Indifferentiable hashing to ordinary elliptic $${\mathbb {F}}_{\!q}$$ F q -curves of $$j=0$$ j = 0 with the cost of one exponentiation in $${\mathbb {F}}_{\!q}$$ F q

Designs, Codes and Cryptography - Let $${\mathbb {F}}_{\!q}$$ be a finite field and $$E_b\!: y^2 = x^3 + b$$ be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that...     

Linear codes over {\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}

In this work, we investigate linear codes over the ring ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linearity of binary codes under the Gray map and give a main class of binary codes as an example of ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes. The duals and the complete weight enumerators for ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over ${\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}$ are obtained.     

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .      

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

Markov subshifts and partial representation of $$\mathbb{F}_{\text n}$$

In this paper we fix a set * of positive elements of the free group (e. g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S : which we prove to be a partial representation of on H under certain conditions studied by Matsumoto.*Supported by Capes.     

Examples of scalar-flat hypersurfaces in $${\mathbb{R}^{n+1}}$$

Given a hypersurface M of null scalar curvature in the unit sphere , n ≥ 4, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in as a normal graph over a truncated cone generated by M. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.     

Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$

We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.     

Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

A contact-stationary Legendrian submanifold of is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S ₀. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S ₀. Two examples of the construction, corresponding to finite cyclic subgroups of U(n + 1) are given. The resulting submanifolds are very symmetric; are geometrically akin to a ‘necklace’ of copies of S ₀ attached to each other by narrow necks and winding a large number of times around before closing up on themselves; and are topologically equivalent to .     

New constructions of permutation polynomials of the form $$x^rh\left( x^{q-1}\right) $$ over $${\mathbb F}_{q^2}$$Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.     

The positive energy conjecture for a class of AHM metrics on R2× Tn-2

We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers(AHM) metrics on R²× T^n-2. This generalizes the previous results of Barzegar et al.(2020) as well as Liang and Zhang(2020).     

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

Forbidden (0,1)-vectors in Hyperplanes of $$\mathbb{R}^{n}$$: The unrestricted case

In this paper, we continue our investigation on “Extremal problems under dimension constraints” introduced [1]. The general problem we deal with in this paper can be formulated as follows. Let be an affine plane of dimension k in . Given determine or estimate .Here we consider and solve the problem in the special case where is a hyperplane in and the “forbidden set” . The same problem is considered for the case, where is a hyperplane passing through the origin, which surprisingly turns out to be more difficult. For this case we have only partial results.AMS Classification: 05C35, 05B30, 52C99     

Linear codes over $$\mathbb {F}_{q}[x]/(x^2)$$ and $$GR(p^2,m)$$GR(p2,m) reaching the Griesmer bound

We construct two series of linear codes C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) and \(GR(p^2,m)\) reaching the Griesmer bound. Moreover, we consider the Gray images of C(G). The results show that the Gray images of C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) are linear and also reach the Griesmer bound in some cases, and many of linear codes over \(\mathbb {F}_{q}\) we constructed have two Hamming (non-zero) weights.     

On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.     

Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Any algebraic surface in which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in ruled in cubics. This work is within the framework of the national research project “Geometry on Algebraic Varieties” Cofin 2006 of MIUR.     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

A new extension theorem for 3-weight modulo q linear codes over $${\mathbb{F}_ q}$$

We prove that every [n, k, d] _q code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and is extendable unless its diversity is for odd q, where .      

Surfaces in $${\mathbb{S}^2\times\mathbb{R}}$$ with a canonical principal direction

We show a way to choose nice coordinates on a surface in and use this to study minimal surfaces. We show that only open parts of cylinders over a geodesic in are both minimal and flat. We also show that the condition that the projection of the direction tangent to onto the tangent space of the surface is a principal direction, is equivalent to the condition that the surface is normally flat in . We present classification theorems under the extra assumption of minimality or flatness. J. Fastenakels is a research assistant of the Research Foundation—Flanders (FWO). J. Van der Veken is a postdoctoral researcher supported by the Research Foundation—Flanders (FWO). This work was partially supported by project G.0432.07 of the Research Foundation—Flanders (FWO).     

Some new estimates of the Fourier-Bessel transform in the space \mathbb{L}_2 \left( {\mathbb{R}_ + } \right)

The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j _p (u) = ((2 ^p Γ(p+1))/(u ^p ))J _p (u) and J _p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity.