The Euler quotient modulo an odd-prime power p r (r > 1) can be uniquely decomposed as a p-adic number of the form \(\frac{{u^{(p - 1)p^{r - 1} } - 1}} {{p^r }} \equiv a_0 (u) + a_1 (u)p + \cdots + a_{r - 1} (u)p^{r - 1} (\bmod p^r ), \gcd (u,p) = 1,\) where 0 ⩽ a j (u) 1. We firstly study certain arithmetic properties of the level sequences (a j (u)) u⩾0 over \(\mathbb{F}_p \) via introducing a new quotient. Then we determine the exact values of linear complexity of (a j (u)) u⩾0 and values of k-error linear complexity for binary sequences defined by (a j (u)) u⩾0.

Linear complexity problems of level sequences of Euler quotients and their related binary sequences

Binary expression of integer sums is investigated. A precise formula is presented for the coefficients of binary expressions of integer sums. This formula is a basis for some other results.

On Coefficients of Binary Expression of Integer Sums

In this paper, we discuss the 0,1 distribution in the highest level sequence a e -1 of primitive sequence over Z 2 e generated by a primitive polynomial of degree n . First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight for e relatively small to n . We also get an estimate which is suitable for e relatively large to n . Combining the two bounds, we obtain an estimate depending only on n , which shows that the larger n is, the closer to 1/2 the proportion of 1 will be.

Shuqin Fan

Papers

0,1 distribution in the highest level sequences of primitive sequences overZ 2e