Tight lower bounds for linear 2-query LCCs over finite fields

Abstract

A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic self-correcting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a δ fraction of the coordinates is corrupted. LCCs are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions.In this work, we show a separation between linear 2-query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form p Ω(δd) on the length of linear 2-query LCCs over Fp, that encode messages of length d. Our bound improves over the known bound of 2 Ω(δd) [8,10,6] which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in theoretical computer science.We also obtain, as corollaries of our main theorem, new results in incidence geometry over finite fields. The first is an improvement to the Sylvester-Gallai theorem over finite fields [14] and the second is a new analog of Beck's theorem over finite fields.The paper also contains an appendix, written by Sergey Yekhanin, showing that there do exist nonlinear LCCs of size 2 O(d) over Fp, thus highlighting the importance of the linearity assumption for our result.

A generalized Sylvester-Gallai type theorem for quadratic polynomials

“…Observe that over positive characteristic we expect the dimension of the set to scale like O(log |Q|), as for such fields a weaker version of Sylvester-Gallai theorem holds. Theorem 6.3 (Corollary 1.3 in [BDSS16]). Let V = {v 1 , .…”

Polynomial time deterministic identity testingalgorithm for $Σ^{[3]}ΠΣΠ^{[2]}$ circuits via Edelstein-Kelly type theorem for quadratic polynomials

“…However, we believe (though we did not check the details) that a similar approach should work over positive characteristic as well. Observe that over positive characteristic we expect the dimension of the set to scale like O(log |Q|), as for such fields a weaker version of Sylvester-Gallai theorem holds (see Corollary 1.3 in [BDSS16]).…”

Untitled

“…The repetition corresponds to the fact that there might be repeated columns in the generator matrix of the code, which may potentially make the property of local correction easier to satisfy. Indeed in the recent lower bounds for 2-query LCCs [8,5], handling the fact that there might be repetitions added significant complexity to the proofs of the lower bounds. In the current paper too we deal with repetitions by treating V as a list.…”

“…Using this to prove a global dimension bound on V (as in Theorem 2.3) is done using a standard amplification lemma (Lemma 10.2) similar to that in [5,8]. For simplicity, we will use big "O" notation to hide constants depending on δ (only for this overview).…”

“…The proofs here use a combination of geometric, analytic and linear-algebraic techniques, and give quantitative form to known qualitative point-line incidence theorems. Tighter bounds of n > p Ω(d) over finite fields of prime order p were proved in [8] using methods from arithmetic combinatorics, matching the trivial construction of taking all vectors in (F q ) d .…”