This paper presents a new method to derive the phase difference between n-tuples of an m-sequence over GF(p) of period pn-1. For the binary m-sequence of the characteristic polynomial f(x)=xn+xd+1 with d=1,2c or n-2c, the explicit formulas of the phase difference from the initial n-tuple are efficiently derived by our method for specific n-tuples such as that consisting of all 1's and that cosisting of one 1 and n-1 0's, although the previously known formula exists only for that consisting of all 1's.

Recently, the small set of nonbinary Kasami sequences was presented and their correlation properties were clarified by Liu and Komo. The family of nonbinary Kasami sequences has the same periodic maximum nontrivial correlation as the family of Kumar-Moreno sequences, but smaller family size. In this paper, first it is shown that each of the nonbinary Kasami sequences is unbalanced. Secondly, a new family of nonbinary sequences obtained from modified Kasami sequences is proposed, and it is shown that the new family has the same maximum nontrivial correlation as the family of nonbinary Kasami sequences and consists of the balanced nonbinary sequences.

In this paper, we will show that some families of periodic sequences over Z4 and Z8 with period multiple of 2r-1 generated by r-th degree basic primitive polynomials assorted by the root of each polynomial, and give the exact distribution of sequences for each family. We also point out such an instability as an extreme increase of their linear complexities for the periodic sequences by one-symbol substitution, i.e., from the minimum value to the maximum value, for all the substitutions except one.

The maximum order complexity (MOC) of a sequence is a very natural generalization of the well-known linear complexity (LC) by allowing nonlinear feedback functions for the feedback shift register which generates a given sequence. It is expected that MOC is effective to reduce such an instability of LC as an extreme increase caused by the minimum changes of a periodic sequence, i. e. , one-symbol substitution, one-symbol insertion or one-symbol deletion per each period. In this paper we will give the bounds (lower and upper bounds) of MOC for the minimum changes of an m-sequence over GF(q) with period qn-1, which shows that MOC is much more natural than LC as a measure for the randomness of sequences in this case.