Linear complexity for multidimensional arrays - a numerical invariant

Linear complexity is a measure of how complex a one dimensional sequence can be. In this paper we extend the concept of linear complexity to multiple dimensions and present a definition that is invariant under well-orderings of the arrays. As a result we find that our new definition for the process introduced in the patent titled “Digital Watermarking” produces arrays with good asymptotic properties.

[1]  Oscar Moreno,et al.  New Optimal Low Correlation Sequences for Wireless Communications , 2012, SETA.

[2]  Ruud Pellikaan,et al.  On the decoding of algebraic-geometric codes , 1995, IEEE Trans. Inf. Theory.

[3]  Harald Niederreiter,et al.  Linear Complexity and Related Complexity Measures for Sequences , 2003, INDOCRYPT.

[4]  Robert A. Scholtz,et al.  Bent-function sequences , 1982, IEEE Trans. Inf. Theory.

[5]  Tom Høholdt,et al.  Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound , 1994, IEEE Trans. Inf. Theory.

[6]  Igor E. Shparlinski,et al.  On the Lower Bound of the Linear Complexity Over of , 2006 .

[7]  Oscar Moreno,et al.  Families of 3D Arrays for Video Watermarking , 2014, SETA.

[8]  A. Nechaev,et al.  Kerdock code in a cyclic form , 1989 .

[9]  Shojiro Sakata,et al.  Finding a Minimal Set of Linear Recurring Relations Capable of Generating a Given Finite Two-Dimensional Array , 1988, J. Symb. Comput..

[10]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[11]  Ingemar J. Cox,et al.  Digital Watermarking , 2003, Lecture Notes in Computer Science.

[12]  Shojiro Sakata,et al.  Extension of the Berlekamp-Massey Algorithm to N Dimensions , 1990, Inf. Comput..

[13]  R. Gold,et al.  Optimal binary sequences for spread spectrum multiplexing (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[14]  Oscar Moreno,et al.  Multi-dimensional arrays for watermarking , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[15]  Rudolf Lide,et al.  Finite fields , 1983 .

[16]  P. Vijay Kumar,et al.  A new family of binary pseudorandom sequences having optimal periodic correlation properties and large linear span , 1989, IEEE Trans. Inf. Theory.

[17]  Martin F. H. Schuurmans,et al.  Digital watermarking , 2002, Proceedings of ASP-DAC/VLSI Design 2002. 7th Asia and South Pacific Design Automation Conference and 15h International Conference on VLSI Design.

[18]  T. Kasami.  WEIGHT DISTRIBUTION FORMULA FOR SOME CLASS OF CYCLIC CODES , 1966 .