arXiv subjects

Anuj Kumar Bhagat

Publications and source records attributed to Anuj Kumar Bhagat.

8 recordsLinked to original sources

arXiv2026

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.

cs.IT↗

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.

cs.IT↗

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 $2^{ω(n)}$.

cs.IT↗

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.

cs.IT↗

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.

cs.IT↗

arXiv2025

$(Θ, Δ_Θ, \mathbf{a})$-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 $Θ$ of $\mathcal{R},$ a $Θ$-derivation $Δ_Θ$ of $\mathcal{R}$ and for a unit $\mathbf{a}$ in $\mathcal{R},$ we study $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ In this direction, we give an algebraic characterization of a $(Θ, Δ_Θ, \mathbf{a})$-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 $(Θ, 0, \mathbf{a})$-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 $(Θ, Δ_Θ, \mathbf{a})$-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.

cs.IT↗

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 $Δ_X:=\{v\in \mathbb{F}_2^m: \Supp(v)\subseteq X\}$ and $D:=(1+u^2)D_1+u^2D_2+(u+u^2)D_3,$ a subset of $\mathcal{R}_2^m,$ where $u=x+\langle x^3-x\rangle, D_1\in \{Δ_L, Δ_L^c\},\, D_2\in \{Δ_M, Δ_M^c\}$ and $ D_3\in \{Δ_N, Δ_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.

cs.IT↗

arXiv2022

On the exponent of cyclic codes

We propose an algorithm to find a lower bound for the number of cyclic codes over any finite field with any given exponent. Besides, we give a formula to find the exponent of BCH codes.

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