## Abstract

In this work we show that high dimensional expansion implies locally testable code. Specifically, we define a notion that we call high-dimensional-expanding-system (HDE-system). This is a set system defined by incidence relations with certain high dimensional expansion relations between its sets. We say that a linear code is modelled over HDE-system, if the collection of linear constraints that the code satisfies could by described via the HDE-system. We show that

a code that can be modelled over HDE-system is locally testable. This implies that high dimensional expansion phenomenon solely implies local testability of

codes. Prior work had to rely to local notions of local testability to get some global forms of testability (e.g. co-systolic expansion from local one, global agreement from local one), while our work infers global testability directly from high dimensional expansion without relying on some local form of testability.

The local testability result that we obtain from HDE-systems is, in fact, stronger than standard one, and we term it amplified local testability. Roughly speaking in an amplified locally testable code, the rejection probability of a corrupted codeword, that is not too far from the code, is amplified by a k factor compared to the guarantee in standard testing, where k is the length of the test (the guarantee for a corrupted codeword that is very far from the code is

the same as in standard testing). Amplified testing is stronger than standard local testability, but weaker than the notion of optimal testing as defined by Bhattacharyya et al., that roughly requires amplified local testability without assuming that the corrupted codeword is not too far from the code. We further show that most of the well studied locally testable codes as Reed-Muller codes

and more generally affine invariant codes with single-orbit property fall into our framework. Namely, it is possible to show that they are modelled over an HDE-system, and hence the family of all p-ary affine invariant codes is amplified locally testable. This yields the strongest known testing results for affine invariant codes with single orbit, strengthening the work of Kaufman and Sudan.

a code that can be modelled over HDE-system is locally testable. This implies that high dimensional expansion phenomenon solely implies local testability of

codes. Prior work had to rely to local notions of local testability to get some global forms of testability (e.g. co-systolic expansion from local one, global agreement from local one), while our work infers global testability directly from high dimensional expansion without relying on some local form of testability.

The local testability result that we obtain from HDE-systems is, in fact, stronger than standard one, and we term it amplified local testability. Roughly speaking in an amplified locally testable code, the rejection probability of a corrupted codeword, that is not too far from the code, is amplified by a k factor compared to the guarantee in standard testing, where k is the length of the test (the guarantee for a corrupted codeword that is very far from the code is

the same as in standard testing). Amplified testing is stronger than standard local testability, but weaker than the notion of optimal testing as defined by Bhattacharyya et al., that roughly requires amplified local testability without assuming that the corrupted codeword is not too far from the code. We further show that most of the well studied locally testable codes as Reed-Muller codes

and more generally affine invariant codes with single-orbit property fall into our framework. Namely, it is possible to show that they are modelled over an HDE-system, and hence the family of all p-ary affine invariant codes is amplified locally testable. This yields the strongest known testing results for affine invariant codes with single orbit, strengthening the work of Kaufman and Sudan.

Original language | English |
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Volume | abs/2107.10488 |

State | Published - 2021 |

### Publication series

Name | CoRR |
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