Binary Reconstruction Codes for Correcting One Deletion and One Substitution

TL;DR

This paper introduces binary reconstruction codes that can correct one deletion and one substitution, providing explicit constructions with low redundancy for various parameters, advancing error correction capabilities in binary sequences.

Contribution

The paper demonstrates the existence of binary reconstruction codes with minimal redundancy capable of correcting one deletion and one substitution, for specific intersection parameters.

Findings

Redundancy can be zero for certain parameters.

Redundancy can be as low as logarithmic in sequence length.

Explicit code constructions are provided for various intersection sizes.

Abstract

In this paper, we investigate binary reconstruction codes capable of correcting one deletion and one substitution. We define the \emph{single-deletion single-substitution ball} function $ \mathcal{B} $ as a mapping from a sequence to the set of sequences that can be derived from it by performing one deletion and one substitution. A binary \emph{$(n,N;\mathcal{B})$-reconstruction code} is defined as a collection of binary sequences of length $ n $ such that the intersection size between the single-deletion single-substitution balls of any two distinct codewords is strictly less than $ N $. This property ensures that each codeword can be uniquely reconstructed from $ N $ distinct elements in its single-deletion single-substitution ball. Our main contribution is to demonstrate that when $ N $ is set to $ 4n - 8 $, $ 3n - 4 $, $2n+9$, $ n+21 $, $31$, and $7$, the redundancy of binary $(n,N;\mathcal{B})$-reconstruction codes can be $0$, $1$, $2$, $ \log\log n + 3 $, $\log n + 1 $, and $ 3\log n + 4 $, respectively, where the logarithm is on base two.