Searcharxiv⌕ Search

arXiv subjects

Madhu Raka

Publications and source records attributed to Madhu Raka.

15 recordsLinked to original sources
arXiv2024

Positive values of non-homogeneous quadratic forms of type (1,4): A conjecture of Bambah, Dumir and Hans-Gill

Let $Q(x_1, \cdots,x_n)$ be a real indefinite quadratic form of the type $(r,s)$, $n=r+s$, signature $σ=r-s$ and determinant $D\neq 0$. Let $Γ_{r,n-r}$ denote the infimum of all numbers $Γ$ such that for any real numbers $c_1, c_2 ,\cdots, c_n$ there exist integers $x_1, x_ 2,\cdots, x_n$ satisfying $$0< Q(x_1+c_1,x_2+c_2,\cdots,x_n+c_n)\leq (Γ|D|)^{1/n}.$$ All the values of $Γ_{r,n-r}$ are known except for $Γ_{1,4}$. Earlier it was shown that $8\leq Γ_{1,4}<12$. It is conjectured that $Γ_{1,4}=8$. Here we shall prove that $Γ_{1,4}=8$, when (i) $c_2 \not \equiv 0 \pmod 1$, (ii) $c_2 \equiv 0 \pmod 1$, $a\geq \frac{1}{2}$, where $a$ is minima of positive definite ternary quadratic forms with determinant $4|D|$, and (iii) in some cases of $c_2 \equiv 0 \pmod 1$, $a< \frac{1}{2}$. We also obtain six critical forms for which the constant 8 is attained. In the remaining cases we prove that $Γ_{1,4}< \frac{32}{3}$.

math.NT↗
arXiv2022

New Quantum codes from constacyclic codes over a general non-chain ring

Let $q$ be a prime power and let $\mathcal{R}=\mathbb{F}_{q}[u_1,u_2, \cdots, u_k]/\langle f_i(u_i),u_iu_j-u_ju_i\rangle$ be a finite non-chain ring, where $f_i(u_i), 1\leq i \leq k$ are polynomials, not all linear, which split into distinct linear factors over $\mathbb{F}_{q}$. We characterize constacyclic codes over the ring $\mathcal{R}$ and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials $f_i(u_i),$ $1 \leq i \leq k$ is irrelevant while constructing quantum codes from constacyclic codes over $\mathcal{R}$, it depends only on their degrees. It is shown that there always exists Quantum MDS code $[[n,n-2,2]]_q$ for any $n$ with $\gcd (n,q)\neq 1.$

cs.IT↗
arXiv2022

Multi-dimensional Constacyclic Codes of Arbitrary Length over Finite Fields

Multi-dimensional cyclic code is a natural generalization of cyclic code. In an earlier paper we explored two-dimensional constacyclic codes over finite fields. Following the same technique, here we characterize the algebraic structure of multi-dimensional constacyclic codes, in particular three-dimensional $(α,β,γ)$- constacyclic codes of arbitrary length $s\ell k$ and their duals over a finite field $\mathbb{F}_q$, where $α,β,γ$ are non zero elements of $\mathbb{F}_q$. We give necessary and sufficient conditions for a three-dimensional $(α,β,γ)$- constacyclic code to be self-dual.

cs.IT↗
arXiv2020

On conjectures of Minkowski and Woods for $n=10$

Let $\mathbb{L}$ be a lattice in $n$-dimensional Euclidean space $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff and having a basis of the form $~(A_1,0,0,\cdots$ $,0),$ ~$(a_{2,1},A_2,0,\cdots,0),\cdots,$ $(a_{n,1},a_{n,2},\cdots,a_{n,n-1},A_n)$. A famous conjecture of Woods in Geometry of Numbers asserts that if $A_1A_2\cdots A_n = 1$ and $A_i\leq A_1$ for each $i$ then any closed sphere in $\mathbb{R}^n$ of radius $\sqrt{n/4}$ contains a point of $\mathbb{L}.$ Together with a result of C. T. McMullen (2005), the truth of Woods' Conjecture for a fixed $n$, implies the long standing classical conjecture of Minkowski on product of $n$ non-homogeneous linear forms for that value of $n$. In an earlier paper `Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, 2016, 501-548' we proved Woods' Conjecture for $n=9$. In this paper, we prove Woods' Conjecture and hence Minkowski's Conjecture for $n=10$.

math.NT↗
arXiv2020

Two Dimensional $\left( α,β\right) $-Constacyclic Codes of arbitrary length over a Finite Field

In this paper we characterize the algebraic structure of two-dimensional $(α,β)$-constacyclic codes of arbitrary length $s.\ell$ and of their duals. For $α,β\in \{1,-1\}$, we give necessary and sufficient conditions for a two-dimensional $(α,β)$-constacyclic code to be self-dual. We also show that a two-dimensional $(α,1 )$-constacyclic code $\mathcal{C}$ of length $n=s.\ell$ can not be self-dual if $\gcd(s,q)= 1$. Finally, we give some examples of self-dual, isodual, MDS and quasi-twisted codes corresponding to two-dimensional $(α,β)$-constacyclic codes.

cs.IT↗
arXiv2019

Some Generalizations of Good Integers and Their Applications in the Study of Self-Dual Negacyclic Codes

Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of $2^β$-good integers, $2^β$-oddly-good integers, and $2^β$-evenly-good integers which are generalizations of good integers. Properties of such integers have been given as well as their applications in characterizing and enumerating self-dual negacyclic codes over finite fields. An alternative proof for the characterization of the existence of a self-dual negacyclic code over finite fields has been given in terms of such generalized good integers. A general enumeration formula for the number of self-dual negacyclic codes of length $n$ over finite fields has been established. For some specific lengths, explicit formulas have been provided as well. Some known results on self-dual negacyclic codes over finite fields can be formalized and viewed as special cases of this work.

cs.IT↗
arXiv2019

Skew constacyclic codes over a non-chain ring $\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle$

Let $f(u)$ and $g(v)$ be two polynomials of degree $k$ and $\ell$ respectively, not both linear, which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),\\uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study $ψ$-skew cyclic and $θ_t$-skew constacyclic codes over the ring $\mathcal{R}$ where $ψ$ and $θ_t$ are two automorphisms defined on $\mathcal{R}$.

cs.IT↗
arXiv2018

Polyadic cyclic codes over a non-chain ring $\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle$

Let $f(u)$ and $g(v)$ be any two polynomials of degree $k$ and $\ell$ respectively ($k$ and $\ell$ are not both $1$), which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring $\mathcal{R}$. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from $\mathcal{R}^n \rightarrow \mathbb{F}^{k\ell n}_q$ which preserves duality. The Gray images of polyadic codes and their extensions over the ring $\mathcal{R}$ lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over $\mathbb{F}_q$. Some examples are also given to illustrate this.

cs.IT↗
arXiv2018

Duadic negacyclic codes over a finite non-chain ring and their Gray images

Let $f(u)$ be a polynomial of degree $m, m \geq 2,$ which splits into distinct linear factors over a finite field $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u]/\langle f(u)\rangle$ be a finite non-chain ring. In an earlier paper, we studied duadic and triadic codes over $\mathcal{R}$ and their Gray images. Here, we study duadic negacyclic codes of Type I and Type II over the ring $\mathcal{R}$, their extensions and their Gray images. As a consequence some self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over $\mathbb{F}_q$ are constructed. Some examples are also given to illustrate this.

cs.IT↗
arXiv2016

Quadratic residue codes over the ring $\mathbb{F}_{p}[u]/\langle u^m-u\rangle$ and their Gray images

Let $m\geq 2$ be any natural number and let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+\cdots+u^{m-1}\mathbb{F}_{p}$ be a finite non-chain ring, where $u^m=u$ and $p$ is a prime congruent to $1$ modulo $(m-1)$. In this paper we study quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^m$ is defined which preserves self duality of linear codes. As a consequence self dual, formally self dual and self orthogonal codes are constructed. To illustrate this several examples of self-dual, self orthogonal and formally self-dual codes are given. Among others a [9,3,6] linear code over $\mathbb{F}_{7}$ is constructed which is self-orthogonal as well as nearly MDS. The best known linear code with these parameters (ref. Magma) is not self orthogonal.

math.NT↗
arXiv2016

$(1-2u^3)$-constacyclic codes and quadratic residue codes over $\mathbb{F}_{p}[u]/\langle u^4-u\rangle$

Let $\mathcal{R}=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}+u^3\mathbb{F}_{p}$ with $u^4=u$ be a finite non-chain ring, where $p$ is a prime congruent to $1$ modulo $3$. In this paper we study $(1-2u^3)$-constacyclic codes over the ring $\mathcal{R}$, their equivalence to cyclic codes and find their Gray images. To illustrate this, examples of $(1-2u^3)$-constacyclic codes of lengths $2^m$ for $p=7$ and of lengths $3^m$ for $p=19$ are given. We also discuss quadratic residue codes over the ring $\mathcal{R}$ and their extensions. A Gray map from $\mathcal{R}$ to $\mathbb{F}_{p}^4$ is defined which preserves self duality and gives self-dual and formally self-dual codes over $\mathbb{F}_{p}$ from extended quadratic residue codes.

math.NT↗
arXiv2015

Refined Estimates on Conjectures of Woods and Minkowski

Let $\wedge$ be a lattice in $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff having a basis of the form $(A_1,0,0,\ldots,0),(a_{2,1},A_2,0,\ldots,0)$, $\ldots,(a_{n,1},a_{n,2},\ldots,a_{n,n-1},A_n)$ where $A_1, A_2,\ldots,A_n$ are all positive. A well known conjecture of Woods in Geometry of Numbers asserts that if $A_{1}A_{2}\cdots A_{n}=1$ and $A_{i}\leqslant A_{1}$ for each $i$ then any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n}/2$ contains a point of $\wedge$. Woods' Conjecture is known to be true for $n\leq 9$. In this paper we give estimates on the Conjecture of Woods for $10\leq n\leq33$, improving the earlier best known results of Hans-Gill et al. These lead to an improvement, for these values of $n$, to the estimates on the long standing classical conjecture of Minkowski on the product of $n$ non-homogeneous linear forms.

math.NT↗
arXiv2014

On Conjectures of Minkowski and Woods for n=9

Let $\mathbb{R}^n$ be the n-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant $1$ such that there is a sphere $|X|<R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly independent points of $\wedge$ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n/4}$ contains a point of $\wedge$. This is known to be true for $n\leq 8$. Here we prove a more general conjecture of Woods for $n=9$ from which this conjecture follows in $\mathbb{R}^9$. Together with a result of C. T. McMullen (2005), the long standing conjecture of Minkowski follows for $n=9$.

math.NT↗