Uniqueness of the distribution of zeroes of primitive level sequences over Z / ( p e ) (II)

• Published in: Finite fields and their applications 2007-04, Vol.13 (2), p.230-248
• Main Authors: Zhu, Xuan-Yong, Qi, Wen-Feng
• Format: Article
• Language: English
• Subjects: Integer residue ring; Level sequence; Linear recurring sequence; Primitive sequence
• ISSN: 1071-5797; 1090-2465
• DOI: 10.1016/j.ffa.2006.02.002

Let p be a prime number, Z / ( p e ) the integer residue ring, e ⩾ 2 . For a sequence a ̲ 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 { 0 , 1 , … , p − 1 } . Let f ( x ) ∈ Z / ( p e ) [ x ] be a primitive polynomial of degree n, a ̲ and b ̲ be sequences generated by f ( x ) over Z / ( p e ) , such that a ̲ ≠ 0 ̲ ( mod p e − 1 ) . This paper shows that the distribution of zero in the sequence a ̲ e − 1 = ( a e − 1 ( t ) ) t ⩾ 0 contains all information of the original sequence a ̲ , that is, if a e − 1 ( t ) = 0 if and only if b e − 1 ( t ) = 0 for all t ⩾ 0 , then a ̲ = b ̲ . Here we mainly consider the case of p = 3 and the techniques used in this paper are very different from those we used for the case of p ⩾ 5 in our paper [X.Y. Zhu, W.F. Qi, Uniqueness of the distribution of zeroes of primitive level sequences over Z / ( p e ) , Finite Fields Appl. 11 (1) (2005) 30–44].