Matrix Construction Using Cyclic Shifts of a Column

TL;DR

This paper introduces a method for constructing matrices with good correlation properties using cyclic shifts of pseudonoise columns, applicable to digital image watermarking and sequence unfolding.

Contribution

It identifies specific shift sequences satisfying the constant difference property for optimal matrix synthesis, expanding options for correlation-optimized matrix construction.

Findings

Matrices with optimal correlation are synthesized using known shift sequences.

Unfolding matrices into sequences preserves correlation with minimal degradation.

The method is applicable to digital image watermarking and sequence generation.

Abstract

This paper describes the synthesis of matrices with good correlation, from cyclic shifts of pseudonoise columns. Optimum matrices result whenever the shift sequence satisfies the constant difference property. Known shift sequences with the constant (or almost constant) difference property are: Quadratic (Polynomial) and Reciprocal Shift modulo prime, Exponential Shift, Legendre Shift, Zech Logarithm Shift, and the shift sequences of some m-arrays. We use these shift sequences to produce arrays for watermarking of digital images. Matrices can also be unfolded into long sequences by diagonal unfolding (with no deterioration in correlation) or row-by-row unfolding, with some degradation in correlation.