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. Introduction The orthogonal transforms, such as the discrete Fourier transform (DFT) and the Walsh-Hadamard transform (WHT), have been widely employed in images processing, feature selection, signal processing, data compressing and coding, and other areas [1-7]. Using orthogonality of the WHT, the interesting orthogonal matrices, such as the element-wise or block-wise inverse Jacket matrices (BIJMs) [8-12], have been developed.More details of these matrices can be referred to [13-19]. Definition 1. An n × n matrix Jn = (αij)n×n is called the element-wise inverse Jacket matrix (EIJM) of order n if its inverse matrix J−1 n can be simply obtained by its element-wise inverse, i.e., J−1 n = 1 n (α−1 ij )n×n, ∀ i, j ∈ Zn := {0, 1, . . . , n − 1}, where the superscript T denotes the transpose. 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 *Correspondence: [email protected] 2Institute of Information and Communication, Chonbuk National University, Jeonju 561-756, Korea Full list of author information is available at the end of the article newly proposed transforms have been widely used in various signal processing, CDMA, cooperative relay MIMO system [20-28]. Recently, the BIJM [ J]n has been investigated while the complex unit exp √−1(2π/p) of the EIJM Jn is substituted for a suitable matrix unit [15-17]. However, the CBIJM does not attract much attention even though the cocyclic matrix has been very useful for the data coding and processing [5,14,29,30]. Definition 2. If G is a finite group of order r with operation ◦ and C is a finite Abelian group of order t, a cocycle is a mapping φ : G × G → C satisfying φ(a, b)φ(a ◦ b, c) = φ(a, b ◦ c)φ(b, c), (1) where a, b, c ∈ G. A square matrixM(φ) whose row a and column b can be indexed by G with entry φ(a, b) ∈ C in position (a, b) under some fixed ordering, i.e., M(φ) = (φ(a, b))a,b∈G , is called a cocyclic matrix. If φ(1, 1) = 1, then it is the normalized cocyclic matrix for the standard usage [5,29,30]. Definition 3. Let Jp = (ω〈i◦j〉p)p×p, ∀ i, j ∈ Zp := {0, 1, . . . , p − 1}, be a matrix of order p, where ω = exp( √−1(2π/p)) and 〈i ◦ j〉p = i× j mod p, i.e., the subscript p implies modulo-p arithmetic for the argument. Then the matrix Jp and its s-fold matrix of order ps Jps = J⊗s p = Jp ⊗ Jp · · · ⊗ Jp } {{ } s © 2012 Guo et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Guo et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:184 Page 2 of 10 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 [1-7], 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(2m)’ reports the CBIJM over finite field GF(2p). 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.,