arXiv subjects

Ameera Chowdhury

Publications and source records attributed to Ameera Chowdhury.

5 recordsLinked to original sources

arXiv2021

Improved Schemes for Asymptotically Optimal Repair of MDS Codes

We consider $(n,k,l)$ MDS codes of length $n$, dimension $k$, and subpacketization $l$ over a finite field $\mathbb{F}$. A codeword of such a code consists of $n$ column-vectors of length $l$ over $\mathbb{F}$, with the property that any $k$ of them suffice to recover the entire codeword. Each of these $n$ vectors may be stored on a separate node in a network. If one of the $n$ nodes fails, we can recover its content by downloading symbols from the surviving nodes, and the total number of symbols downloaded in the worst case is called the repair bandwidth of the code. By the cut-set bound, the repair bandwidth of an $(n,k,l)$ MDS code is at least $l(n{-}1)/(n{-}k)$. There are several constructions of MDS codes whose repair bandwidth meets or asymptotically meets the cut-set bound. For example, Ye and Barg constructed $(n,k,r^{n})$ Reed--Solomon codes that asymptotically meet the cut-set bound, where $r = n-k$. Ye and Barg also constructed optimal-bandwidth and optimal-update $(n,k,r^{n})$ MDS codes. Wang, Tamo, and Bruck constructed optimal-bandwidth $(n, k, r^{n/(r+1)})$ MDS codes, and these codes have the smallest known subpacketization for optimal-bandwidth MDS codes. A key idea in all these constructions is to represent certain integers in base $r$. We show how this technique can be refined to improve the subpacketization of the two MDS code constructions by Ye and Barg, while achieving asymptotically optimal repair bandwidth. Specifically, when $r=s^{m}$ for an integer $s$,we obtain an $(n,k,s^{m+n-1})$ Reed--Solomon code and an optimal-update $(n,k,s^{m+n-1})$ MDS code, both having asymptotically optimal repair bandwidth. We also present an extension of this idea to reduce the subpacketization of the Wang--Tamo--Bruck construction while achieving a repair-by-transfer scheme with asymptotically optimal repair bandwidth.

cs.IT↗

arXiv2016

Inclusion Matrices and the MDS Conjecture

Let F_q be a finite field of order q with characteristic p. An arc is an ordered family of at least k vectors in (F_q)^k in which every subfamily of size k is a basis of (F_q)^k. The MDS conjecture, which was posed by Segre in 1955, states that if k <= q, then an arc in (F_q)^k has size at most q+1, unless q is even and k=3 or k=q-1, in which case it has size at most q+2. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of k when q is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when k <= p, and if q is not prime, for k <= 2p-2. To accomplish this, given an arc G of (F_q)^k and a nonnegative integer n, we construct a matrix M_G^{\uparrow n}, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix M_G^{\uparrow n} to properties of the arc G and may provide new tools in the computational classification of large arcs.

math.CO↗

arXiv2014

The Manickam-Miklós-Singhi Conjectures for Sets and Vector Spaces

More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. We verify this conjecture when $n \geq 8k^{2}$, which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when $k < 10^{45}$. Moreover, our arguments resolve the vector space analogue of this conjecture. Let $V$ be an $n$-dimensional vector space over a finite field. Assign a real-valued weight to each $1$-dimensional subspace in $V$ so that the sum of all weights is zero. Define the weight of a subspace $S \subset V$ to be the sum of the weights of all the $1$-dimensional subspaces it contains. We prove that if $n \geq 3k$, then the number of $k$-dimensional subspaces in $V$ with nonnegative weight is at least the number of $k$-dimensional subspaces in $V$ that contain a fixed $1$-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

math.CO↗

arXiv2014

A New Quadratic Bound for the Manickam-Miklós-Singhi Conjecture

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. We verify this conjecture when $n \geq 8k^2$, which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when $k < 10^{45}$.

math.CO↗

arXiv2005

Colouring Lines in Projective Space

Let $V$ be a vector space of dimension $v$ over a field of order $q$. The $q$-Kneser graph has the $k$-dimensional subspaces of $V$ as its vertices, where two subspaces $α$ and $β$ are adjacent if and only if $α\capβ$ is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when $k=1$ (and the graphs are complete) or when $v<2k$ (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case $k=2$, we show that the chromatic number is $q^2+q$ when $v=4$ and $(q^{v-1}-1)/(q-1)$ when $v > 4$. In both cases we characterise the minimal colourings.

math.CO↗