Hermite Interpolation With Error Correction: Fields of Zero or Large Characteristic and Large Error Rate

Multiplicity code decoders are based on Hermite polynomial interpolation with error correction. In order to have a unique Hermite interpolant one assumes that the field of scalars has characteristic 0 or ≥ 𝓁 +1, where 𝓁 is the maximum order of the derivatives in the list of values of the polynomial and its derivatives which are interpolated. For scalar fields of characteristic 𝓁+1, the minimum number of values for interpolating a polynomial of degree ≤ D is D+1+2E(𝓁+1) when ≤ E of the values are erroneous. Here we give an error-correcting Hermite interpolation algorithm that requires fewer values, that is, that can tolerate more errors, assuming that the characteristic of the scalar field is either 0 or ≥ D+1. Our algorithm requires (𝓁+1)D + 1 - (𝓁+1)𝓁/2 + 2E values.

As an example, we consider 𝓁 = 2. If the error ratio (number of errors)/(number of evaluations) ≤ 0.16, our new algorithm requires (4+7/17),D - (1+8 /17) values, while multiplicity decoding requires 25D+25 values. If the error ratio is ≤ 0.2, our algorithm requires 5D-2 evaluations over fields of characteristic 0 or ≥ D+1, while multiplicity decoding for an error ratio 0.2 over fields of characteristic 3 is not possible for D ≥ 3.

Our algorithm is based on Reed-Solomon interpolation without multiplicities, which becomes possible for Hermite interpolation because of the high redundancy necessary for error-correction.