arXiv subjects

S. Venkitesh

Publications and source records attributed to S. Venkitesh.

15 recordsLinked to original sources

arXiv2026

Covering Weight-Determined Sets of Uniform Grids by Polynomials

We determine the minimum degree of a nonzero polynomial covering "weight-determined" subsets of a uniform grid over a field of characteristic zero. Since such subsets are completely determined by the grid dimensions and the set of weights taken by the points, we see that the minimum degree is also determined by the set of weights. In fact, the minimum degree can be algorithmically computed in time linear in the grid dimensions. Previously, such a result and algorithm was known only for uniform grids that are "strictly unimodal" (Venkitesh, E-JC 2022), which implies the grid cannot be "lopsided". Our result holds in full generality for all uniform grids. Our central tools are the classical algebraic objects called the "finite-degree Zariski closure" and the "affine Hilbert function". Our technical contribution is a combinatorial characterization of both these objects for all weight-determined subsets of a uniform grid. The minimal degree of our covering problem is precisely the threshold degree at which the behavior of the Zariski closure changes sharply.

math.CO↗

arXiv2026

Partial Derandomization for Leakage-Resilient Shamir's Secret Sharing over Composite Order Fields

We make progress on the question of constructing explicit evaluation places for leakage-resilient Shamir's secret sharing, over composite order fields. Previously, Maji et al. (EUROCRYPT 2024) showed that random evaluation places yield Shamir's secret sharing over the composite order field $\mathbb{F}_{p^d}$ that is statistically secure against physical-bit leakage. Later, Nguyen (EUROCRYPT 2025) established a dichotomy that linear code-based secret-sharing scheme over the field $\mathbb{F}_{p^d}$ is either statistically secure or completely insecure against such leakage. Building upon Nguyen's dichotomy, we present a partial derandomization of evaluation places, improving upon the Maji et al. result for a restricted regime of parameters. We replace the random choice of $n$ independent evaluation places by the iterates $x_j = Φ^j(x_0)$ of a simple fixed rational function $Φ$, where the initial point $x_0 \in \mathbb{F}_{p^d}^*$ is randomly chosen. The randomness in the evaluation places thus drops from $nd \log p$ bits to $d\log p$ bits. Our construction is valid for the regime $n = O(d/\log_p d)$, and any reconstruction threshold $k \ge 2$; in fact, the scheme attains perfect security (statistical distance exactly zero) against single-block leakage. Our technique is a partial fraction nondegeneracy argument that exploits the distinct poles of the rational iterates.

cs.CR↗

arXiv2026

Noisy-Syndrome Decoding of Hypergraph Product Codes

Hypergraph product codes are a prototypical family of quantum codes with state-of-the-art decodability properties. In this work we consider the "noisy" syndrome decoding problem and exact recovery problem for hypergraph product codes and show a reduction to the decoding and exact recovery of classical codes in the noisy syndrome setting. Our results hold for a broad class of codes admitting efficient syndrome decoding, including Sipser-Spielman codes and Reed-Solomon codes.

quant-ph↗

arXiv2025

List Recoverable Codes: The Good, the Bad, and the Unknown (hopefully not Ugly)

List recovery is a fundamental task for error-correcting codes, vastly generalizing unique decoding from worst-case errors and list decoding. Briefly, one is given ''soft information'' in the form of input lists S_1,...,S_n of bounded size, and one argues that there are not too many codewords that agree a lot with this soft information. This general problem appears in many guises, both within coding theory and in theoretical computer science more broadly. In this article we survey recent results on list recovery codes, introducing both the ''good'' (i.e., possibility results, showing that codes with certain list recoverability exist), the ''bad'' (impossibility results), and the ''unknown''. We additionally demonstrate that, while list recoverable codes were initially introduced as a component in list decoding concatenated codes, they have since found myriad applications to and connections with other topics in theoretical computer science.

cs.IT↗

arXiv2024

Efficient List-decoding of Polynomial Ideal Codes with Optimal List Size

In a recent breakthrough [BGM23, GZ23, AGL23], it was shown that randomly punctured Reed-Solomon codes are list decodable with optimal list size with high probability, i.e., they attain the Singleton bound for list decoding [ST20, Rot22, GST22]. We extend this result to the family of polynomial ideal codes, a large class of error-correcting codes which includes several well-studied families of codes such as Reed-Solomon, folded Reed-Solomon, and multiplicity codes. More specifically, similarly to the Reed-Solomon setting, we show that randomly punctured polynomial ideal codes over an exponentially large alphabet exactly achieve the Singleton bound for list-decoding; while such codes over a polynomially large alphabet approximately achieve it. Combining our results with the efficient list-decoding algorithm for a large subclass of polynomial ideal codes of [BHKS21], implies as a corollary that a large subclass of polynomial ideal codes (over random evaluation points) is efficiently list decodable with optimal list size. To the best of our knowledge, this gives the first family of codes that can be efficiently list decoded with optimal list size (for all list sizes), as well as the first family of linear codes of rate $R$ that can be efficiently list decoded up to a radius of $1 -R-ε$ with list size that is polynomial (and even linear) in $1/ε$. Our result applies to natural families of codes with algebraic structure such as folded Reed-Solomon or multiplicity codes (over random evaluation points). Our proof follows the general framework of [BGM23, GZ23, AGL23], but several new ingredients are needed. The main two new ingredients are a polynomial-ideal GM-MDS theorem (extending the algebraic GM-MDS theorem of [YH19, Lov21]), as well as a duality theorem for polynomial ideal codes, both of which may be of independent interest.

cs.IT↗

arXiv2024

Polynomials, Divided Differences, and Codes

Multivariate multiplicity codes (Kopparty, Saraf, and Yekhanin, J. ACM 2014) are linear codes where the codewords are described by evaluations of multivariate polynomials (with a degree bound) and their derivatives up to a fixed order, on a suitably chosen affine point set. While good list decoding algorithms for multivariate multiplicity codes were known in some special cases of point sets by a reduction to univariate multiplicity codes, a general list decoding algorithm up to the distance of the code when the point set is an arbitrary finite grid, was obtained only recently (Bhandari et al., IEEE TIT 2023). This required the characteristic of the field to be zero or larger than the degree bound, and this requirement is somewhat necessary, since list decoding this code up to distance with small output list size is not possible when the characteristic is significantly smaller than the degree. In this work, we present an alternate construction, based on divided differences, that closely resembles the classical multiplicity codes but is `insensitive to the field characteristic'. We obtain an efficient algorithm that list decodes this code up to distance, for arbitrary finite grids and over all finite fields. Notably, our construction can be interpreted as a `folded Reed-Muller code', which might be of independent interest. The upshot of our result is that a good `Taylor-like expansion' can be expressed in terms of a good `derivative-like operator' (a divided difference), and this implies that the corresponding code admits good algorithmic list decoding.

cs.IT↗

arXiv2024

Random Reed-Solomon Codes are List Recoverable with Optimal List Size

We prove that Reed-Solomon (RS) codes with random evaluation points are list recoverable up to capacity with optimal output list size, for any input list size. Namely, given an input list size $\ell$, a designated rate $R$, and any $\varepsilon > 0$, we show that a random RS code is list recoverable from $1-R-\varepsilon$ fraction of errors with output list size $L = O(\ell/\varepsilon)$, for field size $q=\exp(\ell,1/\varepsilon) \cdot n^2$. In particular, this shows that random RS codes are list recoverable beyond the "list recovery Johnson bound". Such a result was not even known for arbitrary random linear codes. Our technique follows and extends the recent line of work on list decoding of random RS codes, specifically the works of Brakensiek, Gopi, and Makam (STOC 2023), and of Guo and Zhang (FOCS 2023).

cs.IT↗

arXiv2023

On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

Alon and Füredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem.

math.CO↗

arXiv2023

Covering Symmetric Sets of the Boolean Cube by Affine Hyperplanes

Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least $n$ hyperplanes. We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for `polynomial covers' of `weight-determined' sets of `strictly unimodal uniform' (SU$^2$) grids. As a main tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) closures of symmetric sets of the Boolean cube -- the Z-closure of a symmetric set is symmetric. In fact, we obtain a characterization that concerns, more generally, weight-determined sets of SU$^2$ grids. However, in this generality, our characterization is not of the Z-closures -- unlike over the Boolean cube, the Z-closure of a weight-determined set need not be weight-determined. We introduce a new closure operator exclusively for weight-determined sets -- the `(finite-degree) Z*-closure' -- defined to be the maximal weight-determined set in the Z-closure. (This coincides with the Z-closure over the Boolean cube, for symmetric sets.) We obtain a combinatorial characterization of the finite-degree Z*-closures of weight-determined sets of an SU$^2$ grid. This characterization may also be of independent interest. Indeed, as further applications, we (i) give an alternate proof of a lemma by Alon et al. (IEEE Trans. Inform. Theory, 1988), and (ii) characterize the `certifying degrees' of weight-determined sets. Over the Boolean cube, our above characterization can also be derived using a result of Bernasconi and Egidi (Inf. Comput., 1999). However, our proof is independent of this result, works for all SU$^2$ grids, and could be regarded as being more combinatorial.

math.CO↗

arXiv2022

The Young matroid: A multiset extension of the Catalan matroid to arbitrary Young diagrams

Introduced by Ardila (J. Combin. Theory Ser. A, 2003), the Catalan matroid is obtained by defining the bases of the matroid using Dyck paths from $(0,0)$ to $(n,n)$. Further research has gone into the topic, with variants like lattice path matroids (introduced by Bonin, de Mier, and Noy (J. Combin. Theory Ser. A, 2003)) and shifted matroids (introduced independently by Klivans (2003), and Ardila) being studied intensively. In this short note, we introduce the Young matroid, an extension of the Catalan matroid, where the bases are defined using the standard Young tableaux of a fixed shape. This extension necessarily involves the consideration of independent multisets and multiset bases.

math.CO↗

arXiv2021

On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic

The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass of symmetric sets (subsets that are invariant under permutations of coordinates) of the Boolean cube, in positive characteristic. Our results subsume multiple statements on finite-degree Z-closures that have found applications in extremal combinatorial problems, for instance, pertaining to set systems (Hegedűs, Stud. Sci. Math. Hung. 2010; Hegedűs, arXiv 2021), and Boolean circuits (Hrǔbes et al., ICALP 2019). Our characterization also establishes that for the subclasses of symmetric sets that we consider, the finite-degree Z-closures have low computational complexity. A key ingredient in our characterization is a new variant of finite-degree Z-closures, defined using vanishing conditions on only symmetric polynomials satisfying a degree bound.

math.CO↗

arXiv2021

On the Probabilistic Degree of an $n$-variate Boolean Function

Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log n - O(\log \log n)$. This was improved to a tight $(\log n - O(1))$ bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for \emph{Probabilistic degree}. The function $f$ has probabilistic degree at most $d$ if there is a random real-valued polynomial of degree at most $d$ that agrees with $f$ at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between $(\log n)^{1/2-o(1)}$ and $O(\log n)$ (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of $n$-variate functions $f$, \emph{modulo} our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is $(\log n)^c$, then the minimum possible probabilistic degree of such an $f$ is at least $(\log n)^{c/(c+1)-o(1)}$, and we show this is tight up to $(\log n)^{o(1)}$ factors.

cs.CC↗

arXiv2019

On the Probabilistic Degrees of Symmetric Boolean functions

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions --- specifically symmetric Boolean functions --- have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems. In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

cs.CC↗

arXiv2019

Decoding Downset codes over a finite grid

In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve the unique decoding problem for the general family of Downset codes. Here, a downset code is specified by a family D of monomials closed under taking factors: the corresponding code is the space of evaluations of all polynomials that can be written as linear combinations of monomials from D.

cs.CC↗

arXiv2019

A Fixed-Depth Size-Hierarchy Theorem for AC$^0[\oplus]$ via the Coin Problem

We prove the first Fixed-depth Size-hierarchy Theorem for uniform AC$^0[\oplus]$ circuits; in particular, for fixed $d$, the class $\mathcal{C}_{d,k}$ of uniform AC$^0[\oplus]$ formulas of depth $d$ and size $n^k$ form an infinite hierarchy. For this, we find the first class of explicit functions giving (up to polynomial factor) matching upper and lower bounds for AC$^0[\oplus]$ formulas, derived from the $δ$-Coin Problem, the computational problem of distinguishing between coins that are heads with probability $(1+δ)/2$ or $(1-δ)/2,$ where $δ$ is a parameter going to $0$. We study this problem's complexity and make progress on both upper bounds and lower bounds. Upper bounds. We find explicit monotone AC$^0$ formulas solving the $δ$-coin problem, having depth $d$, size $\exp(O(d(1/δ)^{1/(d-1)}))$, and sample complexity poly$(1/δ)$, for constant $d\ge2$. This matches previous upper bounds of O'Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009) in terms of size and improves the sample complexity. Lower bounds. The upper bounds are nearly tight even for the stronger model of AC$^0[\oplus]$ formulas (which allow NOT and Parity gates): any AC$^0[\oplus]$ formula solving the $δ$-coin problem must have size $\exp(Ω(d(1/δ)^{1/(d-1)})).$ This strengthens a result of Cohen, Ganor and Raz (APPROX-RANDOM 2014), who prove a similar result for AC$^0$, and a result of Shaltiel and Viola (SICOMP 2010), who give a superpolynomially weaker (still exponential) lower bound. The upper bound is a derandomization involving a use of Janson's inequality (as far as we know, the first such use of the inequality) and classical combinatorial designs. For the lower bound, we prove an optimal (up to constant factor) degree lower bound for multivariate polynomials over $\mathbb{F}_2$ solving the $δ$-coin problem, which may be of independent interest.

cs.CC↗