1,721,138 research outputs found
Restauri nel nord Sardegna. Temi e cantieri di restauro architettonico nell’attività della Soprintendenza BeAP di Sassari e Nuoro
No full textThe Möbius function of PSU(3, 22n)
Full text linkLet G be the simple group PSU(3, 22n), n > 0. For any subgroup H of G, we compute the Möbius function μL(H, G) of H in the subgroup lattice L of G, and the Möbius function μL ̄ ([H], [G]) of [H] in the poset L ̄ of conjugacy classes of subgroups of G. For any prime p, we provide the Euler characteristic of the order complex of the poset of non-trivial p-subgroups of G
COPRIME COMMUTATORS in the SUZUKI GROUPS
No full textIn this note we show that every element of a simple Suzuki group is a commutator of elements of coprime orders
On the spectrum of genera of quotients of the Hermitian curve
Full text linkWe investigate the genera of quotient curves Hq∕G of the Fq2 -maximal Hermitian curve Hq, where G is contained in the maximal subgroup Mq ≤ Aut (Hq) fixing a pole-polar pair (P,l) with respect to the unitary polarity associated with Hq. To this aim, a geometric and group-theoretical description of Mq is given. The genera of some other quotients Hq∕G with G≰Mq are also computed. In this way we obtain new values in the spectrum of genera of Fq2 -maximal curves. The complete list of genera g>1 of quotients of Hq is given for q≤29, as well as the genera g of quotients of Hq with g>q2q+30/32 for any q. As a direct application, we exhibit examples of Fq2 -maximal curves which are not Galois covered by Hq when q is not a cube. Finally, a plane model for Hq∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1
Some Ree and Suzuki curves are not Galois covered by the Hermitian curve
Full text linkThe Deligne–Lusztig curves associated to the algebraic groups of type A22, B22, and G22 are classical examples of maximal curves over finite fields. The Hermitian curve Hq is maximal over Fqjavax.xml.bind.JAXBElement@77709645, for any prime power q, the Suzuki curve Sq is maximal over Fqjavax.xml.bind.JAXBElement@99c946, for q=22h+1, h≥1, and the Ree curve Rq is maximal over Fqjavax.xml.bind.JAXBElement@706d12f8, for q=32h+1, h≥0. In this paper we show that S8 is not Galois covered by H64. We also prove an unpublished result due to Rains and Zieve stating that R3 is not Galois covered by H27. Furthermore, we determine the spectrum of genera of Galois subcovers of H27, and we point out that some Galois subcovers of R3 are not Galois subcovers of H27
The complete list of genera of quotients of the Fqjavax.xml.bind.JAXBElement@2b4afbfb-maximal Hermitian curve for q ≡ 1 (mod 4)
No full textLet Fqjavax.xml.bind.JAXBElement@47e531f1 be the finite field with q2 elements. Most of the known Fqjavax.xml.bind.JAXBElement@1831ea17-maximal curves arise as quotient curves of the Fqjavax.xml.bind.JAXBElement@74ff9869-maximal Hermitian curve Hq. After a seminal paper by Garcia, Stichtenoth and Xing, many papers have provided genera of quotients of Hq, but their complete determination is a challenging open problem. In this paper we determine completely the spectrum of genera of quotients of Hq for any q≡1(mod4)
Scattered subspaces and related codes
No full textAfter a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum h-scattered Fq-subspaces of V(r, qn) and maximum rank distance (MRD) codes has been established in the extremal cases h= 1 and h= r- 1. In this paper, we propose a connection for any h∈ { 1 , ... , r- 1 } , extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an h-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces
On the intersection problem for linear sets in the projective line
No full textThe aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two
Generalized Artin–Mumford curves over finite fields
Full text linkLet Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq ̄ be a subfield of Fq. From any two q ̄-linearized polynomials L1,L2∈F ̅q[T] of degree q, we construct an ordinary curve X(Ljavax.xml.bind.JAXBElement@3de21171,Ljavax.xml.bind.JAXBElement@44e73174) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(Ljavax.xml.bind.JAXBElement@265fda03,Ljavax.xml.bind.JAXBElement@5ee20ea3) over the algebraic closure F ̅q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q ̄−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(Ljavax.xml.bind.JAXBElement@4b2f1fff,Ljavax.xml.bind.JAXBElement@2ddc4e9)) over F ̅q; for L1=L2 there exists an extra involution and Aut(X(Ljavax.xml.bind.JAXBElement@2daa9e77,Ljavax.xml.bind.JAXBElement@1c89ae0d))=Σ⋊Δ with a dihedral group Δ of order 2(q ̄−1) containing Γ. Two different choices of the pair L1,L2 may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F ̅q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(Ljavax.xml.bind.JAXBElement@1301e61e,Ljavax.xml.bind.JAXBElement@61aac551) for some separable q ̄-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1]
On a family of linear MRD codes with parameters [8 × 8 , 16 , 7] q
Full text linkIn this paper we consider a family F of 2n-dimensional F-q-linear rank metric codes in F-q(nxn) arising from polynomials of the form x(qs) +delta x(q) (n/2 +s) is an element of F-q(n) [x]. The family F was introduced by Csajbok et al. (JAMA 548:203-220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8, and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8, providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F-q-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far
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