Coordinate-ordering-free Upper Bounds for Linear Insertion-Deletion Codes

TL;DR

This paper establishes new coordinate-ordering-free upper bounds on insdel distances of linear codes, enhancing understanding of their error correction capabilities, applicable to cyclic, algebraic-geometric, and Reed-Muller codes.

Contribution

It introduces stronger, coordinate-ordering-free upper bounds on insdel distances using generalized Hamming weights, applicable to various code families.

Findings

Bounds are stronger than previous results.

Bounds apply to cyclic and algebraic-geometric codes.

Specific bounds are provided for Reed-Muller codes.

Abstract

The insertion-deletion codes were motivated to correct the synchronization errors. In this paper we prove several coordinate-ordering-free upper bounds on the insdel distances of linear codes, which are based on the generalized Hamming weights and the formation of minimum Hamming weight codewords. Our bounds are stronger than some previous known bounds. We apply these upper bounds to some cyclic codes and one algebraic-geometric code with any rearrangement of coordinate positions. Some strong upper bounds on the insdel distances of Reed-Muller codes with special coordinate-ordering are also given.