List decoding subspace codes from insertions and deletions

TL;DR

This paper introduces a novel list decoding algorithm for subspace codes capable of correcting insertions and deletions, utilizing folded Reed-Solomon codes and linearized polynomials, with significant improvements in handling deletions.

Contribution

It presents the first list decoding algorithm for subspace codes that can handle deletions, and improves decoding results for insertion-only scenarios using linearized Reed-Solomon codes.

Findings

First list decoding algorithm for subspace codes with deletions

Handles maximum fraction of insertions and deletions with polynomially small rate

Improves decoding performance for insertion-only subspace codes

Abstract

We present a construction of subspace codes along with an efficient algorithm for list decoding from both insertions and deletions, handling an information-theoretically maximum fraction of these with polynomially small rate. Our construction is based on a variant of the folded Reed-Solomon codes in the world of linearized polynomials, and the algorithm is inspired by the recent linear-algebraic approach to list decoding. Ours is the first list decoding algorithm for subspace codes that can handle deletions; even one deletion can totally distort the structure of the basis of a subspace and is thus challenging to handle. When there are only insertions, we also present results for list decoding subspace codes that are the linearized analog of Reed-Solomon codes (proposed previously, and closely related to the Gabidulin codes for rank-metric), obtaining some improvements over similar results in previous work.