Bounds on List Decoding of Linearized Reed-Solomon Codes

TL;DR

This paper investigates list decoding bounds for Linearized Reed-Solomon codes, revealing exponential list sizes above the Johnson radius and limitations for certain code families beyond the unique decoding radius.

Contribution

It provides the first lower bounds on list size for LRS codes and demonstrates decoding limitations for specific code families.

Findings

Lower bound on list size is exponential above Johnson radius

Some LRS code families cannot be list decoded beyond the unique radius

Results unify understanding of list decoding behavior across sum-rank metric codes

Abstract

Linearized Reed-Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. In the two extreme cases of the sum-rank metric, they coincide with Reed-Solomon codes (Hamming metric) and Gabidulin codes (rank metric). List decoding in these extreme cases is well-studied, and the two code classes behave very differently in terms of list size, but nothing is known for the general case. In this paper, we derive a lower bound on the list size for LRS codes, which is, for a large class of LRS codes, exponential directly above the Johnson radius. Furthermore, we show that some families of linearized Reed-Solomon codes with constant numbers of blocks cannot be list decoded beyond the unique decoding radius.