Chudnovsky's method makes it possible to construct multiplication algorithms in finite fields with good bilinear complexity, i.e. a small number of bilinear multiplications in the base field. However, their total complexity and the difficulty of efficiently constructing these algorithms are unfavorable.

Historically, these are evaluation/interpolation algorithms using rational points of algebraic curves/rational places of function fields. Using asymptotically optimal towers, the existence of algorithms with bilinear complexity that is linear in the degree of extension has been proven. But these algorithms are not constructible in polynomial time.
Another approach is a recursive construction using places of increasing degrees. It provides algorithms with a quasi-linear bilinear complexity, but constructible in polynomial time.

We will introduce a hybrid construction, taking the best of both strategies to obtain algorithms with linear bilinear complexity that can be constructed in polynomial time.

Secondly, we will see that some locally recoverable codes, where the recovery of a corrupted symbol is possible using a small amount of other symbols, appear naturally in the recursive construction of Chudnovsky-type algorithms. We will define and study these codes.