arXiv subjects

Rati Ludhani

Publications and source records attributed to Rati Ludhani.

6 recordsLinked to original sources

arXiv2026

Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes

We study the classification of minimal codewords of projective Reed-Muller codes of order $2$. This problem is equivalent to identifying quadrics over finite fields whose set of rational points is maximal with respect to the inclusion. We prove that except one particular case over $\mathbb{F}_2$, any two absolutely irreducible quadrics whose sets of rational points are contained within one another should be equal as projective varieties. We deduce a precise characterisation of the minimal codewords of projective Reed-Muller codes of order $2$ and further give their exact number for each possible weight.

cs.IT↗

arXiv2025

The Random Variables of the DNA Coverage Depth Problem

DNA data storage systems encode digital data into DNA strands, enabling dense and durable storage. Efficient data retrieval depends on coverage depth, a key performance metric. We study the random access coverage depth problem and focus on minimizing the expected number of reads needed to recover information strands encoded via a linear code. We compute the asymptotic performance of a recently proposed code construction, establishing and refining a conjecture in the field by giving two independent proofs. We also analyze a geometric code construction based on balanced quasi-arcs and optimize its parameters. Finally, we investigate the full distribution of the random variables that arise in the coverage depth problem, of which the traditionally studied expectation is just the first moment. This allows us to distinguish between code constructions that, at first glance, may appear to behave identically.

cs.IT↗

arXiv2024

Projective Nullstellensatz for not necessarily algebraically closed fields

The Nullstellensatz, proved by Hilbert in 1893, is a classical result that holds when the base field is algebraically closed. When the base field is finite, a version of Hilbert's Nullstellensatz is given by Terjanian in 1966. Laksov in 1987 generalized Hilbert's Nullstellensatz to a $K$-Nullstellensatz when the base field $K$ is not necessarily algebraically closed. However, unlike Tarjanian's Nullstellensatz, Laksov's Nullstellensatz is not very explicit. Later, Laksov and Westin in 1990 proposed a strengthening to Laksov's Nullstellensatz in the form of four conjectures. A projective analogue of Nullstellensatz of the classical Nullstellensatz of Hilbert is well-known for projective varieties over algebraically closed fields. For finite fields, the projective analogue of the Nullstellensatz can be derived as an application of Hilbert's Nullstellensatz, though it is not as efficient as Terjanian's Nullstellensatz. Gimenez, Ruano and San-Jos\'e in 2023 strengthened this result by identifying a computationally efficient set. Here, we introduce an even more efficient set that establishes the projective analogue of the Nullstellensatz for finite fields. Additionally, we provide counterexamples to three of the conjectures of Laksov and Westin.

math.AC↗

arXiv2024

Betti Numbers and Higher Weight Spectra of Reed-Muller Codes $RM_q(2,2)$

We determine all the Betti numbers of the $q$-ary second order Reed-Muller codes of length $q^2$, and also of the elongations of matroids associated to these codes. We then use it to determine the higher weight spectra of these codes. As a special case, we recover some results of Kaplan and Matei about counting certain curves over finite fields with prescribed rational intersection points. In geometric terms, our results relate to the affine Veronesean by which we mean the image of the affine plane $\mathbb{A}^2$ under the quadratic Veronese embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$. Indeed, finding the higher weight spectra, of the Reed-Muller code considered here, corresponds to determining the number of $\mathbb{F}_q$-rational points on all possible sections of this affine Veronesean by linear subvarieties of $\mathbb{P}^5$.

math.CO↗

arXiv2023

On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes

We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is then used to determine the number of minimum weight codewords of a projective Reed-Muller code. Various formulas for the dimension of a projective Reed-Muller code, and their equivalences are also discussed.

cs.IT↗

arXiv2021

On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes

Following Johnsen and Verdure (2013), we can associate to any linear code $C$ an abstract simplicial complex and in turn, a Stanley-Reisner ring $R_C$. The ring $R_C$ is a standard graded algebra over a field and its projective dimension is precisely the dimension of $C$. Thus $R_C$ admits a graded minimal free resolution and the resulting graded Betti numbers are known to determine the generalized Hamming weights of $C$. The question of purity of the minimal free resolution of $R_C$ was considered by Ghorpade and Singh (2020) when $C$ is the generalized Reed-Muller code. They showed that the resolution is pure in some cases and it is not pure in many other cases. Here we give a complete characterization of the purity of graded minimal free resolutions of Stanley-Reisner rings associated to generalized Reed-Muller codes of an arbitrary order.

math.AC↗