Testing Tensor Products of Algebraic Codes

TL;DR

This paper investigates the local testability of tensor products of algebraic geometry codes, establishing conditions under which these codes are robustly locally testable, with implications for coding theory and quantum error correction.

Contribution

It demonstrates that tensor products of algebraic geometry codes are robustly locally testable when the block length scales quadratically with the sum of the code's dimension and genus, a novel result for high dual distance codes.

Findings

Tensor product of algebraic geometry codes is robustly locally testable under certain length conditions.

Provides explicit family of high dual distance tensor codes with local testability.

Extends understanding of testability beyond Reed-Solomon codes.

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.