Designs, Codes and Cryptography

Farfalle is a permutation-based construction for building a pseudorandom function which has been proposed by Bertoni et al. in 2017. In this work, we show that by observing suitable inputs to Farfalle, one can derive various constructions of a ...

In this paper, we study a flag-transitive automorphism group G of symmetric 2-$(v,k,\lambda )$ designs, where $\lambda >1$ and the order $n=k-\lambda$ is a prime number. We prove that G is point-primitive, and it is of affine or almost simple type.

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to construct some classes of permutation polynomials and their compositional inverses ...

Threshold public-key encryption (threshold PKE) has various useful applications. A lot of threshold PKE schemes are proposed based on RSA, Diffie–Hellman and lattice, but to the best of our knowledge, code-based threshold PKEs have not been ...

Let $e\ge 2$ and $r\ge 1$ be integers, and let ${\mathcal{R}}_{e,r}$ denote the Galois ring of characteristic $2^{\mathit{e}}$ and cardinality $2^{\mathit{er}}.$ The Teichm$\ddot{u}$ller set ${\mathcal{T}}_r$ of the Galois ring ${\mathcal{R}}_{e,r}$ can be viewed as the finite field of order $2^r$ under the addition operation $\oplus$ and the ...${\mathcal{}}_{}$${\mathcal{}}_{}$$$${\mathcal{}}_{}$$\mathrm{}$$\mathcal{}$${\mathcal{}}_{}$${\mathcal{}}_{\mathrm{}}$$\mathbf{}\mathbf{}\mathrm{}$$\mathbf{}{\mathcal{}}_{\mathrm{}}$$\mathbf{}{\mathcal{}}_{\mathrm{}}$${\mathcal{}}_{}^{}$$$${\mathcal{}}_{}^{}$${\mathcal{}}_{}$${\mathcal{}}_{}$${\mathcal{}}_{}$${\mathtt{}}_{\mathrm{}}{\mathtt{}}_{\mathrm{}}{\mathtt{}}_{}$${\mathcal{}}_{}$${\mathtt{}}_{\mathrm{}}{\mathtt{}}_{\mathrm{}}{\mathtt{}}_{⌊\frac{}{\mathrm{}}⌋}$${\mathcal{}}_{}$${\mathtt{}}_{\mathrm{}}{\mathtt{}}_{\mathrm{}}{\mathtt{}}_{}$$\mathrm{}{\mathtt{}}_{\mathrm{}}\mathrm{}{\mathtt{}}_{\mathrm{}}\mathrm{}{\mathtt{}}_{\mathrm{}}{\mathtt{}}_{\mathrm{}}{\mathtt{}}_{\mathrm{}}{\mathtt{}}_{}$$⌊⌋$${\mathcal{}}_{}$${\mathcal{}}_{}$${\mathcal{}}_{\mathrm{}\mathrm{}}$

We study a new family of metrics, weighted poset block metric, which generalizes the weighted coordinates poset metric introduced by Panek and Pinheiro (IEEE Trans Inf Theory 66(11):6823–6834, 2020) and the metric for linear error-block codes ...

The study of $(n\times m,k,{\lambda }_a,{\lambda }_c)$-geometric orthogonal codes ($(n\times m,k,{\lambda }_a,{\lambda }_c)$-GOCs) is motivated by the application in DNA origami. The central research on GOCs is to determine the value of $\mathrm{\Phi }(n\times m,k,{\lambda }_a,{\lambda }_c)$, i.e., the largest possible size among all $(n\times m,k,{\lambda }_{\mathrm{...}}{}_{}$${}_{}{}_{}$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$${}_{}\mathrm{}$${}_{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}{}_{}\mathrm{}$${}_{}\mathrm{}$

The singleton defect of an [n, k, d] linear code $\mathcal{C}$ is defined as $s(\mathcal{C})=n-k+1-d$. Codes with $s(\mathcal{C})=s({\mathcal{C}}^{\perp })=1$ are called near maximum distance separable (NMDS) codes. It is known that an $[n,3,n-3]$ NMDS code is equivalent to an (n, 3)-arc in PG(2, q). In ...$\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$${\mathrm{}}^{}$$\mathrm{}\mathrm{}\mathrm{}$${\mathbb{}}_{}$${}^{}\mathrm{}\mathrm{}{}^{}\mathrm{}$${\mathbb{}}_{{}^{}}$

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for $d=2$ (Latin squares) and $d=3$ (Latin cubes). Explicit ...$$$$

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics,...$\stackrel{}{}\left({}^{},\mathfrak{},\right)$${\mathbb{}}_{{}^{}}$$$$\mathfrak{}$$$$\stackrel{}{}\left(\right)$

We consider the function $x^{-1}$ that inverses a finite field element $x\in {\mathbb{F}}_{p^n}$ (p is prime, $0^{-1}=0$) and affine ${\mathbb{F}}_p$-subspaces of ${\mathbb{F}}_{p^n}$ such that their images are affine subspaces as well. It is proved that the image of an affine subspace L, $|L|>2$, is an ...${\mathbb{}}_{{}^{}}$${\mathbb{}}_{{}^{}}^{}$$$${\mathbb{}}_{{}^{}}$$$${}^{\mathrm{}}$${}^{\mathrm{}}$$\mathrm{}{}^{}$${\mathbb{}}_{{}^{}}{\mathbb{}}_{{}^{}}$${\mathbb{}}_{{}^{}}^{}$${}^{\mathrm{}}$${\mathbb{}}_{{}^{}}$${\mathbb{}}_{{}^{}}^{}$

An MDS code is a code which achieves equality in the singleton bound. The defect of a code measures how far it is from an MDS code. Amplifying on the relationship between the weight distribution of a code and its dual code as in the well-known ...

Let $\mathcal{C}$ be a 430-cap of $\text{PG}(6,4)$ having two intersection sizes with respect to hyperplanes. We show that no hyperplane of $\text{PG}(6,4)$ intersects $\mathcal{C}$ in a Hill 78-cap. So if it can be shown that the Hill 78-cap of $\text{PG}(5,4)$ is projectively unique, then such a ...${\mathrm{}\mathrm{}}_{{\mathbb{}}_{\mathrm{}}}$