Explicit Constructions of Two-Dimensional Reed-Solomon Codes in High Insertion and Deletion Noise Regime

Résumé : Insertion and deletion (insdel for short) errors are synchronization errors in communication systems caused by the loss of positional information in the message. Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability in the classical setting. This interest also translates to the insdel metric setting, as the Guruswami-Sudan decoding algorithm can be utilized to provide a deletion correcting algorithm in the insdel metric. Nevertheless, there have been few studies on the insdel error-correcting capability of Reed-Solomon codes. Our main contributions in this paper are explicit constructions of two families of 2-dimensional Reed-Solomon codes with insdel error-correcting capabilities asymptotically reaching those provided by the Singleton bound. The first construction gives a family of Reed-Solomon codes with insdel error-correcting capability asymptotic to its length. The second construction provides a family of Reed Solomon codes with an exact insdel error-correcting capability up to its length. Both our constructions improve the previously known construction of 2-dimensional Reed-Solomon codes whose insdel error-correcting capability is only logarithmic on the code length.