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Noise-Tolerant Codebooks for Semi-Quantitative Group Testing: Application to Spatial Genomics

Kok Hao Chen, Duc Tu Dao, Han Mao Kiah, Van Long Phuoc Pham, Eitan Yaakobi

TL;DR

This work develops a formal framework for noise-tolerant codebooks in semi-quantitative group testing, motivated by spatial genomics. It introduces ${\lambda-{\sf ADD}}$-codes that interpolate ${\sf OR}$, ${\sf XOR}$, and ${\sf ADD}$ codes using the ${\boxplus}_{\boldsymbol{\lambda}}$ operation, and provides explicit constructions and bounds in two regimes: constant distance $d$ (where XOR-based constructions yield rates approaching $\tfrac{1}{2}$ for small defectives, e.g., $s=2$) and distance $d=\delta n$ (where computable Gilbert-Varshamov-type bounds apply). The XOR codes receive sharp asymptotic rate characterizations, while the ${\lambda-ADD}$ family enables GV-type lower bounds, with comparisons showing the GV bounds often outperform indirect constructions, and XOR-based methods excel at small $\delta$. Upper bounds for ADD-codes are established via ternary-code embeddings, indicating a fundamental gap between achievable and provable limits and guiding practical code design for spatial-genomics-like settings. Overall, the results yield practical, near-optimal codebooks for robust semi-quantitative testing with potential impact on efficient spatial genomics workflows.

Abstract

Motivated by applications in spatial genomics, we revisit group testing (Dorfman~1943) and propose the class of $λ$-{\sf ADD}-codes, studying such codes with certain distance $d$ and codelength $n$. When $d$ is constant, we provide explicit code constructions with rates close to $1/2$. When $d$ is proportional to $n$, we provide a GV-type lower bound whose rates are efficiently computable. Upper bounds for such codes are also studied.

Noise-Tolerant Codebooks for Semi-Quantitative Group Testing: Application to Spatial Genomics

TL;DR

This work develops a formal framework for noise-tolerant codebooks in semi-quantitative group testing, motivated by spatial genomics. It introduces -codes that interpolate , , and codes using the operation, and provides explicit constructions and bounds in two regimes: constant distance (where XOR-based constructions yield rates approaching for small defectives, e.g., ) and distance (where computable Gilbert-Varshamov-type bounds apply). The XOR codes receive sharp asymptotic rate characterizations, while the family enables GV-type lower bounds, with comparisons showing the GV bounds often outperform indirect constructions, and XOR-based methods excel at small . Upper bounds for ADD-codes are established via ternary-code embeddings, indicating a fundamental gap between achievable and provable limits and guiding practical code design for spatial-genomics-like settings. Overall, the results yield practical, near-optimal codebooks for robust semi-quantitative testing with potential impact on efficient spatial genomics workflows.

Abstract

Motivated by applications in spatial genomics, we revisit group testing (Dorfman~1943) and propose the class of -{\sf ADD}-codes, studying such codes with certain distance and codelength . When is constant, we provide explicit code constructions with rates close to . When is proportional to , we provide a GV-type lower bound whose rates are efficiently computable. Upper bounds for such codes are also studied.
Paper Structure (13 sections, 17 theorems, 29 equations, 1 figure, 1 table)

This paper contains 13 sections, 17 theorems, 29 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. ${\sf OR}$-Codes and ${\sf XOR}$-Codes
  4. $\lambda$-${\sf ADD}$-Codes
  5. Bounds For ${\sf XOR}$-Codes
  6. Bounds for $\lambda$-${\sf ADD}$-Codes
  7. A GV-Type Lower Bound
  8. Comparison of Lower Bounds for ${\lambda-{\sf ADD}}$-Codes
  9. Upper Bound for ${\sf ADD}$-Codes
  10. Conclusion
  11. Acknowledgement
  12. Proof of Proposition \ref{['prop:lower-or']}
  13. Enumerating Hyperedges in ${\mathcal{G}}(n,d)$

Key Result

Theorem 1

There are computable constants $L_s$ and $U_s$ such that $L_s\le \mu_{{\sf OR}}(s)\le U_s$ for $s\ge 2$. Here, we list the first few values of $L_s$ and $U_s$.

Figures (1)

  • Figure 1: Comparison of lower bounds for $\mu_{\lambda-{\sf ADD}}(\delta;2)$. Note that from Proposition \ref{['prop:lamb-others']}, we have that $\mu_{\lambda-{\sf ADD}}(\delta;2)\ge \mu_{\sf ADD}(3\delta;2)\ge \mu_{\sf XOR}(3\delta;2)$ and $\mu_{\lambda-{\sf ADD}}(\delta;2)\ge\mu_{\sf OR}(\delta;2)$.

Theorems & Definitions (27)

  • Theorem 1: see du2000combinatorial or dyachkov2014lectures
  • Proposition 1
  • Theorem 2
  • Corollary 1
  • Theorem 3: see dyachkov2014lectures
  • Theorem 4
  • proof
  • proof : Proof of Upper Bound in Theorem \ref{['thm:main-xor']}
  • Proposition 2
  • proof
  • ...and 17 more