arXiv subjects

Ritumoni Sarma

Publications and source records attributed to Ritumoni Sarma.

At least 19 recordsLinked to original sources

arXiv2026

Linear Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$ associated with Simplicial Complexes, Their Gray Images, and Subfield Codes

In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.

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arXiv2026

New Constructions of Additive MDS TRS Codes

Additive codes over finite fields generalize linear codes, and additive MDS codes provide a natural extension of linear MDS codes. In this article, we study additive twisted Reed--Solomon (TRS) codes and obtain new constructions of additive MDS codes. First, for additive TRS codes with twist $t=2$ and an arbitrary hook, we establish necessary and sufficient conditions for the codes to be additive MDS, thereby generalizing the results in Section 3 of [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. In particular, we show that the existence of an additive MDS TRS code with $t=2$ and hook $h=0$ yields codes of larger lengths than those obtained for $t=2$ and $h=k-1$ in [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. Next, we consider additive TRS codes with twist vector $\mathbf{t}=(1,2)$ and hook vector $\mathbf{h}=(0,0)$, and derive necessary and sufficient conditions for them to be additive MDS. We further establish the existence of such codes. Using the Schur square technique, we obtain mild conditions under which the constructed families are inequivalent to additive Reed--Solomon (RS) codes. Finally, we determine parity-check matrices for both families of additive MDS codes considered in this article.

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arXiv2026

New Quaternary codes with small Plotkin-defects from two-generator simplicial complexes

In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee weight distributions. As a result, we find three quaternary linear code families with Plotkin-defect 1 \& 2 and report at least 32 new or improved parameters having small Plotkin-defects, including 19 projective and 7 optimal parameters. We additionally report 5 quaternary linear codes with best-known parameters that are also projective. Further, we establish necessary and sufficient conditions for their Gray image to be linear, which in turn gives two infinite families of distance-optimal, one infinite family of at least almost dimension-optimal binary linear codes and five infinite families of minimal binary linear codes.

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arXiv2026

Skew Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$

Let $R^t$ denote the finite chain ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle},$ where $p$ is a prime and $t$ is a positive integer. In this article, for a prime $p$ and an automorphism $\theta$ of $\mathbb{F}_{p^m}$, we give the structure of the left ideals of the ring $\frac{R^t[x,\Theta]}{\langle f(x) \rangle},$ where $f(x)$ is in the center of the skew polynomial ring $R^t[x,\Theta]$ and $\Theta$ is an automorphism of $R^t$ that extends $\theta$ with $\Theta(u)=u$. These left ideals are also referred to as skew polycyclic codes associated to $f(x).$ In particular, when the central element $ f(x)$ is $x^{np^s}-\lambda $, where $\lambda=\lambda_0+u\lambda_1+\cdots +u^{t-1}\lambda_{t-1}$ with $\lambda_0\ne0,$ and $ n=1,2 $, we give a more refined form of the left ideals (which are also called skew constacyclic codes). Moreover, the case $\lambda_1 \neq 0$ is analyzed in detail, yielding a simpler form of generators that reveals a more refined structural characterization of the left ideals. As an application, for $n=1,t=3$ and $n=2,t=2$ we give a full description of the left ideals by including certain necessary conditions that were omitted in available literature, preventing the different classes of left ideals from being mutually disjoint and in certain cases, we also compute $i$-th torsion codes.

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arXiv2026

Constacyclic codes of length $np^s$ over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t\rangle}$: Torsions and Cardinalities

The purpose of this article is to study constacyclic codes of length $np^s$ over $R^t:=\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle },$ where $t$ is a natural number and $\gcd(n,p)=1$. We give generators of all the ideals of $R^{t,n}_{\delta}:=\frac{R^t[x]}{\langle x^{np^s}-\delta \rangle},$ where $\delta= \delta_0+u\delta_1+\dots+u^{t-1}\delta_{t-1}$ is a unit in $R^t$. For $n=1,\ 2, \ 3$ and $t=3$, we provide all types of ideals (constacyclic codes) and also give the torsional degrees as well as cardinalities of these codes.

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arXiv2026

Structure of Polycyclic Codes over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}$ and their Cardinalities

The purpose of this article is to study polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}, \,t \geq 1$, and their associated torsion codes. It is shown that if $\phi$ is a surjective ring homomorphism from a commutative ring $A$ to a Noetherian ring $B$ with $ ker(\phi)=\langle \pi\rangle$ then for every ideal $I$ of $A$, there exists $a_1,a_2,\dots,a_n$ in $I$ such that $I=\langle a_1,a_2,\dots,a_n\rangle+\pi(I:\pi)$. Using this, we obtain generators of all ideals of the ring $\frac{\frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x]}{\langle \omega(x)\rangle},$ where $\omega(x)\in \frac{\mathbb{F}_{p^m}[u]}{\langle u^t \rangle}[x] $. For the case when $\omega(x)=f(x)^{p^s}$ where $f(x)$ is an irreducible polynomial in $\mathbb{F}_{p^m}[x]$ and $s$ is a non-negative integer, we obtain several other results, including computation of torsion ideals and their torsional degrees when $t=4$. We use the torsional degree to compute the cardinality of polycyclic codes over the ring $\frac{\mathbb{F}_{p^m}[u]}{\langle u^4 \rangle}$ and illustrate the result with some examples that verify the computed cardinality.

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arXiv2026

Cryptographic Applications of Twisted Goppa Codes

This article defines multi-twisted Goppa (MTG) codes as subfield subcodes of duals of multi-twisted Reed-Solomon (MTRS) codes and examines their properties. We show that if $t$ is the degree of the MTG polynomial defining an MTG code, its minimum distance is at least $t + 1$ under certain conditions. Extending earlier methods limited to single twist at last position, we use the extended Euclidean algorithm to efficiently decode MTG codes with a single twist at any position, correcting up to $\left\lfloor \tfrac{t}{2} \right\rfloor$ errors. This decoding method highlights the practical potential of these codes within the Niederreiter public key cryptosystem (PKC). Furthermore, we establish that the Niederreiter PKC based on MTG codes is secure against partial key recovery attacks. Additionally, we also reduce the public key size by constructing quasi-cyclic MTG codes using a non-trivial automorphism group.

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arXiv2026

On the Euclidean duals of the cyclic codes generated via cyclotomic polynomials

For a natural number $n\ge2$ which is co-prime to Char$(\mathbb{F}_q)$, let $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ denote the cyclic codes of length $n$ over $\mathbb{F}_q$ generated by the $n$-th cyclotomic polynomial $Q_n(x)$ and the polynomial $Q_n(x)Q_1(x)$, respectively. In \cite{BHAGAT2025}, the minimum distances of the codes $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ were determined, and a conjecture regarding the minimum distances of their Euclidean duals was proposed. In this article, we completely describe the structure of these dual codes and as a consequence, we find their minimum distances explicitly as functions of $n$. In fact, we resolve the conjecture in \cite{BHAGAT2025} by proving that the minimum distance of the Euclidean dual of each of $\mathcal{C}_n$ and $\mathcal{C}_{n,1}$ is equal to $2^{\omega(n)}$.

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arXiv2025

$\mathbb{F}_q\mathbb{F}_{q^2}$-additive cyclic codes and their Gray images

We investigate additive cyclic codes over the alphabet $\mathbb{F}_{q}\mathbb{F}_{q^2}$, where $q$ is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of $\mathbb{F}_{q^2}$-additive cyclic codes that satisfy the well-known Singleton bound are constructed. Using a Gray map, we produce certain optimal linear codes over $\mathbb{F}_{3}$. Finally, we obtain a few optimal ternary linear complementary dual (LCD) codes from $\mathbb{F}_{3}\mathbb{F}_{9}$-additive codes.

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arXiv2025

Optimal binary codes from $\mathcal{C}_{D}$-codes over a non-chain ring

In \cite{shi2022few-weight}, Shi and Li studied $\mathcal{C}_D$-codes over the ring $\mathcal{R}:=\mathbb{F}_2[x,y]/\langle x^2, y^2, xy-yx\rangle$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We study the subfield codes $\mathcal{C}_{D}^{(2)}$ of $\mathcal{C}_{D}$-codes over $\mathcal{R},$ where $D$ is as in \cite{shi2022few-weight} and more. We find the Hamming weight distribution and the parameters of $\mathcal{C}_D^{(2)}$ for various $D$, and identify several infinite families of codes that are distance-optimal. Besides, we provide sufficient conditions under which these codes are minimal and self-orthogonal. Two families of strongly regular graphs are obtained as an application of the constructed two-weight codes.

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arXiv2025

Non-RS MDS Codes via Row-Column Twists

This article introduces a new class of codes, called row-column twisted Reed--Solomon (RCTRS) codes, motivated by the constructions in (Beelen et al. \cite{beelen2017twisted}) and (Liu et al. \cite{liu2025column}). Explicit conditions under which RCTRS codes are MDS are established, and their existence is proved by algebraic techniques. By deriving lower bounds on the dimensions of their Schur squares, it is shown that these MDS codes, as well as their extended codes, are not equivalent to Reed--Solomon codes, and hence form new families of non-RS MDS codes. It is further proved that they are not equivalent to column twisted Reed--Solomon codes or their extended versions, and, when the hook lies strictly between the two extreme positions, are not equivalent to twisted Reed--Solomon codes with twist $t = 1$ either. Introducing twists in both a row and a column therefore yields a family of MDS codes distinct from the previously known constructions.

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arXiv2025

$(\Theta, \Delta_\Theta, \mathbf{a})$-cyclic codes over $\mathbb{F}_q^l$ and their applications in the construction of quantum codes

In this article, for a finite field $\mathbb{F}_q$ and a natural number $l,$ let $\mathcal{R}$ denote the product ring $\mathbb{F}_q^l.$ Firstly, for an automorphism $\Theta$ of $\mathcal{R},$ a $\Theta$-derivation $\Delta_\Theta$ of $\mathcal{R}$ and for a unit $\mathbf{a}$ in $\mathcal{R},$ we study $(\Theta, \Delta_\Theta, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ In this direction, we give an algebraic characterization of a $(\Theta, \Delta_\Theta, \mathbf{a})$-cyclic code over $\mathcal{R}$, determine its generator polynomial, and find its decomposition over $\mathbb{F}_q.$ Secondly, we give a necessary and sufficient condition for a $(\Theta, 0, \mathbf{a})$-cyclic code to be Euclidean dual-containing code over $\mathcal{R}.$ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over $\mathbb{F}_q$ as Gray images of $(\Theta, \Delta_\Theta, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.

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arXiv2024

Polycyclic Codes over the Product Ring $\mathbb{F}_q^l$ and their Annihilator Dual

In this article, for the finite field $\mathbb{F}_q$, we show that the $\mathbb{F}_q$-algebra $\mathbb{F}_q[x]/\langle f(x) \rangle$ is isomorphic to the product ring $\mathbb{F}_q^{\deg f(x)}$ if and only if $f(x)$ splits over $\mathbb{F}_q$ into distinct factors. We generalize this result to the quotient of the polynomial algebra $\mathbb{F}_q[x_1, x_2,\dots, x_k]$ by the ideal $\langle f_1(x_1), f_2(x_2),\dots, f_k(x_k)\rangle.$ On the other hand, we establish that every finite-dimensional $\mathbb{F}_q$-algebra $\mathcal{S}$ has an orthogonal basis of idempotents with their sum equal to $1_{\mathcal{S}}$ if and only if $\mathcal{S}\cong\mathbb{F}_q^l$ as $\mathbb{F}_q$-algebras, where $l=\dim_{\mathbb{F}_q} \mathcal{S}$. Instead of studying polycyclic codes over $\mathbb{F}_q$-algebras $\mathbb{F}_q[x_1, x_2,\dots, x_k]/\langle f_1(x_1), f_2(x_2),\dots, f_k(x_k)\rangle$ where $f_i(x_i)$ splits into distinct linear factors over $\mathbb{F}_q,$ which is a subclass of $\mathbb{F}_q^l,$ we study polycyclic codes over $\mathbb{F}_q^l$ and obtain their unique decomposition into polycyclic codes over $\mathbb{F}_q$ for every such orthogonal basis of $\mathbb{F}_q^l$. We refer to it as an $\mathbb{F}_q$-decomposition. An $\mathbb{F}_q$-decomposition enables us to use results of polycyclic codes over $\mathbb{F}_q$ to study polycyclic codes over $\mathbb{F}_q^l$; for instance, we show that the annihilator dual of a polycyclic code over $\mathbb{F}_q^l$ is a polycyclic code over $\mathbb{F}_q^l$. Furthermore, with the help of different Gray maps, we produce a good number of examples of MDS or almost-MDS or/and optimal codes; some of them are LCD over $\mathbb{F}_q$. Finally, we study Gray maps from $(\mathbb{F}_q^l)^n$ to $\mathbb{F}_q^{nl},$ and use it to construct quantum codes with the help of CSS construction.

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arXiv2024

Subfield codes of $C_D$-codes over $\mathbb{F}_2[x]/\langle x^3-x \rangle$ are really nice!

A non-zero $\mathbb{F}$-linear map from a finite-dimensional commutative $\mathbb{F}$-algebra to $\mathbb{F}$ is called an $\mathbb{F}$-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an $\mathbb{F}_2$-valued trace of the $\mathbb{F}_2$-algebra $\mathcal{R}_2:=\mathbb{F}_2[x]/\langle x^3-x\rangle$ to study binary subfield code $\mathcal{C}_D^{(2)}$ of $\mathcal{C}_D:=\{\left(x\cdot d\right)_{d\in D}: x\in \mathcal{R}_2^m\}$ for each defining set $D$ derived from a certain simplicial complex. For $m\in \mathbb{N}$ and $X\subseteq \{1, 2, \dots, m\}$, define $\Delta_X:=\{v\in \mathbb{F}_2^m: \Supp(v)\subseteq X\}$ and $D:=(1+u^2)D_1+u^2D_2+(u+u^2)D_3,$ a subset of $\mathcal{R}_2^m,$ where $u=x+\langle x^3-x\rangle, D_1\in \{\Delta_L, \Delta_L^c\},\, D_2\in \{\Delta_M, \Delta_M^c\}$ and $ D_3\in \{\Delta_N, \Delta_N^c\}$, for $L, M, N\subseteq \{1, 2, \dots, m\}.$ The parameters and the Hamming weight distribution of the binary subfield code $\mathcal{C}_D^{(2)}$ of $\mathcal{C}_D$ are determined for each $D.$ These binary subfield codes are minimal under certain mild conditions on the cardinalities of $L, M$ and $N$. Moreover, most of these codes are distance-optimal. Consequently, we obtain a few infinite families of minimal, self-orthogonal and distance-optimal binary linear codes that are either $2$-weight or $4$-weight. It is worth mentioning that we have obtained several new distance-optimal binary linear codes.

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arXiv2023

Certain binary minimal codes constructed using simplicial complexes

In this manuscript, we work over the non-chain ring $\mathcal{R} = \mathbb{F}_2[u]/\langle u^3 - u\rangle $. Let $m\in \mathbb{N}$ and let $L, M, N \subseteq [m]:=\{1, 2, \dots, m\}$. For $X\subseteq [m]$, define $Δ_X:=\{v \in \mathbb{F}_2^m : \textnormal{Supp}(v)\subseteq X\}$ and $D:= (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3$, an ordered finite multiset consisting of elements from $\mathcal{R}^m$, where $D_1\in \{Δ_L, Δ_L^c\}, D_2\in \{Δ_M, Δ_M^c\}, D_3\in \{Δ_N, Δ_N^c\}$. The linear code $C_D$ over $\mathcal{R}$ defined by $\{\big(v\cdot d\big)_{d\in D} : v \in \mathcal{R}^m \}$ is studied for each $D$. Further, we also consider simplicial complexes with two maximal elements in the above work. We study their binary Gray images and the binary subfield-like codes corresponding to a certain $\mathbb{F}_{2}$-functional of $\mathcal{R}$. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.

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arXiv2023

$\mathbb{F}$-valued trace of a finite-dimensional commutative $\mathbb{F}$-algebra

A non-zero $\mathbb{F}$-valued $\mathbb{F}$-linear map on a finite dimensional $\mathbb{F}$-algebra is called an $\mathbb{F}$-valued trace if its kernel does not contain any non-zero ideals. However, given an $\mathbb{F}$-algebra such a map may not always exist. We find an infinite class of finite-dimensional commutative $\mathbb{F}$-algebras which admit an $\mathbb{F}$-valued trace. In fact, in these cases, we explicitly construct a trace map. The existence of an $\mathbb{F}$-valued trace on a finite dimensional commutative $\mathbb{F}$-algebra induces a non-degenerate bilinear form on the $\mathbb{F}$-algebra which may be helpful both theoretically and computationally. In this article, we suggest a couple of applications of an $\mathbb{F}$-valued trace map of an $\mathbb{F}$-algebra to algebraic coding theory.

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arXiv2023

Minimal and Optimal binary codes obtained using $C_D$-construction over the non-unital ring $I$

In this article, we construct linear codes over the commutative non-unital ring $I$ of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are minimal and self-orthogonal. All codes in this article are few-weight codes. Besides, an infinite class of these binary codes consists of distance optimal codes with respect to the Griesmer bound.

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arXiv2023

Codes over the non-unital non-commutative ring $E$ using simplicial complexes

There are exactly two non-commutative rings of size $4$, namely, $E = \langle a, b ~\vert ~ 2a = 2b = 0, a^2 = a, b^2 = b, ab= a, ba = b\rangle$ and its opposite ring $F$. These rings are non-unital. A subset $D$ of $E^m$ is defined with the help of simplicial complexes, and utilized to construct linear left-$E$-codes $C^L_D=\{(v\cdot d)_{d\in D} : v\in E^m\}$ and right-$E$-codes $C^R_D=\{(d\cdot v)_{d\in D} : v\in E^m\}$. We study their corresponding binary codes obtained via a Gray map. The weight distributions of all these codes are computed. We achieve a couple of infinite families of optimal codes with respect to the Griesmer bound. Ashikhmin-Barg's condition for minimality of a linear code is satisfied by most of the binary codes we constructed here. All the binary codes in this article are few-weight codes, and self-orthogonal codes under certain mild conditions. This is the first attempt to study the structure of linear codes over non-unital non-commutative rings using simplicial complexes.

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