On the distinctness of binary sequences derived from $2$-adic expansion of m-sequences over finite prime fields

Let p be an odd prime with 2-adic expansion ∑k i=0 pi · 2 . For a sequence a = (a(t))t≥0 over Fp, each a(t) belongs to {0, 1, . . . , p − 1} and has a unique 2-adic expansion a(t) = a0(t) + a1(t) · 2 + · · ·+ ak(t) · 2 , with ai(t) ∈ {0, 1}. Let ai denote the binary sequence (ai(t))t≥0 for 0 ≤ i ≤ k. Assume i0 is the smallest index i such that pi = 0 and a and b are two different m-sequences generated by a same primitive characteristic polynomial over Fp. We prove that for i 6= i0 and 0 ≤ i ≤ k, ai = bi if and only if a = b, and for i = i0, ai0 = bi0 if and only if a = b or a = −b. Then the period of ai is equal to the period of a if i 6= i0 and half of the period of a if i = i0. We also discuss a possible application of the binary sequences ai.