## Section: New Results

### Error Locating pairs

Participants : Alain Couvreur, Isabella Panaccione.

Algebraic codes such as Reed–Solomon codes and algebraic geometry
codes benefit from efficient decoding algorithms permitting to
correct errors up to half the minimum distance and sometimes
beyond. In 1992, Pellikaan proved that many **unique** decoding could be
unified using an object called *Error correcting pair*. In
short, given an error correcting code $\mathcal{C}$, an error
correcting pair for $\mathcal{C}$ is a pair of codes
$(\mathcal{A},\mathcal{B})$ whose component wise product
$\mathcal{A}*\mathcal{B}$ is contained in the dual code
${\mathcal{C}}^{\perp}$ and such that $\mathcal{A},\mathcal{B}$ satisfy
some constraints of dimension and minimum distance.

On the other hand, in the late 90's, after the breakthrough of Sudan
and Guruswami Sudan the question of list decoding permitting to
decode beyond half the minimum distance. In a recently submitted
article, A. Couvreur and I. Panaccione [15]
proposed a unified point of view for probabilistic decoding
algorithms decoding beyond half the minimum distance. Similarly to
Pellikaan's result, this framework applies to any code benefiting
from an *error locating pair* which is a relaxed version of
error correcting pairs.