocyclic Jacket matrix with size 𝑝^𝑒1₁𝑝^𝑒2₂⋯𝑝^𝑒𝑠_𝑠.

Lemma 2.8. One has the following equation:(𝐴_𝑖×𝑗⊗𝐵_ℎ×𝑘)(𝐶_𝑗×𝑠⊗𝐷_𝑘×𝑡)=(𝐴_𝑖×𝑗×𝐶_𝑗×𝑠)⊗(𝐵_ℎ×𝑘×𝐷_𝑘×𝑡), (2.24) where ⊗ denotes the Kronecker product [1–3, 5, 6].

Lemma 2.9. One has the following equation:(𝐴_ℎ⊗𝐵_𝑘)⁻¹=𝐴_ℎ⁻¹⊗𝐵_𝑘⁻¹,(𝐴_ℎ⊗𝐵_𝑘)^𝑇=𝐴^𝑇_ℎ⊗𝐵^𝑇_𝑘. (2.25) Now we are going to prove the following theorem.

Theorem 2.10. If 𝐴_𝑢×𝑢 and 𝐵_𝑣×𝑣 are cocyclic Jacket matrices, then 𝐴_𝑢×𝑢⊗𝐵_𝑣×𝑣 is also a cocyclic Jacket matrix with size 𝑢𝑣.

Proof. Since 𝐴_𝑢=[𝑎_𝑖,𝑗]_𝑢×𝑢 and 𝐵_𝑣=[𝑏_𝑠,𝑡]_𝑣×𝑣 are cocyclic Jacket matrices, according to the property of Jacket matrix, we have𝐴_𝑢×𝑢⁻¹=1𝐶_𝑎[𝑎_𝑖,𝑗⁻¹]^𝑇_𝑢×𝑢,𝐵_𝑣×𝑣⁻¹=1𝐶_𝑏[𝑏_𝑠,𝑡⁻¹]^𝑇_𝑣×𝑣. (2.26) Let𝐴_𝑢×𝑢⊗𝐵_𝑣×𝑣=[𝑚_{𝑖𝑢+𝑠,𝑗𝑢+𝑡}]_{𝑢𝑣×𝑢𝑣}, (2.27) where 𝑚_{𝑖𝑢+𝑠,𝑗𝑢+𝑡}=𝑎_𝑖,𝑗𝑏_𝑠,𝑡. On the other hand, from (2.25) and (2.26), we have(𝐴_𝑢×𝑢⊗𝐵_𝑣×𝑣)⁻¹=1𝐶_𝑎𝐶_𝑏[(𝑎_𝑖,𝑗𝑏_𝑠,𝑡)⁻¹]^𝑇_{𝑢𝑣×𝑢𝑣}=1𝐶_𝑎𝐶_𝑏[(𝑚_{𝑖𝑢+𝑠,𝑗𝑢+𝑡})⁻¹]^𝑇_{𝑢𝑣×𝑢𝑣}. (2.28) From (2.27), (2.28), and Definition 1.1, 𝐴_𝑢×𝑢⊗𝐵_𝑣×𝑣 is a Jacket matrix. Next, we will prove that 𝐴_𝑢×𝑢⊗𝐵_𝑣×𝑣 is also a cocyclic matrix.
Assume that 𝐴_𝑢×𝑢 and 𝐵_𝑣×𝑣 are cocyclic under the following row and column index orders:𝑔^(𝐴)_𝑠1≺𝑔^(𝐴)_𝑠2≺⋯≺𝑔^(𝐴)_𝑠𝑢,for𝑔^(𝐴)_𝑠𝑗∈𝐺_𝐴,𝑔^(𝐵)_𝑠1≺𝑔^(𝐵)_𝑠2≺⋯≺𝑔^(𝐵)_𝑠𝑣,for𝑔^(𝐵)_𝑠𝑘∈𝐺_𝐵, (2.29) where 𝑠=𝑟 or 𝑐, 𝑔^(𝐴)_𝑟𝑗 and 𝑔^(𝐴)_𝑐𝑗 denote the 𝑗th row index and 𝑗th column index of matrix 𝐴. Similarly, 𝑔^(𝐵)_𝑟𝑘 and 𝑔^(𝐵)_𝑐𝑘 denote the 𝑘th row index and 𝑘th column index of matrix 𝐵. Then, for matrix 𝐴_𝑢⊗𝐵_𝑣, the row and column index orders are defined as follows:𝑔^(𝐴)_𝑠𝑗𝑔^(𝐵)_𝑠𝑘≺𝑔^(𝐴)_𝑠𝑖𝑔^(𝐵)_𝑠ℎ𝜑_𝐴𝐵,(𝑔^(𝐴)_𝑟𝑖𝑔^(𝐵)_𝑟ℎ,𝑔^(𝐴)_𝑐𝑗𝑔^(𝐵)_𝑐𝑘)=^Δ𝜑_𝐴(𝑔^(𝐴)_𝑟𝑖,𝑔^(𝐴)_𝑐𝑗)𝜑_𝐵(𝑔^(𝐵)_𝑟ℎ,𝑔^(𝐵)_𝑐𝑘). (2.30)(2.31) In order to understand (2.29), (2.30), and (2.31) better, we interpret matrices 𝐴_𝑢×𝑢,𝐵_𝑣×𝑣, and 𝐴_𝑢⊗𝐵_𝑣 as the following three forms shown in Table 2. Since 𝐴_𝑢×𝑢 and 𝐵_𝑣×𝑣 are cocyclic matrices, thus their elements 𝜑_𝐴(𝑔^(𝐴)_𝑟𝑖,𝑔^(𝐴)_𝑐𝑗) and 𝜑_𝐵(𝑔^(𝐵)_𝑟ℎ,𝑔^(𝐵)_𝑐𝑘) should satisfy (1.3). From (2.31), and the above fact, it can be verified that 𝜑_𝐴𝐵(𝑔^(𝐴)_𝑟𝑖𝑔^(𝐵)_𝑟ℎ,𝑔^(𝐴)_𝑐𝑗𝑔^(𝐵)_𝑐𝑘)=^Δ𝜑_𝐴(𝑔^(𝐴)_𝑟𝑖,𝑔^(𝐴)_𝑐𝑗)𝜑_𝐵(𝑔^(𝐵)_𝑟ℎ,𝑔^(𝐵)_𝑐𝑘) is also satisfied (1.3) under the index orders (2.30). Hence, 𝐴_𝑢×𝑢⊗𝐵_𝑣×𝑣 is a cocyclic matrix.

Table 2

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 [𝐽]₃⊗[𝐽]₅=111111𝛽𝛽²𝛽³𝛽⁴1𝛽²𝛽⁴𝛽𝛽³1𝛽³𝛽𝛽⁴𝛽²1𝛽⁴𝛽³𝛽²𝛽111111𝛽𝛽²𝛽³𝛽⁴1𝛽²𝛽⁴𝛽𝛽³1𝛽³𝛽𝛽⁴𝛽²1𝛽⁴𝛽³𝛽²𝛽111111𝛽𝛽²𝛽³𝛽⁴1𝛽²𝛽⁴𝛽𝛽³1𝛽³𝛽𝛽⁴𝛽²1𝛽⁴𝛽³𝛽²𝛽111111𝛽𝛽²𝛽³𝛽⁴1𝛽²𝛽⁴𝛽𝛽³1𝛽³𝛽𝛽⁴𝛽²1𝛽⁴𝛽³𝛽²𝛽𝜇𝜇𝜇𝜇𝜇𝜇𝛽𝜇𝛽²𝜇𝛽³𝜇𝛽⁴𝜇𝜇𝛽²𝜇𝛽⁴𝜇𝛽𝜇𝛽³𝜇𝜇𝛽³𝜇𝛽𝜇𝛽⁴𝜇𝛽²𝜇𝜇𝛽⁴𝜇𝛽³𝜇𝛽²𝜇𝜇𝛽𝜇²𝜇²𝜇²𝜇²𝜇²𝜇²𝛽𝜇²𝛽²𝜇²𝛽³𝜇²𝛽⁴𝜇²𝜇²𝛽²𝜇²𝛽⁴𝜇²𝛽𝜇²𝛽³𝜇²𝜇²𝛽³𝜇²𝛽𝜇²𝛽⁴𝜇²𝛽²𝜇²𝜇²𝛽⁴𝜇²𝛽³𝜇²𝛽²𝜇²𝛽𝜇²111111𝛽𝛽²𝛽³𝛽⁴1𝛽²𝛽⁴𝛽𝛽³1𝛽³𝛽𝛽⁴𝛽²1𝛽⁴𝛽³𝛽²𝛽𝜇²𝜇²𝜇²𝜇²𝜇²𝜇²𝛽𝜇²𝛽²𝜇²𝛽³𝜇²𝛽⁴𝜇²𝜇²𝛽²𝜇²𝛽⁴𝜇²𝛽𝜇²𝛽³𝜇²𝜇²𝛽³𝜇²𝛽𝜇²𝛽⁴𝜇²𝛽²𝜇²𝜇²𝛽⁴𝜇²𝛽³𝜇²𝛽²𝜇²𝛽𝜇²𝜇𝜇𝜇𝜇𝜇𝜇𝛽𝜇𝛽²𝜇𝛽³𝜇𝛽⁴𝜇𝜇𝛽²𝜇𝛽⁴𝜇𝛽𝜇𝛽³𝜇𝜇𝛽³𝜇𝛽𝜇𝛽⁴𝜇𝛽²𝜇𝜇𝛽⁴𝜇𝛽³𝜇𝛽²𝜇𝛽𝜇. (2.32) 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 00≺01≺02≺03≺04≺10≺11≺12≺13≺14≺20≺21≺22≺23≺24,𝑖𝑗∘ℎ𝑘=^Δ⟨𝑖+ℎ⟩₃⟨𝑗+𝑘⟩₅. (2.33) 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 size 𝑝^𝑒1₁𝑝^𝑒2₂⋯𝑝^𝑒𝑠_𝑠, where 𝑝_𝑖, for 𝑖=1,2,…,𝑠, are primes.

Table 3

Definition 2.12. One has the following equation:[𝐽]=_𝑝^𝑒_11𝑝^𝑒_22⋯𝑝^𝑒𝑠_𝑠^Δ[𝐽]_𝑝^𝑒₁₁⊗[𝐽]_𝑝^𝑒₂₂⊗⋯⊗[𝐽]_𝑝^𝑒𝑠_𝑠, (2.34) where [𝐽]=_𝑝^𝑒_𝑖𝑖^Δ[𝐽]_𝑝𝑖⊗[𝐽]_𝑝𝑖⊗⋯⊗[𝐽]_𝑝𝑖for𝑖=1,2,…,𝑠. (2.35)
From Lemma 2.8 and Theorem 2.10, we have the following theorem.

Theorem 2.13. The matrix from Definition 2.12 is a cocyclic Jacket matrix over the complex number field.

Example 2.14. Let us consider 𝑝₁=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 [𝐽]₆=[𝐽]₃⊗[𝐽]₂=1111𝛼𝛼²1𝛼²𝛼⊗=111−11111111−11−11−111𝛼𝛼𝛼²𝛼²1−1−𝛼𝛼𝛼²−𝛼²11𝛼²𝛼²𝛼𝛼1−1𝛼²−𝛼²=𝛼−𝛼1111111−11−11−111𝛽²𝛽²𝛽⁴𝛽⁴1−1𝛽²𝛽⁵𝛽⁴𝛽11𝛽⁴𝛽⁴𝛽²𝛽²1−1𝛽⁴𝛽𝛽²𝛽⁵. (2.36) From [19], we know that [𝐽𝑀]₆=1111111𝛽𝛽²𝛽⁵𝛽⁴−11𝛽²𝛽⁴𝛽⁴𝛽²11𝛽⁵𝛽⁴𝛽𝛽²−11𝛽⁴𝛽²𝛽²𝛽⁴11−11−11−1. (2.37) It can be seen that 100000000001001000000100000010010000×[𝐽𝑀]₆×100000000001000010010000001000000100=[𝐽]₆, (2.38) where [𝐽𝑀]₆ is the generalized Jacket matrix of order 6.

From Lemma 2.9 and the definition of 𝐽_𝑝1^𝑒1_𝑝2^𝑒2_⋯𝑝^𝑒𝑠_𝑠, it can be verified that 𝐽_𝑝1^𝑒1_𝑝2^𝑒2_⋯𝑝^𝑒𝑠_𝑠 is an orthogonal matrix and its inverse matrix can be determined as 𝐽_𝑝⁻¹_{1𝑒1𝑝2𝑒2⋯𝑝𝑠𝑒𝑠}=(𝐽_𝑝1^𝑒1⊗𝐽_𝑝2^𝑒2⊗⋯⊗𝐽_{𝑝𝑠𝑒𝑠})⁻¹𝑝₁^𝑒1𝑝₂^𝑒2⋯𝑝_𝑠^𝑒𝑠=1𝑝₁^𝑒1𝑝₂^𝑒2⋯𝑝_𝑠^𝑒𝑠(𝐽_𝑝⁻¹_1𝑒1⊗𝐽_𝑝⁻²_2𝑒2⊗⋯⊗𝐽_𝑝⁻¹_𝑠𝑒𝑠), (2.39) where 𝐽_𝑝⁻¹_𝑖𝑒𝑖=𝐽_𝑝⁻¹_𝑖⊗𝐽_𝑝⁻¹_𝑖⊗⋯⊗𝐽_𝑝⁻¹_𝑖𝑒𝑖.

Example 2.15. Let us consider 𝐽₄[𝐽]₄=[𝐽]₂⊗[𝐽]₂=[𝐻]₂⊗[𝐻]₂. Thus, we have [𝐽]₄=11111−11−111−1−11−1−11. (2.40) The Jacket matrix 𝐽₄ can be mapped as shown in Table 4. Then 𝐽₄ is also a cocyclic matrix.

Table 4

3. Cocyclic Jacket Matrices over Finite Field

In this section, we will construct the cocyclic Jacket matrices over 𝐺𝐹(2^𝑚). Let 𝛼 be a primitive element of 𝐺𝐹(2^𝑚). Then,𝐺𝐹(2^𝑚)={0,1,𝛼,𝛼²,…,𝛼^2𝑚−2}, (3.1) and we have the following lemma.

Lemma 3.1. One has the following equation:^2𝑚−2_𝑖=0𝛼^𝑟𝑖=1,for𝑟=0,0,for1≤𝑟≤2^𝑚−2. (3.2)

Proof. It is evident that ∑^2𝑚−2_𝑖=0𝛼^𝑟𝑖 contains 2^𝑚−1 terms, that is, odd terms. If 𝑟=0, then ∑^2𝑚−2_𝑖=0𝛼^𝑟𝑖 is a sum of odd 1’s and should be 1. Thus, the first equation is proved.
We now consider the case of 1≤𝑟≤2^𝑚−2. Since 𝛼^{𝑟(2𝑚−2)=1} =1, we have0=𝛼^{𝑟(2𝑚−1)}+1=(𝛼^𝑟+1)(^2𝑚−2_𝑖=0𝑎^𝑟𝑖). (3.3) Since 1≤𝑟≤2^𝑚−2, that is, 𝛼^𝑟+1≠0, we have ∑^2𝑚−2_𝑖=0𝑎^𝑟𝑖=0. The proof is completed.

Let [𝐽𝐹]₂^𝑚₋₁=[𝑚_𝑖,𝑗]_{(2𝑚−1)(2𝑚−1)}, where 𝑚_𝑖𝑗=𝛼^𝑖𝑗for0≤𝑖,𝑗≤2^𝑚−2, (3.4) then, we have the following theorem.

Theorem 3.2. [𝐽𝐹]₂^𝑚₋₁ is a cocyclic Jacket matrix.

Proof. Let[𝐽𝐹]₂^−1𝑚₋₁=[𝑚⁻¹_𝑖,𝑗]^𝑇_{(2𝑚−1)×(2𝑚−1)}, (3.5)where 𝑚_𝑖,𝑗=𝛼^𝑖𝑗. From the definition of [𝐽𝐹]₂^𝑚₋₁ and Lemma 3.1, we have[𝐽𝐹]₂^𝑚₋₁×[𝐽𝐹]₂^−1𝑚₋₁=[𝐽𝐹]₂^−1𝑚₋₁×[𝐽𝐹]₂^𝑚₋₁=𝐼₍₂^𝑚₋₁₎. (3.6) Hence, [𝐽𝐹]₂^𝑚₋₁ is a Jacket matrix. Next, we will prove that [𝐽𝐹]₂^𝑚₋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 2^𝑚−2. From (3.4), we have 𝜑(𝑖,0)=𝜑(0,𝑖)=𝜑(0,0)=𝛼⁰=1,𝜑(𝑖,𝑗)=𝛼^𝑖𝑗,𝜑(𝑖,𝑗∘ℎ)=𝛼^{𝑖(𝑗+𝑘)},𝜑(𝑖,𝑗)𝜑(ℎ,𝑘)=𝛼^{𝑖𝑗+ℎ𝑘}. (3.7) Therefore, for any 𝑔,ℎ,𝑘∈𝑍₂^𝑚₋₁, we have 𝜑(𝑔,ℎ)𝜑(𝑔∘ℎ,𝑘)=𝛼^𝑔ℎ×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}. (3.8) Since ⟨𝑔ℎ+(𝑔+ℎ)𝑘⟩=⟨𝑔(ℎ+𝑘)+ℎ𝑘⟩, we have𝜑(𝑔,ℎ)𝜑(𝑔∘ℎ,𝑘)=𝜑(𝑔,ℎ∘𝑘)𝜑(ℎ,𝑘). (3.9) In terms of (1.3), [𝐽𝐹]₂^𝑚₋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 𝑏_𝑖∈{0,1} for 𝑖=0,1,2 such that 𝛽=𝑏₀+𝑏₁𝛼+𝑏₂𝛼², (3.10) as shown in Table 5.

Table 5 

Binary representation of 𝐺𝐹(2³).

Using Table 5, it can be easily checked that (3.9) is true for 𝐺𝐹(2³). Thus, we have [𝐽𝐹]₇=11111111𝛼𝛼²𝛼³𝛼⁴𝛼⁵𝛼⁶1𝛼²𝛼⁴𝛼⁶𝛼𝛼³𝛼⁵1𝛼³𝛼⁶𝛼²𝛼⁵𝛼𝛼⁴1𝛼⁴𝛼𝛼⁵𝛼²𝛼⁶𝛼³1𝛼⁵𝛼³𝛼𝛼⁶𝛼⁴𝛼²1𝛼⁶𝛼⁵𝛼⁴𝛼³𝛼²𝛼,[𝐽𝐹]₇⁻¹=11111111𝛼⁶𝛼⁵𝛼⁴𝛼³𝛼²𝛼1𝛼⁵𝛼³𝛼𝛼⁶𝛼⁴𝛼²1𝛼⁴𝛼𝛼⁵𝛼²𝛼⁶𝛼³1𝛼³𝛼⁶𝛼²𝛼⁵𝛼𝛼⁴1𝛼²𝛼⁴𝛼⁶𝛼𝛼³𝛼⁵1𝛼𝛼²𝛼³𝛼⁴𝛼⁵𝛼⁶, (3.11) and index mapping of order-8 Cocyclic Jacket matrix (see Table 6).

Table 6 

Index mapping of order-8 Cocyclic Jacket matrix.

It 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 of 𝐺𝐹(3²).

Using this table, it is easy to deduce that (3.2) is true for 𝐺𝐹(3²) (change 2^𝑚 to 3^𝑚). Thus, we have [𝐽𝐹]₈=111111111𝛼𝛼²𝛼³𝛼⁴𝛼⁵𝛼⁶𝛼⁷1𝛼²𝛼⁴𝛼⁶1𝛼²𝛼⁴𝛼⁶1𝛼³𝛼⁶𝛼𝛼⁴𝛼⁷𝛼²𝛼⁵1𝛼⁴1𝛼⁴1𝛼⁴1𝛼⁴1𝛼⁵𝛼²𝛼⁷𝛼⁴𝛼𝛼⁶𝛼³1𝛼⁶𝛼⁴𝛼²1𝛼⁶𝛼⁴𝛼²1𝛼⁷𝛼⁶𝛼⁵𝛼⁴𝛼³𝛼²𝛼,[𝐽𝐹]₈⁻¹=111111111𝛼⁷𝛼⁶𝛼⁵𝛼⁴𝛼³𝛼²𝛼1𝛼⁶𝛼⁴𝛼²1𝛼⁶𝛼⁴𝛼²1𝛼⁵𝛼²𝛼⁷𝛼⁴𝛼𝛼⁶𝛼³1𝛼⁴1𝛼⁴1𝛼⁴1𝛼⁴1𝛼³𝛼⁶𝛼𝛼⁴𝛼⁷𝛼²𝛼⁵1𝛼²𝛼⁴𝛼⁶1𝛼²𝛼⁴𝛼⁶1𝛼𝛼²𝛼³𝛼⁴𝛼⁵𝛼⁶𝛼⁷, (3.12) and index mapping of order-9 Cocyclic Jacket matrix (see Table 8).

Table 8 

Index mapping of order-9 Cocyclic Jacket matrix.

It 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 field 𝐹_𝑞 and first-order 𝑞-ary Reed-Muller codes 𝑅𝑀_𝑝(1,𝑛) [20], where 𝐹_𝑞={𝛼₁=0,𝛼₂,…,𝛼_𝑞} is a finite field of 𝑞 elements, 𝑞=𝑝^𝑚, and 𝑝 is a prime number. The way of construction is described by the following lemma.

Lemma 3.6. The cocyclic Jacket matrix with order 𝑁=𝑝^𝑛 is 𝐽_𝑝𝑛=[𝜔^{→𝑖∘→𝑗}], where 𝜔=exp(2𝜋√−1/𝑝), and one defines ^→𝑖⋅^→𝑗=(𝑖₀,𝑖₁,…,𝑖_𝑛−1)⋅(𝑗₀,𝑗₁,…,𝑗_𝑛−1)=⟨𝑖₀×𝑗₀⟩_𝑝+⟨𝑖₁×𝑗₁⟩_𝑝+⋯+⟨𝑖_𝑛−1×𝑗_𝑛−1⟩_𝑝 (3.13) for 0≤𝑖_𝑘, 𝑗_𝑘≤𝑝−1 (0≤𝑘≤𝑛−1).

Example 3.7. Let 𝑝=3, 𝑛=2, and 𝑚=1, the finite field of 𝑞=𝑝¹=3 elements 𝐹₃={0,1,2},𝑅𝑀₃(1,2) is as follows: 𝑅𝑀₃(1,2)=000012021000012021000012021000012021111120102222201210000012021222201210111120102. (3.14) The entries of 𝐽₃² are shown in Table 9.

Table 9 

The correspondence between the indices and the entries of 𝐽₃².

From Table 9, we can see when 𝑖=2, 𝑗=4, we have 𝜔^(02)∘(11)=𝜔^{⟨0×1⟩3+⟨2×1⟩3}=𝜔². (3.15) 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 matrix 𝑈 is a unitary matrix if 𝑈⁻¹=𝑈^𝐻, where 𝑈^𝐻 denote the conjugate transpose and 𝑈⁻¹ is the matrix inverse.

Proposition 4.2. The matrix 𝑈_𝑛=(1/√𝑐)𝐽_𝑛 is a unitary matrix where 𝐽_𝑛 is the cocyclic Jacket matrix, 𝑐 is the normalized value for 𝐽_𝑛.

Proof. From the definition of Jacket matrix, we have 𝐽_𝑛=[𝑗]_𝑛⁻¹=(1/𝑐)[𝑗⁻¹]^𝑇_𝑛, and the entries in cocyclic Jacket matrices also satisfy ‖𝑗‖=1, we have 𝑗⁻¹=𝑗, then [𝑗]_𝑛⁻¹=(1/𝑐)[𝑗]^𝑇_𝑛,[𝑗]^𝑇_𝑛=[𝑗]^𝐻_𝑛=𝑐[𝑗]_𝑛⁻¹. Certainly,𝑈_𝑛⋅𝑈^𝐻_𝑛=1√𝑐[𝑗]_𝑛⋅1√𝑐[𝑗]^𝐻_𝑛=1𝑐[𝑗]_𝑛⋅[𝑗]^𝐻_𝑛=1𝑐[𝑗]_𝑛⋅𝑐[𝑗]_𝑛⁻¹=𝐼_𝑛. (4.1)

Example 4.3. Based on Example 2.14, we have[𝐽]₆=[𝐽]₃⊗[𝐽]₂=1111111−11−11−111𝛼𝛼𝛼²𝛼²1−1−𝛼𝛼𝛼²−𝛼²11𝛼²𝛼²𝛼𝛼1−1𝛼²−𝛼²𝛼−𝛼, (4.2) where 𝛼=𝑒^{√−1(2𝜋/3)}=𝑒^𝑖𝜃, 𝑖=√1, 𝜃=2𝜋/3, then[𝑈]₆⋅[𝑈]^𝐻₆=16[𝐽]₆⋅[𝐽]^𝐻₆=161111111−11−11−111𝑒^𝑖𝜃𝑒^𝑖𝜃𝑒^𝑖2𝜃𝑒^𝑖2𝜃1−1−𝑒^𝑖𝜃𝑒^𝑖𝜃𝑒^𝑖2𝜃−𝑒^𝑖2𝜃11𝑒^𝑖2𝜃𝑒^𝑖2𝜃𝑒^𝑖𝜃𝑒^𝑖𝜃1−1𝑒^𝑖2𝜃−𝑒^𝑖2𝜃𝑒^𝑖𝜃−𝑒^𝑖𝜃1111111−11−11−111𝑒^−𝑖𝜃−𝑒^−𝑖𝜃𝑒^−2𝑖𝜃𝑒^−2𝑖𝜃1−1𝑒^−𝑖𝜃𝑒^−𝑖𝜃𝑒^−2𝑖𝜃−𝑒^−2𝑖𝜃11𝑒^−2𝑖𝜃𝑒^−2𝑖𝜃𝑒^−𝑖𝜃𝑒^−𝑖𝜃1−1𝑒^−2𝑖𝜃−𝑒^−2𝑖𝜃𝑒^−𝑖𝜃−𝑒^−𝑖𝜃=[𝐼]₆. (4.3) A special feature of cocyclic Jacket matrices has been introduced in [21]. If the cocyclic Jacket matrices with order 𝑁=𝑝₁𝑝₂⋯𝑝_𝑛, 𝑝_𝑖 is the prime number, then 𝐽_𝑁=𝐽_𝑝1⊗𝐽_𝑝2⊗⋯⊗𝐽_𝑝𝑛=𝐴¹_𝑝1𝐴²_𝑝2⋯𝐴^𝑛_𝑝𝑛, (4.4)where𝐴^𝑚_𝑝𝑚=𝐼_𝑝1⊗𝐼_𝑝2⊗⋯𝐼_{𝑝𝑚−1𝑚−1}⊗𝐽_𝑝𝑚⊗𝐼_𝑝𝑚+1⊗𝐼_𝑝𝑚+2⊗⋯⊗𝐼_{𝑝𝑛𝑛−𝑚}, (4.5) and based on this characteristic of cocyclic Jacket matrices, we can easy decompose the unitary matrices with sparse matrices𝑈_𝑁=1√𝑁𝐽_𝑁=1√𝑁𝐴¹_𝑝1𝐴²_𝑝2⋯𝐴^𝑛_𝑝𝑛. (4.6) From (4.6), the 𝑈₆ can be decomposed as 𝑈₆=1√6(𝐼₃⊗𝐽₂)(𝐽₃⊗𝐼₂1)=√6(𝐼₃⊗√2𝑈₂)(√3𝑈₃⊗𝐼₂)=(𝐼₃⊗𝑈₂)(𝑈₃⊗𝐼₂)=1√61100001−10000001100001−10000001100001−110101001010110𝑒^𝑖𝜃0𝑒^𝑖2𝜃0010𝑒^𝑖𝜃0𝑒^𝑖2𝜃10𝑒^𝑖2𝜃0𝑒^𝑖𝜃0010𝑒^𝑖2𝜃0𝑒^𝑖𝜃. (4.7) 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 order 𝑝^𝑘 directly, for the other orders 𝑁=𝑝^𝑘1₁𝑝^𝑘2₂⋯𝑝^𝑘𝑛_𝑛, they can be obtained from the Kronecker product with some matrices whose orders are 𝑝^𝑘𝑖_𝑖. 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.

Acknowledgments

This work was supported by the Ministry of Knowledge Economy, the IT Foreign Specialist Inviting Program supervised by IITA: C1012-0801-0001, KRF-2007-521-D00330, Small and Medium Business Administration, South Korea.