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Anuradha Sharma

Publications and source records attributed to Anuradha Sharma.

15 recordsLinked to original sources
arXiv2026

A recursive approach to the construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic

Let $\mathcal{R}_{e,m}$ be a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 2,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathcal{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$ for some even positive integer $ κ\leq e.$ In this paper, we provide a recursive method to construct a self-orthogonal code $\mathcal{C}_e$ of type $\{λ_1, λ_2, \ldots, λ_e\}$ and length $n$ over $\mathcal{R}_{e,m}$ from a chain $\mathcal{D}^{(1)}\subseteq \mathcal{D}^{(2)} \subseteq \cdots \subseteq \mathcal{D}^{(\lceil \frac{e}{2} \rceil)}$ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, where $\dim \mathcal{D}^{(i)}=λ_1+λ_2+\cdots+λ_i$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil,$ the codes $\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor-κ)},\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor -κ+1)},\ldots,\mathcal{D}^{(\lfloor \frac{e}{2}\rfloor-\lfloor \fracκ{2} \rfloor)}$ satisfy certain additional conditions, and $λ_1,λ_2,\ldots,λ_e$ are non-negative integers satisfying $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e.$ This construction guarantees that $Tor_i(\mathcal{C}_e)=\mathcal{D}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ By employing this recursive construction method, together with the results from group theory and finite geometry, we derive explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathcal{R}_{e,m}.$ We also demonstrate these results through examples.

cs.IT↗
arXiv2026

Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic

Let $\mathscr{R}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 3,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathscr{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^κ\rangle \setminus \langle u^{κ+1}\rangle$ for some odd integer $κ$ with $3 \leq κ\leq e.$ In this paper, we first develop a recursive method to construct a self-orthogonal code $\mathscr{D}_e$ of type $\{λ_1, λ_2, \ldots, λ_e\}$ and length $n$ over $\mathscr{R}_{e,m}$ from a chain $\mathcal{C}^{(1)}\subseteq \mathcal{C}^{(2)} \subseteq \cdots \subseteq \mathcal{C}^{(\lceil \frac{e}{2} \rceil)} $ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, subject to certain conditions, where $λ_1,λ_2,\ldots,λ_e$ are non-negative integers satisfying $2λ_1+2λ_2+\cdots+2λ_{e-i+1}+λ_{e-i+2}+λ_{e-i+3}+\cdots+λ_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e,$ and $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions, respectively. This construction ensures that $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ With the help of this recursive construction method and by applying results from group theory and finite geometry, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathscr{R}_{e,m}.$ We also illustrate these results with some examples.

cs.IT↗
arXiv2025

Linear codes over a mixed-alphabet ring and their Gray images with applications to projective and locally repairable codes

Let $m \geq 2$ be an integer, and let $\mathbb{F}_q$ be the finite field of prime power order $q.$ Let $\mathcal{R}=\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}\times \mathbb{F}_q$ be the mixed-alphabet ring, where $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ is the quasi-Galois ring with maximal ideal $\langle u\rangle$ of nilpotency index $2$ and residue field $\mathbb{F}_q.$ In this paper, we construct four infinite families of linear codes over the ring $\frac{\mathbb{F}_q[u]}{\langle u^2 \rangle}$ whose defining sets are certain non-empty subsets of $\mathcal{R}^m$ associated with three simplicial complexes of $\mathbb{F}_q^m,$ each possessing a single maximal element. We explicitly determine the parameters and Lee weight distributions of these codes. We also study their Gray images and identify several infinite families of few-weight codes over $\mathbb{F}_q,$ as well as an infinite family of minimal, near-Griesmer and distance-optimal codes over $\mathbb{F}_q.$ We also observe that their Gray images are self-orthogonal codes for $q=2$ or $3.$ We determine spanning matrices of these codes. Leveraging this result, we provide two constructions of infinite families of projective few-weight codes over $\mathbb{F}_q$ with new parameters. As an application of our newly constructed minimal codes over $\mathbb{F}_q,$ we examine the minimal access structures of Masseys secret sharing schemes based on their duals and determine the number of dictatorial participants in these schemes. Finally, we investigate the locality properties of our newly constructed projective codes and show that these codes have locality either $2$ or $3.$ As a consequence, we obtain four infinite families of $q$-ary locally repairable codes (LRCs) with locality $2,$ and two infinite families of binary LRCs with locality $3.$

cs.IT↗
arXiv2025

Some remarks on the results derived by Ramy Takieldin and Patrick Solé (2025)

The purpose of this note is to rectify a typographical error in the statements of Theorems 5.5 and 5.6 of Sharma, Chauhan and Singh[3] and further analyze and discuss the significance of the results derived in Takieldin and Solé [4]. In our opinion, several claims made by the authors in [4] are either factually incorrect or lack adequate substantiation, which may confuse the readers about the contributions of [1,3]. Our remarks on the work [4] intend to provide the clarity and inform about the true contributions and findings of our research.

cs.IT↗
arXiv2024

On Eisenstein additive codes over chain rings and linear codes over mixed alphabets

Let $\mathcal{R}_e=GR(p^e,r)[y]/\langle g(y),p^{e-1}y^t\rangle$ be a finite commutative chain ring, where $p$ is a prime number, $GR(p^e,r)$ is the Galois ring of characteristic $p^e$ and rank $r,$ $t$ and $k$ are positive integers satisfying $1\leq t\leq k$ when $e \geq 2,$ while $t=k$ when $e=1,$ and $g(y)=y^k+p(g_{k-1}y^{k-1}+\cdots+g_1y+g_0)\in GR(p^e,r)[y]$ is an Eisenstein polynomial with $g_0$ as a unit in $GR(p^e,r).$ In this paper, we first establish a duality-preserving 1-1 correspondence between additive codes over $\mathcal{R}_e$ and $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, where the character-theoretic dual codes of additive codes over $\mathcal{R}_e$ correspond to the Euclidean dual codes of $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes, and vice versa. This correspondence gives rise to a method for constructing additive codes over $\mathcal{R}_e$ and their character-theoretic dual codes, as unlike additive codes over $\mathcal{R}_e,$ $\mathbb{Z}_{p^e}\mathbb{Z}_{p^{e-1}}$-linear codes can be completely described in terms of generator matrices. We also list additive codes over the chain ring $\mathbb{Z}_4[y]/\langle y^2-2,2y \rangle$ achieving the Plotkin's bound for homogeneous weights, which suggests that additive codes over $\mathcal{R}_e$ is a promising class of error-correcting codes to find optimal codes with respect to the homogeneous metric.

cs.IT↗
arXiv2024

On Galois LCD codes and LCPs of codes over mixed alphabets

Let $\mathtt{R}$ be a finite commutative chain ring with the maximal ideal $γ\mathtt{R}$ of nilpotency index $e\geq 2,$ and let $\check{\mathtt{R}}=\mathtt{R}/γ^{s}\mathtt{R}$ for some positive integer $ s< e.$ In this paper, we study and characterize Galois $\mathtt{R}\check{\mathtt{R}}$-LCD codes of an arbitrary block-length. We show that each weakly-free $\mathtt{R}\check{\mathtt{R}}$-linear code is monomially equivalent to a Galois $\mathtt{R}\check{\mathtt{R}}$-LCD code when $|\mathtt{R}/γ\mathtt{R}|>4,$ while it is monomially equivalent to a Euclidean $\mathtt{R}\check{\mathtt{R}}$-LCD code when $|\mathtt{R}/γ\mathtt{R}|>3.$ We also obtain enumeration formulae for all Euclidean and Hermitian $\mathtt{R}\check{\mathtt{R}}$-LCD codes of an arbitrary block-length. With the help of these enumeration formulae, we classify all Euclidean $\mathbb{Z}_4 \mathbb{Z}_{2}$-LCD codes and $\mathbb{Z}_9 \mathbb{Z}_{3}$-LCD codes of block-lengths $(1,1),$ $(1,2),$ $(2,1),$ $(2,2),$ $(3,1)$ and $(3,2)$ and all Hermitian $\frac{\mathbb{F}_{4}[u]}{\langle u^2\rangle} \;\mathbb{F}_{4}$-LCD codes of block-lengths $(1,1),$ $(1,2),$ $(2,1)$ and $(2,2)$ up to monomial equivalence. Apart from this, we study and characterize LCPs of $\mathtt{R}\check{\mathtt{R}}$-linear codes. We further study a direct sum masking scheme constructed using LCPs of $\mathtt{R}\check{\mathtt{R}}$-linear codes and obtain its security threshold against fault injection and side-channel attacks. We also discuss another application of LCPs of $\mathtt{R}\check{\mathtt{R}}$-linear codes in coding for the noiseless two-user adder channel.

cs.IT↗
arXiv2018

A class of constacyclic codes over $\mathbb{F}_{p^m}[u]/\left $

Let $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~(\text{mod }4).$ Let $\mathbb{F}_{p^m}$ be the finite field with $p^m$ elements, and let $R=\mathbb{F}_{p^m}[u]/\left $ be the finite commutative chain ring with unity. In this paper, we determine all constacyclic codes of length $4p^s$ over $R$ and their dual codes, where $s$ is a positive integer. We also determine their sizes and list some isodual constacyclic codes of length $4p^s$ over $R.$

cs.IT↗
arXiv2017

Repeated-root constacyclic codes over finite commutative chain rings and their distances

Let $\mathcal{R}_e$ be a finite commutative chain ring with nilpotency index $e \geq 2.$ In this paper, all repeated-root constacyclic codes of arbitrary lengths over $\mathcal{R}_{2},$ their sizes and their dual codes are determined. As an application, some isodual constacyclic codes over $\mathcal{R}_{2}$ are also listed. Moreover, Hamming distances, Rosenbloom-Tsfasman distances and Rosenbloom-Tsfasman weight distributions of all repeated-root constacyclic codes over $\mathcal{R}_{2}$ and some repeated-root constacyclic codes over $\mathcal{R}_{e}$ are determined.

math.AC↗
arXiv2017

Repeated-root constacyclic codes over the finite chain ring $\mathbf{ \mathbb{F}_{p^m}[u]/\langle u^3 \rangle }$

Let $\mathcal{R}=\mathbb{F}_{p^m}[u]/\langle u^3 \rangle $ be the finite commutative chain ring with unity, where $p$ is a prime, $m$ is a positive integer and $\mathbb{F}_{p^m}$ is the finite field with $p^m$ elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over $\mathcal{R},$ their sizes and their dual codes. As an application, we list some isodual constacyclic codes over $\mathcal{R}.$ We also determine Hamming distances, RT distances, and RT weight distributions of some repeated-root constacyclic codes over $\mathcal{R}.$

math.NT↗
arXiv2017

Multi-twisted codes over finite fields and their dual codes

Let $\mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,\cdots,m_{\ell}$ be positive integers satisfying $\gcd(m_i,q)=1$ for $1 \leq i \leq \ell,$ and let $n=m_1+m_2+\cdots+m_{\ell}.$ Let $Λ=(λ_1,λ_2,\cdots,λ_{\ell})$ be fixed, where $λ_1,λ_2,\cdots,λ_{\ell}$ are non-zero elements of $\mathbb{F}_{q}.$ In this paper, we study the algebraic structure of $Λ$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ and their dual codes with respect to the standard inner product on $\mathbb{F}_{q}^n.$ We provide necessary and sufficient conditions for the existence of a self-dual $Λ$-multi-twisted code of length $n$ over $\mathbb{F}_{q},$ and obtain enumeration formulae for all self-dual and self-orthogonal $Λ$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}.$ We also derive some sufficient conditions under which a $Λ$-multi-twisted code is LCD. We determine the parity-check polynomial of all $Λ$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some $Λ$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ from the generating sets of the codes. Besides this, we provide a trace description for all $Λ$-multi-twisted codes of length $n$ over $\mathbb{F}_{q}$ by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.

math.AC↗
arXiv2014

Construction of self-dual codes over $\mathbb{Z}_{2^m}$

Self-dual codes (Type I and Type II codes) play an important role in the construction of even unimodular lattices, and hence in the determination of Jacobi forms. In this paper, we construct both Type I and Type II codes (of higher lengths) over the ring $\mathbb{Z}_{2^m}$ of integers modulo $2^m$ from shadows of Type I codes of length $n$ over $\mathbb{Z}_{2^m}$ for each positive integer $n;$ and obtain their complete weight enumerators. Using these results, we also determine some Jacobi forms on the modular group $Γ(1) = SL(2; \mathbb{Z}).$ Besides this, for each positive integer $n$; we also construct self-dual codes (of higher lengths) over $\mathbb{Z}_{2^m}$ from the generalized shadow of a self-dual code $\mathcal{C}$ of length $n$ over $\mathbb{Z}_{2^m}$ with respect to a vector $s\in \mathbb{Z}_{2^m}^n\setminus \mathcal{C}$ satisfying either $s\cdot s \equiv 0 (mod 2^m)$ or $s\cdot s \equiv 2^{m-1} (mod 2^m).$

math.NT↗
arXiv2013

An Approach Of Substitution Method Based On ASCII Codes In Encryption Technique

In poly alphabetic substitution the plain texts letters are enciphered differently according to their position. The name poly alphabetic suggests that there are more than one key so we have used two keys combination instead of just one, in order to produce the cipher text. We can also use three or more keys to make the enciphering process more complicated. In this paper have produced ASCII Codes of the plain text and then we have reversed it said reverse ASCII Codes and then we have generated two keys K1 is generated by addition of reverse ASCII Codes and K2 is generated by addition of ASCII Codes. Then these K1 and K2 Keys are alternatively applied on Reverse ASCII codes in order to produce cipher text. On the Destination hand Deciphering is used to produce the plain text again. Our technique generates random cipher text for the same plain text and this is the major advantage of our technique.

cs.CR↗
arXiv2011

MacWilliams type identities for some new $m$-spotty weight enumerators

Past few years have seen an extensive use of high-density RAM chips with wide I/O data (e.g., 16, 32, 64 bits) in computer memory systems. These chips are highly vulnerable to a special type of byte error, called an $m$-spotty byte error, which can be effectively detected or corrected using byte error-control codes. In this paper, we present joint $m$-spotty weight enumerator and split $m$-spotty weight enumerator for byte error-control codes over the ring of integers modulo $\ell$ ($\ell \geq 2$ is an integer) and over arbitrary finite fields. We also derive MacWilliams type identities for each of the aforementioned enumerators and discuss some of their applications.

cs.IT↗