Research proof index

For a complex-valued function on the ambient finite group and a nonzero even integer $n$, ordinary self-convolution has no larger normalized $L^n$ norm than self-difference-convolution: $\|f*f\|_n\le\|f\mathbin{\circleddash}f\|_n$.

The $2n$-th Fourier moment of the indicator of a finite set equals its order-$n$ additive energy: $\|\widehat{1_s}\|_{2n}^{2n}=E_n(s)$. This is the standard bridge between Fourier norms and additive tuple counts.

A non-trivial code (a code with at least two distinct codewords) must have a minimum distance greater than 0.

A word u is close to a code C within an absolute error bound e if and only if it is close within the equivalent relative error bound e / n.

For finite nonempty codes, the computable distance equals the noncomputable distance.

A stronger version of distFromCode_eq_of_lt_half_dist: If two codewords v and w are both within the uniqueDecodingRadius of u (i.e. 2 * Δ₀(u, v) < ‖C‖₀ and 2 * Δ₀(u, w) < ‖C‖₀), then they must be equal.