Uniqueness of the distribution of zeroes of primitive level sequences over Z/( pe)

Abstract

Let p be a prime number, p⩾5, Z/( p e ) the integer residue ring, e⩾2, Γ={0,1,…, p−1}. For a sequence ā over Z/( p e ), there is a unique decomposition a ̄ = a ̄ 0+ a ̄ 1·p+⋯+ a ̄ e−1·p e−1 , where a ̄ i be the sequence over Γ. Let f( x) be a primitive polynomial with degree n over Z/( p e ), ā and b̄ sequences generated by f( x) over Z/( p e ), a ̄ ≠0( mod p e−1) ; we prove that the distribution of zeroes in the sequence a ̄ e−1=(a e−1(t)) t⩾0 contains all information of the original sequence ā, that is, if a e−1 ( t)=0 if and only if b e−1 ( t)=0 for all t⩾0, then a ̄ = b ̄ . As a consequence, we have the following results: (i) two different primitive level sequences are linearly independent over Z/( p); (ii) for all positive integer k, a ̄ e−1 k= b ̄ e−1 k if and only if a ̄ = b ̄ .