Code-based quantum fingerprint

For $n\ge 1$ and $N\ge 1$, let $E:\{0,1{\}}^n\to \{0,1{\}}^N$ be a binary code map. Define its relative distance by $\delta (E):=\frac{1}{N}{\min }_{x\mathrm{\ne }y}{d}_H(E(x),E(y)),$ where $d_H(u,v):=\mathrm{\mid }\{i<N:u_i\mathrm{\ne }{v}_i\}\mathrm{\mid }$ and the minimum is over distinct inputs. Thus duplicate codewords are allowed and give relative distance $0$, and $\delta (E)\in [0,1]$; the minimum is over a nonempty finite set because $n\ge 1$. For every such map, with no positive-distance assumption, let $m=\lceil {\log }_{2}N\rceil$ and use the ordered computational basis of the $m$-qubit index register, padding unused indices by zero amplitude. The fingerprint state of $x$ is $\mathrm{\mid }{h}_x⟩:=\frac{1}{\sqrt{N}}{\sum }_{i=0}^{N-1}\mathrm{\mid }i⟩\mathrm{\mid }{E}_i(x)⟩,$ where $E_i(x)$ is bit $i$ of $E(x)$, indexed from $0$ to $N-1$. It is a vector in the $(m+1)$-qubit index-and-data register. The following theorem proves it is normalized and computes its overlap. This is a state definition; it makes no uniformity or encoder-time claim.