Evaluation codes are defined as follows: let F be a finite field and let P1, ... , Pn be points in F × F; fix some positive integer m. The code is then the subspace of F × ... × F, consisting of all vectors of the form (f(P1), ... , f(Pn)), where f(X, Y) is a polynomial in commuting indeterminates of degree at most m over F. Such codes play an important role in the theory of error-correcting codes, and their value depends on the proper selection of the points P1, ... , Pn.

In this paper, the authors consider the situation where the points are chosen to be singular points of algebraic differential equations. Using sophisticated techniques from algebraic geometry over finite fields, they are able to study the various parameters of such codes, including giving estimates for the minimal distance of evaluation codes constructed in this manner. This allows for new methods of construction for error-correcting codes with pre-specified minimal distances.

The paper will be quite interesting for those who have the mathematical background to follow the sophisticated proofs. This line of research seems very promising, and is sure to lead to even more interesting results in the future. Those working with error-correcting codes in practice should keep an eye on this research.