An explicit construction of fast cocyclic jacket transform on the finite field with any size

An orthogonal cocyclic framework of the block-wise inverse Jacket transform (BIJT) is proposed over the finite field. Instead of the conventional block-wise inverse Jacket matrix (BIJM), we investigate the cocyclic block-wise inverse Jacket matrix (CBIJM), where the high-order CBIJM can be factorized into the low-order sparse CBIJMs with a successive block architecture. It has a recursive fashion that leads to a fast algorithm concerned for reducing computational load. The fast transforms are also developed for the two-dimensional cocyclic block-wise inverse Jacket transform (CBIJT). The present CBIJM may be used for many matrix-based applications, such as the DFT signal processing, combinatorics, and the Reed-Muller code design.

Many interesting orthogonal matrices, say the Hadamard matrices and the DFT matrices, belong to the Jacket matrix family. With the rapid technological development, different forms of such transforms were improved and generalized. It has been discovered that the http://asp.eurasipjournals.com/content/2012/1/184 are the conventional cocyclic element-wise inverse Jacket matrices (CEIJM), where ⊗ denotes the Kronecker product and p is a prime number.
As a generation of the Hadamard matrix, the BIJM inherits the merits of the Hadamard matrix, at the same time, without the restriction that entries must be '±1' . On the other hand, this matrix has very amicable properties, such as reciprocal orthogonality. The inverse transform can be easily obtained by the reciprocal relationships and the fast algorithms. However, the versions of cocyclic block-wise inverse Jacket matrix (CBIJM) are still absent since the existence of the CEIJM has attracted minor attention in the existing literature [8,21]. The purpose of this article is to develop the CBIJM and its generalizations, instead of the CEIJM. In addition, the present CBIJM has some potential practical applications in signal sequence transforms [1][2][3][4][5][6][7], coding design for wireless networks [22,27,28], and cryptography [31].
This article is organized as follows. Section 'Cocyclic block-wise inverse Jacket transforms' presents a simple framework of the fast CBIJT. Section 'Designs of the CBIJM over finite field GF(2 m )' reports the CBIJM over finite field GF (2 p ). Section 'Two-dimensional fast CBIJM' proposes the structure of the two-dimensional CBIJM. Finally, conclusions are drawn in Section 'Conclusion' .

Cocyclic block-wise inverse Jacket transforms
In this section, we show that the EIJM can be generalized for the constructions of the CBIJT.
Based on the one-dimensional BIJM [ J] p of order p, which can be partitioned to the p × p block matrix, we can transform a suitable vector x into another vector y through a BIJT, i.e., In order to derive the CBIJT, we denote a matrix unit by α such that α p = I p for a given prime number p, where I p denotes the p × p identity matrix. As an example, let α be a square matrix of size 2 × 2 defined as It is easy to prove that α 2 = I 2 . Actually, matrix α in (3) has been employed for the existence of the BIJM [15][16][17]. Fortunately, it will be shown that the s-fold block Jacket matrix [ J] 2 s α ⊗s is also a CBIJM.
In what follows we illustrate the cocyclicity of the BIJM [ J] p s based on the matrix unit α of size p × p. In particular for the given prime number p, we define the matrix unit where j +h p = j +h mod p, ∀ i, j, h ∈ Z p := {0, 1, . . . , p− 1}. It can be shown that A := {α h : h ∈ Z p } forms an Abelian group with the traditional matrix multiplication. Namely, for the given number p, one obtains the matrix units as follows It is obvious that Z p with the multiplication operation a · b p is a finite field of order p. For ∀ a, x ∈ Z p , we define an multiplication function f a (x) over Z p , i.e., With the aid of the multiplication function f a (x), we define a block matrix of size p × p 2 by concatenating p matrices α h i of size p × p, ∀ h i ∈ Z p , i.e., and hence The proof of Lemma 1 is illustrated in Appendix. http://asp.eurasipjournals.com/content/2012/1/184 Example 2. Let us consider α with p = 2 in (3). It is obvious that α 2 = I is an identity matrix of size 2 × 2. Let [ β] = α 0 , α 1 , then we have It is straightforward to show that The p-order CBIJM In [15][16][17], Lee et al. expanded the EIJM to the BIJM.
where c is the normalized value and [ α ij ] p×p denotes a matrix unit of size p × p.
Definition 5. For a given prime number p, let α be a p × p matrix unit such that α p = I and Define the p-order BIJM [ J] p of size p 2 × p 2 as follows and thus its inverse Consequently, we have Example 3. Taking [ β 0 ] and [ β 1 ] for p = 2, we have and its inverse Actually, we have where α 0 + α 1 = 0 since α 2 = I and α = I over the finite field.
We note that the above-mentioned BIJM was first proposed by Lee and Hou [13] for the proof of existence of Jacket matrices over the finite field. Next, we illustrate that this BIJM is also a CBIJM in essence. (15) whose rows and columns are both indexed in G under the increasing order (i.e., 0 ≺ 1 ≺ · · · ≺ p − 1) and entries φ(a, b) in position (a, b) is the normalized CBIJM.
The proof of Theorem 1 is illustrated in Appendix.

The multi-fold CBIJM
In order to derive the high-order recursive CBIJM [ J] p s for any prime number p and nonnegative integer s, let us introduce some lemmas [1][2][3][4][5].
Lemma 2. Let A, B, C, and D are matrices with suitable sizes. Then we have be two CBIJMs of order p that corresponds to the matrix units α and γ such that α p = I and γ p = I, respectively. Then the two-fold Kronecker product matrix is a two-fold CBIJM of order p 2 .
The proof of Theorem 2 is shown in Appendix.

Corollary 1. For any prime number p and non-negative integer number s, let [ J] p s =[ J] ⊗s
p be an s-fold block matrix, i.e., Then the block matrix [ J] p s is a CBIJM of order p s .

Example 5.
For p = 2 and s = 2, we consider a matrix unit α of size 2 × 2 in (3). Thus we have the four-order BIJM [ J] 2 2 given by Similarly, we have an index order matrix in Table 2, where the row and column index orders are 00 ≺ 01 ≺ 10 ≺ 11 (27) and for ∀ a 1 , b 1 , a 2 , b 2 ∈ Z 2 , As an example, if a = 2 and b = 3, then we have It can be easily verified that the two-fold matrix [ J] 2 2 in (26) is a four-order CBIJM of size 8 × 8. In addition, using the same index mapping in Table 1, we obtain the index matrix I 4 as follows which is a generator matrix of the first order binary Reed-Muller code [3]. We note that this phenomena exists in the generalized s-fold CBIJM [ J] p s of order p s for any prime number p. Actually, the two-fold CBIJM [ J] 2 2 in (26) based on the factorization algorithm can be rewritten as Namely, we have The comparison between the direct computation and fast transform in terms of operations (i.e., additions and multiplications) is illustrated in the greater efficiency for computation than that of the direct approach. (23), we have p = 3, s = 2 and

Example 6. From Equation
which can be factorized as with the signal flow graph in Figure 1. It is obvious that which can be used for the generalization of the first order 3-ary Reed-Muller code [3].
Consequently, the s-fold CBIJM [ J] p s of order p s can be generated from the following factorization algorithm where I p i denotes the identity matrix of size p i × p i and I p 0 = 1 for the simple description.
where p is any prime number and s is a nonnegative integer number.
The proof of Corollary 2 is shown in Appendix. http://asp.eurasipjournals.com/content/2012/1/184 In order to show the factorization of the generalized CBIJM [ J] n of order p s with any prime number p, we propose several construction approaches in Table 4. In this table, the second column is the decomposition for the numbers (order) of the CBIJM, and the third column is the construction for CBIJM. It shows that the largeorder CBIJM can be designed on the basis of the lower order CBIJM [ J] p with sparse matrices in the successive architecture.

Low-density of the CBIJM
In what follows, we consider the density of 1's in the s-fold According to the afore-mentioned CBIJM [ J] p , it is known that matrix [ J] p whose matrix unit is α in (4) is a p 2 × p 2 binary matrix. The total number of 1's is p in each matrix unit α h , ∀ h ∈ Z p . Then the density of 1's in α h is Therefore the density of 1's in [ J] p is calculated as   Table 5.

Designs of the CBIJM over finite field GF(2 m )
In this section, we consider the generalized CBIJM over finite field GF(2 m ) and derive the high-order CBIJM for p = 2 m − 1.
Let α be a matrix unit of size p×p over GF(2 m ) such that α 2 m −1 = I and α = I. Then we obtain the (2 m − 1)-order CBIJM [ J] 2 m −1 as follows.
The proof of Theorem 3 are shown in Appendix.

Example 7.
We consider the seven-order block matrix [ J] 2 3 −1 with the primitive polynomial x 3 + x + 1 = 0 over GF (2 3 ). Let α be an arbitrary matrix unit such that α 7 = I and α = I. Then any matrix element β over GF(2 3 ) can be represented as a binary vector (b 0 , b 1 , b 2 ), ∀ b i ∈ Z 2 and i ∈ {0, 1, 2}, such that By the Table 6, it is straightforward that Theorem 3 is true over GF (2 3 ). Then we obtain the BIJM [ J] 7 and its inverse [ J] −1 7 , i.e., Actually, according to the index mapping of the present matrix in Table 7, it can be shown that matrix [ J] 7 in (38) is a seven-order CBIJM over GF(2 3 ).

Two-dimensional fast CBIJM
In the previous section, we consider the one-dimensional CBIJT based on the CBIJM. Now we extend it to the version of the two-dimensional CBIJT. The fast two-dimensional CBIJM can be similarly derived from the two-dimensional Jacket transform [15]    [ Example 8. We consider the two-dimensional fourorder CBIJM It is shown in the previous section that block matrix [ J] 2 2 is a four-order CBIJM that can be constructed in the recursive fashion on the basis of [ J] 2 with fast algorithm. Therefore, the two-dimensional CBIJM [ J] 2 4 can be similarly designed in the recursive fashion with fast algorithm based on two-fold four-order CBIJT [ J] 2 2 , as shown in Figure 2. Compared with the fast algorithm of the one-dimensional CBIJM [ J] 3 2 in Figure 1, the present fast algorithm needs four steps for calculations, instead of two steps for the factorizing decomposition.

Conclusion
A simple method of developing the fast CBIJM is proposed over finite field. This method is presented for its simplicity and clarity, which decomposes the high-order CBIJM into multiple sparse matrices with the lower-order CBIJMs, instead of the conventional BIJMs or EIJMs. This factorization algorithm is valid for the generalized s-fold CBIJM of order p s over finite field with a suitable matrix unit α of size p × p. Also, the present CBIJM is useful for developing the fast two-dimensional CBIJM based on http://asp.eurasipjournals.com/content/2012/1/184 sparse matrices in the recursive forms. It may have potential applications in combinatorial designs (CD) [8], spacetime block codes [23,27], and odd-order code design [20] thanks to its successive architecture.
Also the entries of [ J] p 2 are defined on the basis of [ J] p as As for the entries φ p (a i , a j ) and Therefore, it can be easily verified that It shows that block matrix [ J] p 2 is also a CBIJM under the indexed order in (51). This completes the proof of this theorem.

Proof of Corollary 2
We deploy induction on index s.
Combining (56) and (58), we obtain This completes the proof of this corollary.

Proof of Theorem 3
In order to prove Theorem 3, we introduce a lemma as follows.
Proof. It is evident that 2 m −2 i=0 α ir contains 2 m −1 terms. If r = 0, then 2 m −2 i=0 α ir is a sum of 2 m − 1 identity matrices. Thus the first equation is proved. We now consider the case of 1 ≤ r ≤ 2 m −2 such that α r = I, i.e., α r −I = 0. It shows that block matrix [ J] 2 m −1 is a BIJM. In order to prove that it is a CBIJM, we let φ(i, j) be an entry in position (i, j), where the order of rows and columns is from http://asp.eurasipjournals.com/content/2012/1/184 0 to 2 m −2 over Z 2 m −1 . Consequently, for i, j, h, k ∈ Z 2 m −1 we have Then we achieve and It is obvious to verify which implies that the BIJM [ J] 2 m −1 is a CBIJM over GF(2 m ).