Search

In this paper we consider the Hermitian curve yq0+y=xq0+1over the field Fq(q:=q02). The automorphism group of this curve is known to be the projective unitary group PGU(3,q)=2A2(q)with q03(q03+1)(q02-1)elements. We follow the construction done for the Suzuki code in Eid et al. (Designs Codes Cryptogr 81(3):413–425, 2016, https://doi.org/10.1007/s10623-015-0164-5). We construct algebraic geometry codes over Fq3from an 2A2(q)-invariant divisor D, give an explicit basis for the Riemann–Roch space L(ℓD)for 0<ℓ≤q03-1. These families of codes have good parameters and information rate close to one. In addition, they are explicitly constructed. The dual codes of these families are of the same kind if q03-2g+1≤ℓ≤q03-1.

This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schrödinger equation. We examine the critical point structure of the kinematic landscapes JF(U)=\Vert(U-W)A\Vert^{2} and JP(U)=\Vert A\Vert^{4}-\vertTr(AA^{\dagger}W^{\dagger}U)\vert^{2} defined on the unitary group U(\cal H) of a finite-dimensional Hilbert space \cal H. The parameter operator A\in\cal B(\cal H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of $U$ and the target W$ on a subspace of state space, namely the column space of A$. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse–Bott functions on U(\cal H). Landscapes based on the intrinsic (geodesic) distance on U(\cal H) and the projective unitary group are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes. [ABSTRACT FROM AUTHOR]

For the Hermitian curve H defined over the finite field [image omitted] , we give a complete classification of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves [image omitted] -covered by H arising from subgroups of the special linear group SL(2,Fq) or from subgroups in the normaliser of the Singer group of the projective unitary group [image omitted] . Since curves [image omitted] -covered by H are maximal over [image omitted] , our results provide some classification and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q2, both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the "third largest genus problem" including some new results and a detailed account of current work is given.