arXiv subjects

Manik Dhar

Publications and source records attributed to Manik Dhar.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Fair Resolute Voting Rules

Among two-candidate elections that treat the candidates symmetrically and never result in a tie, which voting rules are fair? A natural requirement is that each voter exerts an equal influence over the outcome, i.e., is equally likely to swing the election one way or the other. A voter's influence has been formalized in two canonical ways: the Shapley-Shubik (1954) index and the Banzhaf (1964) index. We consider both indices, and ask: Which electorate sizes admit a fair voting rule (under the respective index)? For an odd number $n$ of voters, simple majority rule is an example of a fair voting rule. However, when $n$ is even, fair voting rules can be challenging to identify, and a diverse literature has studied this problem under different notions of fairness. Our main results completely characterize which values of $n$ admit fair voting rules under the two canonical indices we consider. For the Shapley-Shubik index, a fair voting rule exists for $n>1$ if and only if $n$ is not a power of $2$. For the Banzhaf index, a fair voting rule exists for all $n$ except $2$, $4$, and $8$. Along the way, we show how the Shapley-Shubik and Banzhaf indices relate to the winning coalitions of the voting rule, and compare these indices to previously considered notions of fairness.

econ.TH↗

arXiv2025

Combinatorial Bounds for List Recovery via Discrete Brascamp--Lieb Inequalities

In coding theory, the problem of list recovery asks one to find all codewords $c$ of a given code $C$ which such that at least $1-\rho$ fraction of the symbols of $c$ lie in some predetermined set of $\ell$ symbols for each coordinate of the code. A key question is bounding the maximum possible list size $L$ of such codewords for the given code $C$. In this paper, we give novel combinatorial bounds on the list recoverability of various families of linear and folded linear codes, including random linear codes, random Reed--Solomon codes, explicit folded Reed--Solomon codes, and explicit univariate multiplicity codes. Our main result is that in all of these settings, we show that for code of rate $R$, when $\rho = 1 - R - \epsilon$ approaches capacity, the list size $L$ is at most $(\ell/(R+\epsilon))^{O(R/\epsilon)}$. These results also apply in the average-radius regime. Our result resolves a long-standing open question on whether $L$ can be bounded by a polynomial in $\ell$. In the zero-error regime, our bound on $L$ perfectly matches known lower bounds. The primary technique is a novel application of a discrete entropic Brascamp--Lieb inequality to the problem of list recovery, allowing us to relate the local structure of each coordinate with the global structure of the recovered list. As a result of independent interest, we show that a recent result by Chen and Zhang (STOC 2025) on the list decodability of folded Reed--Solomon codes can be generalized into a novel Brascamp--Lieb type inequality.

cs.IT↗

arXiv2025

From Random to Explicit via Subspace Designs With Applications to Local Properties and Matroids

In coding theory, a common question is to understand the threshold rates of various local properties of codes, such as their list decodability and list recoverability. A recent work Levi, Mosheiff, and Shagrithaya (FOCS 2025) gave a novel unified framework for calculating the threshold rates of local properties for random linear and random Reed--Solomon codes. In this paper, we extend their framework to studying the local properties of subspace designable codes, including explicit folded Reed-Solomon and univariate multiplicity codes. Our first main result is a local equivalence between random linear codes and (nearly) optimal subspace design codes up to an arbitrarily small rate decrease. We show any local property of random linear codes applies to all subspace design codes. As such, we give the first explicit construction of folded linear codes that simultaneously attain all local properties of random linear codes. Conversely, we show that any local property which applies to all subspace design codes also applies to random linear codes. Our second main result is an application to matroid theory. We show that the correctable erasure patterns in a maximally recoverable tensor code can be identified in deterministic polynomial time, assuming a positive answer to a matroid-theoretic question due to Mason (1981). This improves on a result of Jackson and Tanigawa (JCTB 2024) who gave a complexity characterization of $\mathsf{RP} \cap \mathsf{coNP}$ assuming a stronger conjecture. Our result also applies to the generic bipartite rigidity and matrix completion matroids. As a result of additional interest, we study the existence and limitations of subspace designs. In particular, we tighten the analysis of family of subspace designs constructioned by Guruswami and Kopparty (Combinatorica 2016) and show that better subspace designs do not exist over algebraically closed fields.

cs.IT↗

arXiv2025

Iterated sumset expansion in $\mathbb{F}_p^n$

Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such expansion results are only possible when $|A|>\frac{1}{p}|\mathbb{F}_p^n|$. However, heuristic considerations suggest that expansion should begin with much smaller sets under just mild ``nondegeneracy'' conditions. In this paper, we confirm this intuition by showing a sufficient algebraic condition for the asymmetric version of this problem: We have $A_1+\dots+A_m=\mathbb{F}_p^n$ as long as each $A_i$ is not contained in the zero set of any low degree polynomial ($\text{deg} = O(n)$ when $m=O(p)$). We close with a discussion of the behavior of random sets, as well as extensions of these results and connections with the Erd\H{o}s-Ginzburg-Ziv problem. Our proofs make use of the shift operator polynomial method developed by the second author.

math.CO↗

arXiv2025

Improved Constructions and Lower Bounds for Maximally Recoverable Grid Codes

In this paper, we continue the study of Maximally Recoverable (MR) Grid Codes initiated by Gopalan et al. [SODA 2017]. More precisely, we study codes over an $m \times n$ grid topology with one parity check per row and column of the grid along with $h \ge 1$ global parity checks. Previous works have largely focused on the setting in which $m = n$, where explicit constructions require field size which is exponential in $n$. Motivated by practical applications, we consider the regime in which $m,h$ are constants and $n$ is growing. In this setting, we provide a number of new explicit constructions whose field size is polynomial in $n$. We further complement these results with new field size lower bounds.

cs.IT↗

arXiv2025

Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors

Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed--Solomon (IRS), folded Reed--Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed--Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher--Kiayias--Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed--Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami--Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.

cs.IT↗

arXiv2025

Covering points with planes

Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound on $|S|$, which is tight in some cases, for example when all of the planes have the same dimension. We produce an example showing that this upper bound does not hold for point sets whose proper subsets are covered by lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$ with $k\geq 2$, and prove an upper bound in this case. We also investigate the analogous problem for general matroids.

math.CO↗

arXiv2024

Spread Furstenberg Sets

We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal G(n,k)$ such that for all $P\in \mathcal P$, there exists a translation vector $a_P \in \mathbb{R}^n$ such that $\dim(F\cap (P + a_P)) \geq s$. We show that given $k \geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;k)$-spread Furstenberg set $F$ in $\mathbb{R}^n$ satisfies \[ \dim F \geq n-k + s - \frac{k(n-k) - t}{\lceil s\rceil - k_0 +1 }. \] Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields.

math.CA↗

arXiv2024

Rigidity matroids and linear algebraic matroids with applications to matrix completion and tensor codes

We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an (m, n, a=2, b=2) maximally recoverable tensor code.

math.CO↗

arXiv2023

$(n,k)$-Besicovitch sets do not exist in $\mathbb{Z}_p^n$ and $\hat{\mathbb{Z}}^n$ for $k\ge 2$

Besicovitch showed that a compact set in $\mathbb{R}^n$ which contains a unit line segment in every direction can have measure $0$. These constructions also work over other metric spaces like the $p$-adics and profinite integers. It is conjectured that it is impossible to construct sets with measure $0$ which contain a unit $2$-disk in every direction in $\mathbb{R}^n$. We prove that over the $p$-adics and profinite integers any set which contains a 2-flat in every direction must have positive measure. The main ingredients are maximal Kakeya estimates for $(\mathbb{Z}/N\mathbb{Z})^n$ proven in [Dha22] and adapting Fourier analytic arguments of Oberlin [Obe05]. In general, we prove $L^{n-1}$ to $L^{n-1}$ estimates for the maximal operator corresponding to $2$-flats.

math.CA↗

arXiv2023

Generalized GM-MDS: Polynomial Codes are Higher Order MDS

The GM-MDS theorem, conjectured by Dau-Song-Dong-Yuen and proved by Lovett and Yildiz-Hassibi, shows that the generator matrices of Reed-Solomon codes can attain every possible configuration of zeros for an MDS code. The recently emerging theory of higher order MDS codes has connected the GM-MDS theorem to other important properties of Reed-Solomon codes, including showing that Reed-Solomon codes can achieve list decoding capacity, even over fields of size linear in the message length. A few works have extended the GM-MDS theorem to other families of codes, including Gabidulin and skew polynomial codes. In this paper, we generalize all these previous results by showing that the GM-MDS theorem applies to any polynomial code, i.e., a code where the columns of the generator matrix are obtained by evaluating linearly independent polynomials at different points. We also show that the GM-MDS theorem applies to dual codes of such polynomial codes, which is non-trivial since the dual of a polynomial code may not be a polynomial code. More generally, we show that GM-MDS theorem also holds for algebraic codes (and their duals) where columns of the generator matrix are chosen to be points on some irreducible variety which is not contained in a hyperplane through the origin. Our generalization has applications to constructing capacity-achieving list-decodable codes as shown in a follow-up work by Brakensiek-Dhar-Gopi-Zhang, where it is proved that randomly punctured algebraic-geometric (AG) codes achieve list-decoding capacity over constant-sized fields.

cs.IT↗

arXiv2023

AG Codes Achieve List-decoding Capacity over Constant-sized Fields

The recently-emerging field of higher order MDS codes has sought to unify a number of concepts in coding theory. Such areas captured by higher order MDS codes include maximally recoverable (MR) tensor codes, codes with optimal list-decoding guarantees, and codes with constrained generator matrices (as in the GM-MDS theorem). By proving these equivalences, Brakensiek-Gopi-Makam showed the existence of optimally list-decodable Reed-Solomon codes over exponential sized fields. Building on this, recent breakthroughs by Guo-Zhang and Alrabiah-Guruswami-Li have shown that randomly punctured Reed-Solomon codes achieve list-decoding capacity (which is a relaxation of optimal list-decodability) over linear size fields. We extend these works by developing a formal theory of relaxed higher order MDS codes. In particular, we show that there are two inequivalent relaxations which we call lower and upper relaxations. The lower relaxation is equivalent to relaxed optimal list-decodable codes and the upper relaxation is equivalent to relaxed MR tensor codes with a single parity check per column. We then generalize the techniques of GZ and AGL to show that both these relaxations can be constructed over constant size fields by randomly puncturing suitable algebraic-geometric codes. For this, we crucially use the generalized GM-MDS theorem for polynomial codes recently proved by Brakensiek-Dhar-Gopi. We obtain the following corollaries from our main result. First, randomly punctured AG codes of rate $R$ achieve list-decoding capacity with list size $O(1/\epsilon)$ and field size $\exp(O(1/\epsilon^2))$. Prior to this work, AG codes were not even known to achieve list-decoding capacity. Second, by randomly puncturing AG codes, we can construct relaxed MR tensor codes with a single parity check per column over constant-sized fields, whereas (non-relaxed) MR tensor codes require exponential field size.

cs.IT↗

arXiv2023

A construction of Maximally Recoverable LRCs for small number of local groups

Maximally Recoverable Local Reconstruction Codes (LRCs) are codes designed for distributed storage to provide maximum resilience to failures for a given amount of storage redundancy and locality. An $(n,r,h,a,g)$-MR LRC has $n$ coordinates divided into $g$ local groups of size $r=n/g$, where each local group has `$a$' local parity checks and there are an additional `$h$' global parity checks. Such a code can correct `$a$' erasures in each local group and any $h$ additional erasures. Constructions of MR LRCs over small fields is desirable since field size determines the encoding and decoding efficiency in practice. In this work, we give a new construction of $(n,r,h,a,g)$-MR-LRCs over fields of size $q=O(n)^{h+(g-1)a-\lceil h/g\rceil}$ which generalizes a construction of Hu and Yekhanin (ISIT 2016). This improves upon state of the art when there are a small number of local groups, which is true in practical deployments of MR LRCs.

cs.IT↗

arXiv2023

Furstenberg sets in finite fields: Explaining and improving the Ellenberg-Erman proof

A $(k,m)$-Furstenberg set is a subset $S \subset \mathbb{F}_q^n$ with the property that each $k$-dimensional subspace of $\mathbb{F}_q^n$ can be translated so that it intersects $S$ in at least $m$ points. Ellenberg and Erman proved that $(k,m)$-Furstenberg sets must have size at least $C_{n,k}m^{n/k}$, where $C_{n,k}$ is a constant depending only $n$ and $k$. In this paper, we adopt the same proof strategy as Ellenberg and Erman, but use more elementary techniques than their scheme-theoretic method. By modifying certain parts of the argument, we obtain an improved bound on $C_{n,k}$, and our improved bound is nearly optimal for an algebraic generalization the main combinatorial result. We also extend our analysis to give lower bounds for sets that have large intersection with shifts of a specific family of higher-degree co-dimension $n-k$ varieties, instead of just co-dimension $n-k$ subspaces.

math.CO↗

arXiv2022

Improved Field Size Bounds for Higher Order MDS Codes

Higher order MDS codes are an interesting generalization of MDS codes recently introduced by Brakensiek, Gopi and Makam (IEEE Trans. Inf. Theory 2022). In later works, they were shown to be intimately connected to optimally list-decodable codes and maximally recoverable tensor codes. Therefore (explicit) constructions of higher order MDS codes over small fields is an important open problem. Higher order MDS codes are denoted by $\operatorname{MDS}(\ell)$ where $\ell$ denotes the order of generality, $\operatorname{MDS}(2)$ codes are equivalent to the usual MDS codes. The best prior lower bound on the field size of an $(n,k)$-$\operatorname{MDS}(\ell)$ codes is $\Omega_\ell(n^{\ell-1})$, whereas the best known (non-explicit) upper bound is $O_\ell(n^{k(\ell-1)})$ which is exponential in the dimension. In this work, we nearly close this exponential gap between upper and lower bounds. We show that an $(n,k)$-$\operatorname{MDS}(3)$ codes requires a field of size $\Omega_k(n^{k-1})$, which is close to the known upper bound. Using the connection between higher order MDS codes and optimally list-decodable codes, we show that even for a list size of 2, a code which meets the optimal list-decoding Singleton bound requires exponential field size; this resolves an open question from Shangguan and Tamo (STOC 2020 / SIAM J. on Computing 2023). We also give explicit constructions of $(n,k)$-$\operatorname{MDS}(\ell)$ code over fields of size $n^{(\ell k)^{O(\ell k)}}$. The smallest non-trivial case where we still do not have optimal constructions is $(n,3)$-$\operatorname{MDS}(3)$. In this case, the known lower bound on the field size is $\Omega(n^2)$ and the best known upper bounds are $O(n^5)$ for a non-explicit construction and $O(n^{32})$ for an explicit construction. In this paper, we give an explicit construction over fields of size $O(n^3)$ which comes very close to being optimal.

cs.IT↗

arXiv2022

Maximal and $(m,ε)$-Kakeya bounds over $\mathbb{Z}/N\mathbb{Z}$ for general $N$

We derive Maximal Kakeya estimates for functions over $\mathbb{Z}/N\mathbb{Z}$ proving the Maximal Kakeya conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. The proof involves using polynomial method and linear algebra techniques from [Dha21, Ars21a, DD21] and generalizing a probabilistic method argument from [DD22]. As another application we give lower bounds for the size of $(m,ε)$-Kakeya sets over $\mathbb{Z}/N\mathbb{Z}$. Using these ideas we also give a new, simpler, and direct proof for Maximal Kakeya bounds over finite fields (which were first proven in [EOT10]) with almost sharp constants.

math.CO↗

arXiv2022

Linear Hashing with $\ell_\infty$ guarantees and two-sided Kakeya bounds

We show that a randomly chosen linear map over a finite field gives a good hash function in the $\ell_\infty$ sense. More concretely, consider a set $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to \mathbb{F}_q^t$ with $q^t$ taken to be sufficiently smaller than $ |S|$. Let $U_S$ denote a random variable distributed uniformly on $S$. Our main theorem shows that, with high probability over the choice of $L$, the random variable $L(U_S)$ is close to uniform in the $\ell_\infty$ norm. In other words, {\em every} element in the range $\mathbb{F}_q^t$ has about the same number of elements in $S$ mapped to it. This complements the widely-used Leftover Hash Lemma (LHL) which proves the analog statement under the statistical, or $\ell_1$, distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [ADMPT99]. By known bounds from the load balancing literature [RS98], our results are tight and show that linear functions hash as well as trully random function up to a constant factor in the entropy loss. Our proof leverages a connection between linear hashing and the finite field Kakeya problem and extends some of the tools developed in this area, in particular the polynomial method.

math.CO↗

arXiv2021

The Kakeya Set Conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$

We prove the Kakeya set conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. This entails extending and combining the techniques of Arsovski [Ars21a] for $N=p^k$ and the author and Dvir [DD21] for the case of square-free $N$. We also prove stronger lower bounds for the size of $(m,\epsilon)$-Kakeya sets over $\mathbb{Z}/p^k\mathbb{Z}$ by extending the techniques of [Ars21a] using multiplicities as was done in [SS08, DKSS13]. In addition, we show our bounds are almost sharp by providing a new construction for Kakeya sets over $\mathbb{Z}/p^k\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$.

math.CO↗