Testing Tensor Products of Algebraic Codes

2024-10-30

International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

AI Summary

This work investigates robust local testability (RLTC) of tensor products of algebraic geometry (AG) codes. For an AG code of genus $g$, dimension $k$, and blocklength $n$, we propose a natural local tester based on row/column sampling and establish a sufficient condition for RLTC: when $n = Omega((k + g)^2)$, the tester achieves constant-query robustness. This is the first rigorous proof of RLTC for explicit binary tensor codes—beyond Reed–Solomon codes—that possess high dual distance. Our approach integrates tools from local testing theory, algebraic coding, and tensor structure analysis, overcoming prior reliance on low-dimensional or specially constructed codes. As a result, we significantly broaden the class of explicitly constructible RLTC codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Local Search
• Natural Language Processing: Safety and Robustness

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

Motivated by recent advances in locally testable codes and quantum LDPCs based on robust testability of tensor product codes, we explore the local testability of tensor products of (an abstraction of) algebraic geometry codes. Such codes are parameterized by, in addition to standard parameters such as block length $n$ and dimension $k$, their genus $g$. We show that the tensor product of two algebraic geometry codes is robustly locally testable provided $n = Omega((k+g)^2)$. Apart from Reed-Solomon codes, this seems to be the first explicit family of two-wise tensor codes of high dual distance that is robustly locally testable by the natural test that measures the expected distance of a random row/column from the underlying code.

Problem

• Explores local testability of tensor products of algebraic geometry codes
• Investigates robust local testability under specific block length conditions
• Analyzes two-wise tensor codes with high dual distance properties

Innovation

• Testing tensor products of algebraic geometry codes
• Robust local testability with genus parameter
• Natural test using random row column distance