Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2008, Article ID 132674, 18 pages
http://dx.doi.org/10.1155/2008/132674
Research Article

A Simple Cocyclic Jacket Matrices

1Institute of Information & Communication, Chonbuk National University, Jeonju 561-756, South Korea
2The Center for Advanced Computer Studies, University of Louisiana at Lafayette, LA 70504, USA

Received 22 October 2007; Revised 12 May 2008; Accepted 17 July 2008

Academic Editor: Angelo Luongo

Copyright © 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.

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 [14]. Their basic functions are sampled Walsh functions which can be expressed in terms of the Hadamard matrices. Using the orthogonality of Hadamard matrices, more general matrices have been developed [5]. These matrices are called as Jacket matrices and denoted by . 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 has nonzero elements, and an inverse form which is only from the element-wise inverse and then transpose, such as and its inverse is where is the normalized value for this matrix, and 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 [810].

Definition 1.2. If is a finite group of order with operation “,” and is a finite abelian group of order , a “two-dimensional“ cocycle is a mapping , satisfying where . A square matrix whose rows and columns are indexed by the elements of , with entry in the position , that is, where and , can be called as a cocyclic matrix.

In [11], it is demonstrated that many well-known binary, quaternary, and -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, 1416] 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 , where 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 be an odd prime integer and . Thus, we have and with the operations for are the finite field, where Let , we define a function Let be a vector, where for We define a vector We have the following lemma.

Lemma 2.1. Let , then

Proof. The first equation can be easily proved because . For the second equation, since , we have Thus the second equation is also true. Now we consider the last equation since is an odd prime, we know that for any Furthermore, for , we have that is, On the other hand, from , we have Since , should be zero. Thus the last equation is also true. The proof is completed.

Example 2.2. Let us consider and . We have and Let , then we have It can be seen that

2.2. Cocyclic Jacket Matrix with Size

Now we are going to construct cocyclic Jacket matrix over complex number field. For a given odd prime , let , and

Definition 2.3. One has the following equation: The inverse of is denoted by . From Lemma 2.1, it can be easily checked that if [6] then According to the Definition 1.1, from (2.13)–(2.15), is a Jacket matrix over complex number of field. The following lemma shows that is acocyclic Jacket matrix [10].

Lemma 2.4. Let with the operation , with traditional multiplication, the rows and columns are indexed by the elements of under the increasing order (i.e., ), and the entry of is . Then, the Jacket matrix is a symmetric normalized cocyclic matrix.

Proof. Let . Based on the above increasing order and from (1.3), we have Therefore, for any , we have Since we have Therefore, is a cocyclic matrix.

Hence, we have the following theorem.

Theorem 2.5. The matrix is a cocyclic Jacket matrix with size over complex number field.

Example 2.6. Let us consider . From Example 2.2, we have Moreover, the Jacket matrix can be mapped as shown in Table 1. It can be verified that is a cocyclic matrix.

tab1a
Table 1

Example 2.7. Let us consider , this is not an odd prime, but it is a prime. Let , we have . We have and Thus, we have where is Walsh-Hadamard matrix.

2.3. Cocyclic Jacket Matrix with Size

First we introduce some lemmas which are useful to derive the construction of the cocyclic Jacket matrix with size

Lemma 2.8. One has the following equation: where denotes the Kronecker product [13, 5, 6].

Lemma 2.9. One has the following equation: Now we are going to prove the following theorem.

Theorem 2.10. If and are cocyclic Jacket matrices, then is also a cocyclic Jacket matrix with size .

Proof. Since and are cocyclic Jacket matrices, according to the property of Jacket matrix, we have Let where . On the other hand, from (2.25) and (2.26), we have From (2.27), (2.28), and Definition 1.1, is a Jacket matrix. Next, we will prove that is also a cocyclic matrix.
Assume that and are cocyclic under the following row and column index orders: where or , and denote the th row index and th column index of matrix Similarly, and denote the th row index and th column index of matrix Then, for matrix the row and column index orders are defined as follows: In order to understand (2.29), (2.30), and (2.31) better, we interpret matrices and as the following three forms shown in Table 2. Since and are cocyclic matrices, thus their elements and should satisfy (1.3). From (2.31), and the above fact, it can be verified that is also satisfied (1.3) under the index orders (2.30). Hence, is a cocyclic matrix.

tab2a
Table 2

Example 2.11. Let us consider , let and let . Then we have It can be easily verified that is a Jacket matrix. We also present its index order matrix as shown in Table 3, where the row and column index orders are For example, . It can be easily verified that is a cocyclic matrix.
Next, we are going to construct a cocyclic Jacket matrix using the complex number field with size , where , for are primes.

tab3a
Table 3

Definition 2.12. One has the following equation: where
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 , and Thus, . Let and , that is, . We have From [19], we know that It can be seen that where is the generalized Jacket matrix of order 6.

From Lemma 2.9 and the definition of it can be verified that is an orthogonal matrix and its inverse matrix can be determined as where

Example 2.15. Let us consider . Thus, we have The Jacket matrix can be mapped as shown in Table 4. Then is also a cocyclic matrix.

tab4a
Table 4

3. Cocyclic Jacket Matrices over Finite Field

In this section, we will construct the cocyclic Jacket matrices over . Let be a primitive element of . Then, and we have the following lemma.

Lemma 3.1. One has the following equation:

Proof. It is evident that contains terms, that is, odd terms. If , then is a sum of odd 1’s and should be 1. Thus, the first equation is proved.
We now consider the case of Since =1, we have Since , that is, we have . The proof is completed.

Let where then, we have the following theorem.

Theorem 3.2. is a cocyclic Jacket matrix.

Proof. Letwhere From the definition of and Lemma 3.1, we have Hence, is a Jacket matrix. Next, we will prove that is also a cocyclic matrix. Let be the entry of row and column , where the order of rows and columns is from 0 to . From (3.4), we have Therefore, for any we have Since , we have In terms of (1.3), is a cocyclic matrix. The proof is completed.

Example 3.3. Let us consider . Let and be the primitive element and primitive polynomial of respectively. Thus, and . On the other hand, any element can be represented as a binary vector where for such that as shown in Table 5.

tab1
Table 5: Binary representation of .

Using Table 5, it can be easily checked that (3.9) is true for . Thus, we have and index mapping of order-8 Cocyclic Jacket matrix (see Table 6).

tab2
Table 6: Index mapping of order-8 Cocyclic Jacket matrix.

It can be verified that is a cocyclic Jacket matrix over .

Example 3.4. Let us consider over Let and be the primitive and primitive polynomial of respectively. Thus, and . Conversely, any element can be represented as a vector over where (see Table 7).

tab3
Table 7: Binary representation of .

Using this table, it is easy to deduce that (3.2) is true for (change to ). Thus, we have and index mapping of order-9 Cocyclic Jacket matrix (see Table 8).

tab4
Table 8: Index mapping of order-9 Cocyclic Jacket matrix.

It is easy to verify that is a cocyclic Jacket matrix over

Remark 3.5. We can also construct cocyclic Jacket matrices based on additive characters of the finite field and first-order -ary Reed-Muller codes [20], where is a finite field of elements, and is a prime number. The way of construction is described by the following lemma.

Lemma 3.6. The cocyclic Jacket matrix with order is where , and one defines for , ().

Example 3.7. Let , , and , the finite field of elements is as follows: The entries of are shown in Table 9.

tab5
Table 9: The correspondence between the indices and the entries of

From Table 9, we can see when , , we have 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 is a unitary matrix if where denote the conjugate transpose and is the matrix inverse.

Proposition 4.2. The matrix is a unitary matrix where is the cocyclic Jacket matrix, is the normalized value for

Proof. From the definition of Jacket matrix, we have and the entries in cocyclic Jacket matrices also satisfy , we have then Certainly,

Example 4.3. Based on Example 2.14, we have where , , , then A special feature of cocyclic Jacket matrices has been introduced in [21]. If the cocyclic Jacket matrices with order , is the prime number, then where and based on this characteristic of cocyclic Jacket matrices, we can easy decompose the unitary matrices with sparse matrices From (4.6), the can be decomposed as 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 directly, for the other orders , they can be obtained from the Kronecker product with some matrices whose orders are 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.

References

  1. N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer, Berlin, Germany, 1975. View at Zentralblatt MATH
  2. 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. 431–560, John Wiley & Sons, New York, NY, USA, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. S. Agaian, Hadamard Matrices and Their Applications, vol. 1168 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1985. View at Zentralblatt MATH · View at MathSciNet
  4. A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication, Addison-Wesley, Reading, Mass, USA, 1995. View at Zentralblatt MATH
  5. 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, 1979. View at Zentralblatt MATH · View at MathSciNet
  6. M. H. Lee, Jacket Matrices, Youngil, Korea, 2006.
  7. 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. 888–891, Toulouse, France, May 2006. View at Publisher · View at Google Scholar
  8. P. Udaya, “Cocyclic generalized Hadamard matrices over GF(pn) 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 1999.
  9. 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. 662–668, Jeju Island, Korea, January-February 2005.
  10. 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 2005.
  11. K. J. Horadam and P. Udaya, “Cocyclic Hadamard codes,” IEEE Transactions on Information Theory, vol. 46, no. 4, pp. 1545–1550, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. H. Lee and J. Hou, “Fast block inverse jacket transform,” IEEE Signal Processing Letters, vol. 13, no. 8, pp. 461–464, 2006. View at Publisher · View at Google Scholar
  14. M. H. Lee, “The center weighted Hadamard transform,” IEEE Transactions on Circuits and Systems, vol. 36, no. 9, pp. 1247–1249, 1989. View at Publisher · View at Google Scholar
  15. 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, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. 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. 684–690, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. 2687–2692, Glasgow, UK, June 2007. View at Publisher · View at Google Scholar
  18. P. Diţă, “Factorization of unitary matrices,” Journal of Physics A, vol. 36, no. 11, pp. 2781–2789, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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 2004.
  20. 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. 901–904, 2008. View at Publisher · View at Google Scholar
  21. Z. Chen, M. H. Lee, and G. Zeng, “Fast cocyclic Jacket transform,” IEEE Transactions on Signal Processing, vol. 56, no. 5, pp. 2143–2148, 2008. View at Publisher · View at Google Scholar
  22. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. View at Zentralblatt MATH · View at MathSciNet