Coset Correlation of LR m-Arrays

The normal form and the coset correlation function of an arbitrary linear recurring m-array over F2 are introduced. It is proved that the normal form can be determined by its coset correlation function. Introduction Nguyen [3] studied the coset correlation of m-sequences and pointed out that the coset correlation provides an attractive technique for detecting the normal form when m-sequences are corrupted by noise. In this paper we will generalize the concept on coset correlation of m-sequences to linear recurring m-arrays. Furthermore we will calculate the coset correlation function of any linear recurring m-array. In Section 1, we will give the basic concepts and properties of linear recurring m-arrays which we need; in Section 2, discuss the coset correlation on linear recurring m-arrays; in Section 3, show the proof of the main theorem. 1. Basic concepts and properties of LR m-arrays An array A of period (I, s) means an infinite matrix A = (ai, j)i > 0, j 2 O over the finite field F2 such that %+r, j = ai,j’ai.j+s, i 27 O,j 2 0, (1) where r, s are the smallest positive integers for which the condition (1) is satisfied. AnmxnsubmatrixA(i,j) = (ai+~~,j+j~)o~i~<m,O~j,<nofAiscalledanm~nwindow or state of A at (i,j). Correspondence to: Professor M. Liu, Institute of Systems Science, Academia Sinica, Beijing 100080, China. 0166-218X/93/$06.00