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arXiv:2410.22606 · Leibniz international proceedings in informatics

Testing Tensor Products of Algebraic Codes

Sumegha Garg, Madhu Sudan, Wu, Gabriel

Published Oct 30, 2024• 0 citations
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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.

DOI
10.4230/lipics.approx/random.2025.59

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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.
We study the robust local testability of tensor products of two Algebraic-Geometry (AG) codes. In particular, we prove that constant rate AG codes are robust locally testable. This significantly generalizes the seminal result of Polishchuk-Spielman (1994), which proved robust local testability of Reed-Solomon codes. We establish an algebraic-geometric framework that enables us to geometrically interpret codewords in tensor products of AG codes. Thereby, we use tools from intersection theory of algebraic surfaces to prove a divisibility criterion for AG codes, that generalizes the bivariate divisibility result of Polishchuk-Spielman.Over the years, robust local testability of tensor products has played a key role in the development of classical locally testable codes (LTCs) as well as quantum Low Density Parity Check (qLDPC) codes and quantum Locally Testable Codes (qLTCs). To the best of our knowledge, after Reed-Solomon codes, our result provides the first explicit family of robustly locally testable codes with constant rate and linear dual-distance. Moreover, our result, when combined with Golowich-Guruswami (2024), yields new explicit families of good quantum CSS codes of length N which are locally testable with locality O(N)O(\sqrt N )O(N​) and constant soundness.
We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.
We show that the tensor product of two random linear codes is robustly testable with high probability. This implies that one can obtain pairs of linear codes such that their product and the product of their dual codes are simultaneously robustly testable. Such two-sided robustly testable codes (with a much weaker form of robustness) were the key ingredient in the recent constructions of asymptotically good quantum LDPC codes, which ensured their linear minimum distance. We hope that the existence of such codes with a stronger form of robustness, shown here, can be used to simplify the proofs and provide better distance bounds in these constructions. We also give new very simple examples of non-robustly testable codes. We show that if the parity-checks of two codes are mutually orthogonal, then their product is not robustly testable. In particular, this implies that the product of a code with its dual can never be robustly testable. We also study a property of a collection of linear codes called product-expansion, which can be viewed as a coboundary expansion of the cochain complex naturally associated with the product of these codes. We show that this property is related with the robust testability and the agreement testability of the products of codes.
We investigate the coboundary expansion property of tensor product codes, known as product expansion, which plays an important role in recent constructions of good quantum LDPC codes and classical locally testable codes. Prior research has shown that this property is equivalent to agreement testability and robust testability for products of two codes with linear distance. However, for products of more than two codes, product expansion is a strictly stronger property. In this paper, we prove that a collection of an arbitrary number of random codes over a sufficiently large field has good product expansion. We believe that, in the case of four codes, the same ideas can be used to construct good quantum locally testable codes, in a way similar to the current constructions that use only products of two codes.