[Paper Review] Guess & Check Codes for Deletions, Insertions, and Synchronization

Serge Kas Hanna, Salim El Rouayheb|arXiv (Cornell University)|May 24, 2017

DNA and Biological Computing17 references3 citations

TL;DR

This paper introduces Guess & Check (GC) codes, a new family of systematic, explicitly constructed codes that correct up to a constant number of deletions or insertions with high probability, using logarithmic redundancy. The method leverages polynomial hashing over finite fields and probabilistic decoding with bounded failure probability, enabling efficient file synchronization in distributed systems with minimal communication overhead.

ABSTRACT

We consider the problem of constructing codes that can correct $\delta$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes, dating back to 1965, are zero-error single deletion $(\delta=1)$ correcting codes, and have an asymptotically optimal redundancy. Finding similar codes for $\delta \geq 2$ deletions remains an open problem. In this work, we relax the standard zero-error (i.e., worst-case) decoding requirement by assuming that the positions of the $\delta$ deletions (or insertions) are independent of the codeword. Our contribution is a new family of explicit codes, that we call Guess & Check (GC) codes, that can correct with high probability up to a constant number of $\delta$ deletions (or insertions). GC codes are systematic; and have deterministic polynomial time encoding and decoding algorithms. We also describe the application of GC codes to file synchronization.

Motivation & Objective

• To design systematic codes that correct multiple deletions with low redundancy for practical file synchronization.
• To relax the worst-case zero-error decoding requirement by allowing a small, bounded probability of decoding failure.
• To enable efficient remote file synchronization by minimizing transmitted parity bits.
• To construct explicit, polynomial-time encodable and decodable codes for δ deletions, where δ is a constant.
• To generalize VT codes beyond single deletion correction while maintaining system-level efficiency.

Proposed method

• Uses a systematic encoding scheme where message bits are grouped into blocks and mapped to elements in a finite field GF(q) with q = 2^ℓ.
• Employs polynomial-based parity checks over GF(q) to generate redundancy, with coefficients derived from MDS code generator matrices.
• Applies a 'guess and check' decoding strategy: the decoder guesses the positions of δ deletions and verifies consistency using polynomial equations.
• Bounds the number of possible deletion position combinations via combinatorial and algebraic constraints in GF(q), showing it is upper bounded by 2^{δ²}.
• Uses the total law of probability to derive a failure probability bound that vanishes asymptotically when ℓ = Ω(log k).
• Employs finite field arithmetic to ensure unique mapping of binary vectors to field elements, enabling unique solution recovery for given bit types.

Experimental results

Research questions

• RQ1 Can systematic codes be constructed for multiple deletions with logarithmic redundancy and polynomial-time encoding/decoding?
• RQ2 What is the maximum number of deletion positions that can be handled with bounded decoding failure probability?
• RQ3 How can the system-level efficiency of file synchronization be improved by reducing transmitted parity bits?
• RQ4 Can the decoding failure probability be made to vanish asymptotically under reasonable assumptions on message and deletion distributions?
• RQ5 What is the trade-off between redundancy, decoding complexity, and failure probability in probabilistic deletion correction codes?

Key findings

• GC codes achieve logarithmic redundancy, specifically O(δ log n) for δ deletions, which is asymptotically optimal for constant δ.
• The probability of decoding failure is upper bounded by O((k/ℓ)^δ / 2^{ℓ(c−δ)}), which vanishes as k → ∞ when ℓ = Ω(log k).
• The number of possible deletion position combinations is upper bounded by 2^{δ²}, independent of block size k.
• Encoding and decoding run in deterministic polynomial time, with encoding complexity O(kℓ) and decoding complexity O(k^{δ+1}/ℓ^{δ−1}).
• GC codes are systematic and explicitly constructible, enabling direct application to remote file synchronization.
• The method generalizes VT codes beyond single deletion correction while maintaining low redundancy and efficient decoding.