On the linear span of binary sequences obtained from q-ary m-sequences, q odd

A class of periodic binary sequences that are obtained from q -ary m-sequences is defined, and a general method to determine their linear spans (the length of the shortest linear recursion over the Galois field GF(2) satisfied by the sequence) is described. The results imply that the binary sequences under consideration have linear spans that are comparable with their periods, which can be made very long. One application of the results shows that the projective and affine hyperplane sequences of odd order both have full linear span. Another application involves the parity sequence of order n, which has period p^m-1, where p is an odd prime. The linear span of a parity sequence of order n is determined in terms of the linear span of a parity sequence of order 1, and this leads to an interesting open problem involving primes