New dimension-independent upper bounds on linear insdel codes

Abstract

To determine the insdel distances of linear codes is a very challenging problem. The half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li is a basic upper bound on the insertion-deletion error-correcting capabilities of linear codes. In this paper we give several new coordinate ordering-free and coordinate ordering-depending upper bounds for the insdel distances of linear codes. These upper bounds do not depend on dimensions of linear codes and only depend on the formations of codewords. It is shown that for many natural well-known linear codes including binary simplex codes, Kasami codes, many linear binary codes with few non-zero weights and some algebraic geometry codes, the new coordinate ordering-free upper bounds on their insdel distances are strictly smaller than the half-Singleton bound and the direct upper bound. On the other hand for many linear binary codes, the ordering-depending upper bound on their insdel distance is 2. We also give ordering-depending upper bounds on the insdel distances of some linear ternary codes with few non-zero weights and some binary and ternary algebraic geometry codes.