Compressing Mappings on Primitive Sequences over Z/(2 e) and Its Galois Extension

Let f( x) be a strongly primitive polynomial of degree n over Z/(2 e ), η( x 0, x 1,…, x e−2 ) a Boolean function of e−1 variables and ϕ( x 0, x 1,…, x e−1 )= x e−1 + η( x 0, x 1,…, x e−2 ) G ( f( x), Z/(2 e )) denotes the set of all sequences over Z/(2 e ) generated by f( x), F 2 ∞ the set of all sequences over the binary field F 2, then the compressing mapping Φ : G(f(x),Z/(2 e))→F 2 ∞, a= a 0+ a 12+···+ a e−12 e−1↦ϕ( a 0, a 1,…, a e−1) mod 2 is injective, that is, for a , b ∈ G( f( x), Z/(2 e )), a = b if and only if Φ( a )= Φ( b ), i.e., ϕ( a 0,…, a e−1 )= ϕ( b 0,…, b e−1 ) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.