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Optimal Two-Dimensional Reed--Solomon Codes Correcting Insertions and Deletions

Roni Con, Amir Shpilka, Itzhak Tamo

TL;DR

The paper resolves the minimal field-size question for constructing $[n,2]_q$ Reed–Solomon codes that can correct $n-3$ insdel errors by giving explicit constructions over the cubic extension field $\mathbb{F}_{q^3}$ with $q = O(n^3)$. It uses an algebraic Vandermonde-type determinant condition to certify insdel-correcting capability and provides two variants that work for all characteristics and, for odd characteristic, yield longer codes. It also proves a near-tight lower bound $q \ge \binom{n}{3}-1$, establishing near-optimality of the cubic-field-size construction. This advances practical insdel-robust RS-code designs and tightens the field-size bounds for two-dimensional RS codes under synchronization errors.

Abstract

Constructing Reed--Solomon (RS) codes that can correct insertions and deletions (insdel errors) has been considered in numerous recent works. For the special case of two-dimensional RS-codes, it is known [CST23] that an $[n,2]_q$ RS-code that can correct from $n-3$ insdel errors satisfies that $q=Ω(n^3)$. On the other hand, there are several known constructions of $[n,2]_q$ RS-codes that can correct from $n-3$ insdel errors, where the smallest field size is $q=O(n^4)$. In this short paper, we construct $[n,2]_q$ Reed--Solomon codes that can correct $n-3$ insdel errors with $q=O(n^3)$, thereby resolving the minimum field size needed for such codes.

Optimal Two-Dimensional Reed--Solomon Codes Correcting Insertions and Deletions

TL;DR

The paper resolves the minimal field-size question for constructing Reed–Solomon codes that can correct insdel errors by giving explicit constructions over the cubic extension field with . It uses an algebraic Vandermonde-type determinant condition to certify insdel-correcting capability and provides two variants that work for all characteristics and, for odd characteristic, yield longer codes. It also proves a near-tight lower bound , establishing near-optimality of the cubic-field-size construction. This advances practical insdel-robust RS-code designs and tightens the field-size bounds for two-dimensional RS codes under synchronization errors.

Abstract

Constructing Reed--Solomon (RS) codes that can correct insertions and deletions (insdel errors) has been considered in numerous recent works. For the special case of two-dimensional RS-codes, it is known [CST23] that an RS-code that can correct from insdel errors satisfies that . On the other hand, there are several known constructions of RS-codes that can correct from insdel errors, where the smallest field size is . In this short paper, we construct Reed--Solomon codes that can correct insdel errors with , thereby resolving the minimum field size needed for such codes.
Paper Structure (10 sections, 6 theorems, 21 equations, 1 table)

This paper contains 10 sections, 6 theorems, 21 equations, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions and notation
  3. Related work
  4. Linear codes against insdel errors.
  5. RS-codes against insdel errors.
  6. Our results
  7. An algebraic condition
  8. Explicit construction for $k=2$ with cubic field size
  9. Construction for any characteristic
  10. Improved construction for odd characteristics

Key Result

Theorem 1.3

Every linear insdel code which is capable of correcting a $\delta$ fraction of deletions has rate at most $(1-\delta)/2 + o(1)$.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 1.2: Reed-Solomon codes
  • Theorem 1.3: Half-Singleton bound: Corollary 5.1 in cheng2020efficient
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 2.1
  • Proposition 2.1
  • ...and 5 more