Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance

We give two finite-circle constructions of binary strings whose least ℓ${}_1$ distortion is $\exp (\Omega (\sqrt{\log d\log \log d}))$, where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d, including the empty word.