Operation counts for Walsh–Hadamard circuits
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The Walsh–Hadamard transform combines inputs using fixed patterns of plus and minus signs. Circuit complexity asks how many arithmetic steps are needed, with the permitted operation and its cost stated explicitly.
Why it matters and what counts as progress
Why it matters: Exact circuit counts separate algebraic savings from measured program speed.
What counts as progress: A new construction or justified bound with the transform, coefficient domain, input lengths and operation costs specified.
AI contribution
Explicit circuits for the Walsh–Hadamard transform below the textbook operation count
For every commutative ring R and every k≥0, the exact, unscaled Walsh–Hadamard transform on N=2^k inputs has a straight-line circuit with at most (3/4) N log₂ N + N/4 lines. Each line creates u+c·v or c·v−u from available values, with a fixed ring constant c, and costs one. Reuse, output selection and permutations are free.
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Explicit circuits for the Walsh–Hadamard transform below the textbook operation count
Explicit circuits for the Walsh–Hadamard transform below the textbook operation count