Error-correcting codes are characterized by minimum distance — the smallest number of errors that can convert one valid codeword into another. This single number determines how many errors can be corrected. But in analog computation, where values are real numbers rather than bits, errors come in sizes. A small perturbation (magnitude ≤ δ) is tolerable; a large outlier (magnitude > Δ) must be detected or located. Minimum distance cannot distinguish between these regimes.
Ravagnani, Rini, and Wachter-Zeh (arXiv:2602.20366) introduce the height profile — a function that captures the full error-handling capability of a real-valued linear code, not just the worst case. The height profile maps each error position to the threshold at which the error transitions from tolerable to detectable. Different positions can have different thresholds. The profile is finer than minimum distance because it tracks how individual components contribute to the code's robustness.
The computation of the height profile reduces to a linear program — solvable in polynomial time. For codes whose generator matrix columns share the same norm, the height profile has a geometric interpretation as distances between hyperplane arrangements. The profile can be computed exactly for several classical code families.
The result: a complete characterization of when analog computation can tolerate bounded noise while detecting outliers. The profile tells you not just how many errors the code handles, but which errors at which positions and which magnitudes.
The general observation: when a system faces heterogeneous perturbations — some tolerable, some catastrophic — a single robustness metric is insufficient. A profile that maps perturbation type to response threshold captures the system's behavior more faithfully. The move from scalar to functional characterization reveals structure that the scalar hides.