Bhattacharyya, Arnab and Gopi, Sivakanth (2017) Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs. In: ACM Transactions on Computation Theory, 9 (2). pp. 1-17. ISSN 1942-3454
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Abstract
Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C ⊂ ΣKn is an r-query affine invariant locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp(OK,r,|Σ|(nr-1)). Also, we show that if C ⊂ ΣKn is an r-query affine invariant locally testable code (LTC), then the number of codewords in C is at most exp(OK,r,|Σ|(nr-2)). The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty, and Sudan (ITCS'13) constructed affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM'11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems, which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, up to a small error in the Gowers norm.
| Item Type: | Journal Article |
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| Publication: | ACM Transactions on Computation Theory |
| Publisher: | Association for Computing Machinery |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Keywords: | Affine invariance; Gowers uniformity norm; Locally correctable code; Locally testable code |
| Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |
| Date Deposited: | 14 Jun 2022 06:04 |
| Last Modified: | 14 Jun 2022 06:04 |
| URI: | https://eprints.iisc.ac.in/id/eprint/73466 |
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