  This paper examines the construction and properties of binary Parseval
frames. We address two questions: When does a binary Parseval frame have a
complementary Parseval frame? Which binary symmetric idempotent matrices are
Gram matrices of binary Parseval frames? In contrast to the case of real or
complex Parseval frames, the answer to these questions is not always
affirmative. The key to our understanding comes from an algorithm that
constructs binary orthonormal sequences that span a given subspace, whenever
possible. Special regard is given to binary frames whose Gram matrices are
circulants.
