List Decoding of Deletions Using Guess & Check Codes

TL;DR

This paper analyzes the list decoding performance of Guess & Check codes for multiple deletions, showing that the list size tends to one on average and remains bounded for large message lengths, with supporting simulations.

Contribution

It investigates the list decoding capabilities of GC codes, revealing their asymptotic optimality and bounded list size for constant deletions, which was not previously established.

Findings

Average list size approaches 1 as message length increases

Maximum list size can be bounded independently of message length

Numerical simulations support theoretical results

Abstract

Guess & Check (GC) codes are systematic binary codes that can correct multiple deletions, with high probability. GC codes have logarithmic redundancy in the length of the message $k$, and the encoding and decoding algorithms of these codes are deterministic and run in polynomial time for a constant number of deletions $\delta$. The unique decoding properties of GC codes were examined in a previous work by the authors. In this paper, we investigate the list decoding performance of these codes. Namely, we study the average size and the maximum size of the list obtained by a GC decoder for a constant number of deletions $\delta$. The theoretical results show that: (i) the average size of the list approaches $1$ as $k$ grows; and (ii) there exists an infinite sequence of GC codes indexed by $k$, whose maximum list size in upper bounded by a constant that is independent of $k$. We also provide numerical simulations on the list decoding performance of GC codes for multiple values of $k$ and $\delta$.