Research Article A Simple Cocyclic Jacket Matrices

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1 Mathematical Problems in Engineering Volume 2008, Article ID , 18 pages doi:101155/2008/ Research Article A Simple Cocyclic Jacket Matrices Moon Ho Lee, 1 Gui-Liang Feng, 2 and Zhu Chen 1 1 Institute of Information & Communication, Chonbuk National University, Jeonju , South Korea 2 The Center for Advanced Computer Studies, University of Louisiana at Lafayette, LA 70504, USA Correspondence should be addressed to Zhu Chen, chenzhu@chonbukackr Received 22 October 2007; Revised 12 May 2008; Accepted 17 July 2008 Recommended by Angelo Luongo 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 Copyright q 2008 Moon Ho Lee et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited 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 Matrix Definition 11 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 0,0 j 0,1 j 0,m 1 j 1,0 j 1,1 j 1,m 1 J m m, 11 j m 1,0 j m 1,1 j m 1,m 1

2 2 Mathematical Problems in Engineering and its inverse is J 1 m m 1 C j 1 0,0 j 1 0,1 j 1 0,m 1 j 1 1,0 j 1 1,1 j 1 1,m 1 j 1 m 1,0 j 1 m 1,1 j 1 m 1,m 1 T, 12 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 12 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, 13 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, thatis,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, 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 l p, 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 21 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 Δ a mod p 21

3 Moon Ho Lee et al 3 Let a F p, we define a function f a x Δ a x 22 Let V α v 0,α v 1,,α v p 1 be a vector, where v i F p for i 0, 1,,p 1 We define a vector V a ( α f a v 0,α f a v 1,,α f a v p 1 ) 23 We have the following lemma Lemma 21 Let V 1,α 1,α 2,,α p 1,then V 0 V 0 T p, V a V b T p, for a b 0, 24 V a V b T 0, for a b / 0 Proof The first equation can be easily proved because V 0 equation, since a b 0, we have 1, 1,,1 For the second ( fa ( vi ) fb ( vi )) mod p a b vi mod p 0 25 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 } 26 Furthermore, for a b / 0, we have {0, 1, 2,,p 1} { 0, a b, 2 a b,, p 1 a b }, 27 that is, p 1 V a V b α i i 0 28 On the other hand, from α p 1, we have p 1 0 α p 1 α 1 α i i 0 29 Since α 1 / 0, p 1 i 0 αi should be zero Thus the last equation is also true The proof is completed

4 4 Mathematical Problems in Engineering Example 22 Let us consider p 5andα e 1 2π/5 We have α 5 1and F 5 {0, 1, 2, 3, 4} 210 Let V {1,α,α 2,α 3,α 4 }, then we have V 0 ( ), V 1 ( 1 α α 2 α 3 α 4), V 2 ( 1 α 2 α 4 α α 3), 211 V 3 ( 1 α 3 α α 4 α 2), V 4 ( 1 α 4 α 3 α 2 α ) It can be seen that V 0 V 0 T 5, V a V b T 5, for a b 0, 212 V a V b T 0, for a b / 0 22 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,letα e 1 2π/p,and V { 1,α,α 2,,α p 2,α p 1} 213 Definition 23 One has the following equation: J p V 0 V 1 V p The inverse of J p is denoted by J 1 p FromLemma 21, it can be easily checked that if 6 J 1 p 1 p [ V 0 T V 1 T V p 1 T ] 1 p J T p, 215

5 Moon Ho Lee et al 5 then J p J 1 p I p J 1 p J p, J p J T p p I p 216 According to the Definition 11, from , J p is a Jacket matrix over complex number of field The following lemma shows that J p is acocyclic Jacket matrix 10 Lemma 24 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 (ie, 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 13, 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, 217 ϕ g,h ϕ i, k α gh ik 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 218 Since gh g h k g h k hk, 219 we have ϕ g,h ϕ g h, k ϕ g,h k ϕ h, k 220 Therefore, J p is a cocyclic matrix Hence, we have the following theorem Theorem 25 The matrix J p is a cocyclic Jacket matrix with size p over complex number field

6 6 Mathematical Problems in Engineering Table 1 g \ h α α 2 α 3 α α 2 α 4 α α α 3 α α 4 α α 4 α 3 α 2 α Example 26 Let us consider p 5 From Example 22, we have V 0 ( ), V 1 ( 1 α α 2 α 3 α 4), V 2 ( 1 α 2 α 4 α α 3), V 3 ( 1 α 3 α α 4 α 2), V 4 ( 1 α 4 α 3 α 2 α ), α α 2 α 3 α 4 J 5 1 α 2 α 4 α α 3, 1 α 3 α α 4 α 2 1 α 4 α 3 α 2 α ( J 5 ) α 4 α 3 α 2 α 1 α 3 α α 4 α 2 1 α 2 α 4 α α 3 1 α α 2 α 3 α 4 1 [ j 1 ] T 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 Example 27 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 α 2 1 We have V 1,α 2 and V 0 1, 1, V 1 ( 1,α 2) 1, 1 222

7 Moon Ho Lee et al 7 Thus, we have J [ ] [ ] J 2 H 1α 2 2, 1 1 [ ] [ ] α H J T 2, 223 where H 2 is Walsh-Hadamard matrix 23 Cocyclic Jacket matrix with size p e 1 1 pe 2 2 pe s s First we introduce some lemmas which are useful to derive the construction of the cocyclic Jacket matrix with size p e 1 1 pe 2 2 pe s s Lemma 28 One has the following equation: ( )( ) ( ) ( ) Ai j B h k Cj s D k t Ai j C j s Bh k D k t, 224 where denotes the Kronecker product [1 3, 5, 6] Lemma 29 One has the following equation: ( ) 1 Ah B k A 1 h B 1 k, ( ) 225 T Ah B k A T h BT k Now we are going to prove the following theorem Theorem 210 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 have 1 1 [ A ] u u 1 T ai,j C u u, a B v v 1 1 C b [ bs,t 1 ] v v T 226 Let A u u B v v [ m iu s,ju t ]uv uv, 227 where m iu s,ju t a i,j b s,t On the other hand, from 225 and 226, we have ( ) 1 1 [( ) 1 T 1 [( ) 1 T Au u B v v ai,j b s,t C a C ]uv uv miu s,ju t b C a C ]uv uv b 228 From 227, 228,andDefinition 11, 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

8 8 Mathematical Problems in Engineering Table 2 a g \ h g A cj g A ri ϕ A g A ri,g A cj b g \ h ck ϕ rh B rh,g B ck c g \ h g A cj ck g A ri ϕ rh AB g A ri rh,g A cj ck Assume that A u u and B v v are cocyclic under the following row and column index orders: g A s1 s1 g A s2 s2 g A su, sv, for g A sj G A, for sk G B, 229 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, and g B denote the kth row index and kth column index of matrix B Then, for rk ck matrix A u B v, the row and column index orders are defined as follows: ϕ AB ( g A ri rh,g A cj g A sj sk ) Δ ( ϕa ck g A si sh, g A ri,g A cj 230 ( ) )ϕ B rh,g B 231 ck In order to understand 229, 230, and 231 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 should satisfy rh,g B ck 13 From 231, and the above fact, it can be verified that ϕ AB g A ri rh,g A cj ck ϕ A g A ri,g A cj ϕ B rh,g B ck is also satisfied 13 under the index orders 230 Hence, A u u B v v is a cocyclic matrix Δ

9 Moon Ho Lee et al 9 Table 3 g \ h β β 2 β 3 β 4 1 β β 2 β 3 β 4 1 β β 2 β 3 β β 2 β 4 β β 3 1 β 2 β 4 β β 3 1 β 2 β 4 β β β 3 β β 4 β 2 1 β 3 β β 4 β 2 1 β 3 β β 4 β β 4 β 3 β 2 β 1 β 4 β 3 β 2 β 1 β 4 β 3 β 2 β μ μ μ μ μ μ 2 μ 2 μ 2 μ 2 μ β β 2 β 3 β 4 μ βμ β 2 μ β 3 μ β 4 μ μ 2 βμ 2 β 2 μ 2 β 3 μ 2 β 4 μ β 2 β 4 β β 3 μ β 2 μ β 4 μ βμ β 3 μ μ 2 β 2 μ 2 β 4 μ 2 βμ 2 β 3 μ β 3 β β 4 β 2 μ β 3 μ βμ β 4 μ β 2 μ μ 2 β 3 μ 2 βμ 2 β 4 μ 2 β 2 μ β 4 β 3 β 2 β μ β 4 μ β 3 μ β 2 μ βμ μ 2 β 4 μ 2 β 3 μ 2 β 2 μ 2 βμ μ 2 μ 2 μ 2 μ 2 μ 2 μ μ μ μ μ 21 1 β β 2 β 3 β 4 μ 2 βμ 2 β 2 μ 2 β 3 μ 2 β 4 μ 2 μ βμ β 2 μ β 3 μ β 4 μ 22 1 β 2 β 4 β β 3 μ 2 β 2 μ 2 β 4 μ 2 βμ 2 β 3 μ 2 μ β 2 μ β 4 μ βμ β 3 μ 23 1 β 3 β β 4 β 2 μ 2 β 3 μ 2 βμ 2 β 4 μ 2 β 2 μ 2 μ β 3 μ βμ β 4 μ β 2 μ 24 1 β 4 β 3 β 2 β μ 2 β 4 μ 2 β 3 μ 2 β 2 μ 2 βμ 2 μ β 4 μ β 3 μ β 2 μ βμ Example 211 Let us consider J 3 J 5,letβ e 1 2π/5, and let μ e 1 2π/3 Then we have β β 2 β 3 β 4 1 β β 2 β 3 β 4 1 β β 2 β 3 β 4 1 β 2 β 4 β β 3 1 β 2 β 4 β β 3 1 β 2 β 4 β β 3 1 β 3 β β 4 β 2 1 β 3 β β 4 β 2 1 β 3 β β 4 β 2 1 β 4 β 3 β 2 β 1 β 4 β 3 β 2 β 1 β 4 β 3 β 2 β μ μ μ μ μ μ 2 μ 2 μ 2 μ 2 μ 2 1 β β 2 β 3 β 4 μ βμ β 2 μ β 3 μ β 4 μ μ 2 βμ 2 β 2 μ 2 β 3 μ 2 β 4 μ 2 J 3 J 5 1 β 2 β 4 β β 3 μ β 2 μ β 4 μ βμ β 3 μ μ 2 β 2 μ 2 β 4 μ 2 βμ 2 β 3 μ 2 1 β 3 β β 4 β 2 μ β 3 μ βμ β 4 μ β 2 μ μ 2 β 3 μ 2 βμ 2 β 4 μ 2 β 2 μ 2 1 β 4 β 3 β 2 β μ β 4 μ β 3 μ β 2 μ βμ μ 2 β 4 μ 2 β 3 μ 2 β 2 μ 2 βμ μ 2 μ 2 μ 2 μ 2 μ 2 μ μ μ μ μ 1 β β 2 β 3 β 4 μ 2 βμ 2 β 2 μ 2 β 3 μ 2 β 4 μ 2 μ βμ β 2 μ β 3 μ β 4 μ 1 β 2 β 4 β β 3 μ 2 β 2 μ 2 β 4 μ 2 βμ 2 β 3 μ 2 μ β 2 μ β 4 μ βμ β 3 μ 1 β 3 β β 4 β 2 μ 2 β 3 μ 2 βμ 2 β 4 μ 2 β 2 μ 2 μ β 3 μ βμ β 4 μ β 2 μ 1 β 4 β 3 β 2 β μ 2 β 4 μ 2 β 3 μ 2 β 2 μ 2 βμ 2 μ β 4 μ β 3 μ β 2 μ βμ 232 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 , ij hk Δ i h 3 j k For example, 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 e 1 1 pe 2 2 pe s s, where p i,fori 1, 2,,s,are primes

10 10 Mathematical Problems in Engineering Definition 212 One has the following equation: where J p e i i J e p 1 1 pe 2 2 pes s Δ J e p 1 J e 1 p 2 J 2 p es, 234 s Δ J pi J pi J pi for i 1, 2,,s 235 From Lemma 28 and Theorem 210, we have the following theorem Theorem 213 The matrix from Definition 212 is a cocyclic Jacket matrix over the complex number field Example 214 Let us consider p 1 3, p 2 2, and e 1 e 2 1 Thus, N Let β e 1 π/3 and α e 1 2π/3,thatis,α β 2 We have [ ] J 6 J 3 J α α α α α α β 1 α α α α α α 2 β 2 β 4 β β 1 1 α 2 α 2 β 5 β 4 β α α 1 1 β 4 β 4 β 2 β α 2 α 2 α α 1 1 β 4 β β 2 β From 19, we know that β β 2 β 5 β β JM 6 2 β 4 β 4 β β 5 β 4 β β β 4 β 2 β 2 β It can be seen that JM J , where JM 6 is the generalized Jacket matrix of order 6 From Lemma 29 and the definition of J p e1, it can be verified that J 1 pe2 2 pes s p e1 is an 1 pe2 2 pes s orthogonal matrix and its inverse matrix can be determined as J 1 p e1 1 pe2 2 pes s ( Jp e1 1 J p e2 2 ) 1 J p es s p e1 1 pe2 2 pes s where J 1 J p ei p 1 i Jp 1 i Jp 1 i i }{{} ei 1 p e1 1 pe2 2 pes s ( J 1 p e1 1 J 2 p e2 2 J 1 ) ps, 239 es

11 Moon Ho Lee et al 11 Table 4 g \ h Example 215 Let us consider J 4 J 4 J 2 J 2 H 2 H 2 Thus, we have J The Jacket matrix J 4 can be mapped as shown in Table 4 Then J 4 is also a cocyclic matrix 3 Cocyclic Jacket matrices over finite field In this section, we will construct the cocyclic Jacket matrices over GF 2 m Letα be a primitive element of GF 2 m Then, GF ( 2 m) { 0, 1,α,α 2,,α 2m 2 }, 31 and we have the following lemma Lemma 31 One has the following equation: 2 m 2 1, for r 0, α ri i 0 0, for 1 r 2 m 2 32 Proof It is evident that 2 m 2 i 0 αri contains 2 m 1 terms, that is, odd terms If r 0, then 2 m 2 i 0 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 have 0 α r 2m 1 1 ( α r 1 )( 2 m 2 a ) ri 33 Since 1 r 2 m 2, that is, α r 1 / 0, we have 2 m 2 i 0 ari 0 The proof is completed Let JF 2 m 1 m i,j 2 m 1 2 m 1, where i 0 αri m ij α ij for 0 i, j 2 m 2, 34 then, we have the following theorem Theorem 32 JF 2 m 1 is a cocyclic Jacket matrix

12 12 Mathematical Problems in Engineering Table 5: Binary representation of GF 2 3 Elements Binary representation α 0 10 α α α α α 6 Proof Let JF 1 2 m 1 [ m 1 i,j ] T 2 m 1 2 m 1, 35 where m i,j α ij From the definition of JF 2 m 1 and Lemma 31, we have JF 2 m 1 JF 1 2 m 1 JF 1 2 m 1 JF 2 m 1 I 2 m 1 36 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 34, we have ϕ i, 0 ϕ 0,i ϕ 0, 0 α 0 1, ϕ i, j α ij, ϕ i, j h α i j k, ϕ i, j ϕ h, k α ij hk 37 Therefore, for any g,h,k Z 2 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 38 Since gh g h k g h k hk, we have ϕ g,h ϕ g h, k ϕ g,h k ϕ h, k 39 In terms of 13, JF 2 m 1 is a cocyclic matrix The proof is completed Example 33 Let us consider JF 7 JF Letα and x3 x 1 0 be the primitive element and primitive polynomial of GF 2 3, respectively Thus, GF 2 3 {α, α 2,α 3,α 4,α 5,α 6 } and α 7 1 On the other hand, any element β GF 2 3 can be represented as a binary vector b 0,b 1,b 2, where b i {0, 1} for i 0, 1, 2 such that β b 0 b 1 α b 2 α 2, 310 as shown in Table 5

13 Moon Ho Lee et al 13 Table 6: Index mapping of order-8 Cocyclic Jacket matrix g \ h α α 2 α 3 α 4 α 5 α α 2 α 4 α 6 α α 3 α α 3 α 6 α 2 α 5 α α α 4 α α 5 α 2 α 6 α α 5 α 3 α α 6 α 4 α α 6 α 5 α 4 α 3 α 2 α Table 7: Binary representation of GF 3 2 Elements Binary representation α 0 0 α α α α α α 7 Using Table 5, it can be easily checked that 39 is true for GF 2 3 Thus, we have α α 2 α 3 α 4 α 5 α 6 1 α 2 α 4 α 6 α α 3 α 5 JF 7 1 α 3 α 6 α 2 α 5 α α 4, 1 α 4 α α 5 α 2 α 6 α 3 1 α 5 α 3 α α 6 α 4 α 2 1 α 6 α 5 α 4 α 3 α 2 α α 6 α 5 α 4 α 3 α 2 α 1 α 5 α 3 α α 6 α 4 α 2 JF α 4 α α 5 α 2 α 6 α 3, 1 α 3 α 6 α 2 α 5 α α 4 1 α 2 α 4 α 6 α α 3 α 5 1 α α 2 α 3 α 4 α 5 α and index mapping of order-8 Cocyclic Jacket matrix see Table 6 It can be verified that JF 7 is a cocyclic Jacket matrix over GF 2 3 Example 34 Let us consider JF JF 8 over GF 3 2 Let α and x 2 x 2 0 be the primitive and primitive polynomial of GF 3 2, respectively Thus, GF 3 2 {0, 1,α,α 2,α 3,α 4,α 5,α 6,α 7 } and α 8 1 Conversely, any element β GF 3 2 can be represented as a vector over GF 3 : β b 0 b 1 α, where b 0,b 1 {0, 1, 2} see Table 7

14 14 Mathematical Problems in Engineering Table 8: Index mapping of order-9 Cocyclic Jacket matrix g \ h α α 2 α 3 α 4 α 5 α 6 α α 2 α 4 α 6 1 α 2 α 4 α α 3 α 6 α α 4 α 7 α 2 α α 4 1 α 4 1 α 4 1 α α 5 α 2 α 7 α 4 α α 6 α α 6 α 4 α 2 1 α 6 α 4 α α 7 α 6 α 5 α 4 α 3 α 2 α Using this table, it is easy to deduce that 32 is true for GF 3 2 change 2 m to 3 m Thus, we have α α 2 α 3 α 4 α 5 α 6 α 7 1 α 2 α 4 α 6 1 α 2 α 4 α 6 1 α JF 8 3 α 6 α α 4 α 7 α 2 α 5 1 α 4 1 α 4 1 α 4 1 α 4, 1 α 5 α 2 α 7 α 4 α α 6 α 3 1 α 6 α 4 α 2 1 α 6 α 4 α 2 1 α 7 α 6 α 5 α 4 α 3 α 2 α α 7 α 6 α 5 α 4 α 3 α 2 α 1 α 6 α 4 α 2 1 α 6 α 4 α 2 JF 1 1 α 8 5 α 2 α 7 α 4 α α 6 α 3 1 α 4 1 α 4 1 α 4 1 α 4, 1 α 3 α 6 α α 4 α 7 α 2 α 5 1 α 2 α 4 α 6 1 α 2 α 4 α 6 1 α α 2 α 3 α 4 α 5 α 6 α 7 and index mapping of order-9 Cocyclic Jacket matrix see Table 8 It is easy to verify that JF 8 is a cocyclic Jacket matrix over GF Remark 35 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 {α 1 0,α 2,,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 36 The cocyclic Jacket matrix with order N p n is J p n ω i j, where ω exp 2π 1/p, and one defines i j ( i 0,i 1,,i n 1 ) ( j0,j 1,,j n 1 ) i 0 j 0 p i 1 j 1 p i n 1 j n 1 p 313 for 0 i k, j k p 1 (0 k n 1)

15 Moon Ho Lee et al 15 Table 9: The correspondence between the indices and the entries of J 3 2 j/ j i/ j ω 0 ω 0 ω 0 ω 0 ω 0 ω 0 ω 0 ω 0 ω ω 0 ω 1 ω 2 ω 0 ω 1 ω 2 ω 0 ω 1 ω ω 0 ω 2 ω 1 ω 0 ω 2 ω 1 ω 0 ω 2 ω ω 0 ω 0 ω 0 ω 1 ω 1 ω 1 ω 2 ω 2 ω ω 0 ω 1 ω 2 ω 1 ω 2 ω 0 ω 2 ω 0 ω ω 0 ω 2 ω 1 ω 1 ω 0 ω 2 ω 2 ω 1 ω ω 0 ω 0 ω 0 ω 2 ω 2 ω 2 ω 1 ω 1 ω ω 0 ω 1 ω 2 ω 2 ω 0 ω 1 ω 1 ω 2 ω ω 0 ω 2 ω 1 ω 2 ω 1 ω 0 ω 1 ω 0 ω 2 Example 37 Let p 3, n 2, and m 1, the finite field of q p 1 3 elements F 3 {0, 1, 2},RM 3 1, 2 is as follows: RM 3 1, The entries of J 3 2 are shown in Table 9 From Table 9, we can see when i 2, j 4, we have ω ω ω The other entries can be obtained using the same fashion, perfectly 4 The factorization of cocyclic Jacket matrices and unitary matrices Definition 41 A square matrix U is a unitary matrix if U 1 conjugate transpose and U 1 is the matrix inverse U H, where U H denote the Proposition 42 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 1 n in cocyclic Jacket matrices also satisfy j 1, we have j 1 j, then j 1 n j H n c j 1 n Certainly, 1/c j 1 T n, and the entries 1/c j T n, j T n U n Un H 1 1 j c n j H c n 1 c j n j H n 1 j n c j 1 n c I n 41

16 16 Mathematical Problems in Engineering Example 43 Based on Example 214, we have α α α J 6 J 3 J 2 α α α α α 2, α 2 α 2 α α 1 1 α 2 α 2 α α where α e 1 2π/3 e iθ, i 1, θ 2π/3, then U 6 U H J 6 J H e iθ e iθ e i2θ e i2θ 1 1 e iθ e iθ e 2iθ e 2iθ e iθ e iθ e i2θ e i2θ 1 1 e 1 1 e i2θ e i2θ e iθ e iθ iθ e iθ e 2iθ e 2iθ I e 2iθ e 2iθ e iθ e iθ 1 1 e i2θ e i2θ e iθ e iθ 1 1 e 2iθ e 2iθ e iθ e iθ A special feature of cocyclic Jacket matrices has been introduced in 21 If the cocyclic Jacket matrices with order N p 1 p 2 p n, p i is the prime number, then where A m p m 43 J N J p1 J p2 J pn A 1 p 1 A 2 p 2 A n p n, 44 I p1 I p2 I pm 1 J pm I pm 1 I pm 2 I pn, 45 }{{}}{{} m 1 n m and based on this characteristic of cocyclic Jacket matrices, we can easy decompose the unitary matrices with sparse matrices U N 1 N J N 1 N A 1 p 1 A 2 p 2 A n p n 46 From 46, theu 6 can be decomposed as U ( I3 J 2 )( J3 I 2 ) 1 6 ( I3 2U 2 )( 3U3 I 2 ) ( I3 U 2 )( U3 I 2 ) e iθ 0 e i2θ e iθ 0 e i2θ e i2θ 0 e iθ e i2θ 0 e iθ 47 Clearly, 47 is the new factorization matrix

17 Moon Ho Lee et al 17 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 k 1 1 pk 2 2 pk n n, they can be obtained from the Kronecker product with some matrices whose orders are p k i 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: C , KRF D00330, Small and Medium Business Administration, South Korea References 1 N Ahmed and K R Rao, Orthogonal Transforms for Digital Signal Processing, Springer, Berlin, Germany, J Seberry and M Yamada, Hadamard matrices, sequences, and block designs, in Contemporary Design Theory: A Collection of Surveys, Wiley-Interscience Series in Discrete Mathematics and Optimization, chapter 11, pp , John Wiley & Sons, New York, NY, USA, S S Agaian, Hadamard Matrices and Their Applications, vol 1168 of Lecture Notes in Mathematics, Springer, Berlin, Germany, A J Viterbi, CDMA: Principles of Spread Spectrum Communication, Addison-Wesley, Reading, Mass, USA, A V Geramita and J Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, vol 45 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, M H Lee, Jacket Matrices, Youngil, Korea, J Hou, M H Lee, D C Park, and K J Lee, Simple element inverse DCT/DFT hybrid architecture algorithm, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 06), vol 3, pp , Toulouse, France, May P Udaya, Cocyclic generalized Hadamard matrices over GF p n and their related codes, in Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms and Error- Correcting Codes (AAECC-13), pp 35 36, Honolulu, Hawaii, USA, November J Hou and M H Lee, Cocyclic Jacket matrices and its application to cryptography systems, in Proceedings of the International Conference on Information Networking (ICOIN 05), vol 3391 of Lecture Notes in Computer Science, pp , Jeju Island, Korea, January-February G L Feng and M H Lee, An explicit construction of co-cyclic Jacket matrices with any size, in Proceedings of the 5th Shanghai Conference in Combinatorics, Shanghai, China, May K J Horadam and P Udaya, Cocyclic Hadamard codes, IEEE Transactions on Information Theory, vol 46, no 4, pp , K Finlayson, M H Lee, J Seberry, and M Yamada, Jacket matrices constructed from Hadamard matrices and generalized Hadamard matrices, The Australasian Journal of Combinatorics, vol 35, pp 83 87, M H Lee and J Hou, Fast block inverse jacket transform, IEEE Signal Processing Letters, vol 13, no 8, pp , M H Lee, The center weighted Hadamard transform, IEEE Transactions on Circuits and Systems,vol 36, no 9, pp , M H Lee, A new reverse jacket transform and its fast algorithm, IEEE Transactions on Circuits and Systems II, vol 47, no 1, pp 39 47, M H Lee, B Sundar Rajan, and J Y Park, A generalized reverse jacket transform, IEEE Transactions on Circuits and Systems II, vol 48, no 7, pp , 2001

18 18 Mathematical Problems in Engineering 17 G Zeng and M H Lee, Fast block jacket transform based on Pauli matrices, in Proceedings of the IEEE International Conference on Communications (ICC 07), pp , Glasgow, UK, June P Diţă, Factorization of unitary matrices, Journal of Physics A, vol 36, no 11, pp , J Hou and M H Lee, On cocyclic jacket matrices, in Proceedings of the 3rd WSEAS International Conference on Applied Mathematics and Computer Science (AMCOS 04), pp 12 15, Rio de Janeiro, Brazil, October M H Lee and Y L Borissov, Fast decoding of the p-ary first-order Reed-Muller codes based on jacket transform, IEICE Transaction on Fundamentals of Electronics, vol E91-A, no 3, pp , Z Chen, M H Lee, and G Zeng, Fast cocyclic Jacket transform, IEEE Transactions on Signal Processing, vol 56, no 5, pp , M A Nielsen and I L Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000

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