Testing Tensor Products of Algebraic Codes

TL;DR

This work studies robust local testability for tensor products of algebraic geometry codes, motivated by locally testable codes and quantum LDPCs. It introduces an abstraction of AG codes with genus $g$ and a Hadamard-product closure, and defines a two-dimensional robustness notion for tensor-product tests that measure row/column proximity to component codes. The main result shows that for $(q,n,g)$-AG code sequences, if $\ell>\max\{c_0,g\}$ and $n>c_0(\ell+g)^2$, then the pair $(\mathcal{C}_1(\ell),\mathcal{C}_2(\ell))$ is $\rho$-robust for some $\rho>0$, providing an explicit, high-dual-distance two-wise tensor code that is robustly locally testable. This advances beyond Reed-Solomon codes by offering a broad, explicit family of AG-based tensor codes with robust testability, with implications for PCPs, LTCs, and quantum LDPC constructions, while highlighting a quadratic-length requirement in the dimension regime.

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 $n = Ω((k+g)^2)$. 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.

Theorems & Definitions (13)

• Definition 2.1: (Abstract) Algebraic Geometry Code
• Definition 2.2: Robustness of Tensor Product
• Proposition 2.3
• Theorem 2.4
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Claim 3.3
• proof