Pattern distributions of Legendre sequences

Legendre sequences have a number of interesting randomness properties and are closely related with quadratic residue codes. We give lower and upper bounds on the number of patterns distributed in a cycle of the Legendre sequences and establish the relationship between the weight distribution of quadratic residue codes and the pattern distribution of Legendre sequences. Our result shows that Legendre sequences have an ideal distribution of patterns of length s, when s is not large compared with log₂N, where N is the prime used to define the sequence