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This paper examines the construction and properties of binary Parsevalframes. We address two questions: When does a binary Parseval frame have acomplementary Parseval frame? Which binary symmetric idempotent matrices areGram matrices of binary Parseval frames? In contrast to the case of real orcomplex Parseval frames, the answer to these questions is not alwaysaffirmative. The key to our understanding comes from an algorithm thatconstructs binary orthonormal sequences that span a given subspace, wheneverpossible. Special regard is given to binary frames whose Gram matrices arecirculants.

What is odd about binary Parseval frames?