313 research outputs found
Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4
It is shown that the maximum size of a binary subspace code of packet length
, minimum subspace distance , and constant dimension is ;
in Finite Geometry terms, the maximum number of planes in
mutually intersecting in at most a point is .
Optimal binary subspace codes are classified into
isomorphism types, and a computer-free construction of one isomorphism type is
provided. The construction uses both geometry and finite fields theory and
generalizes to any , yielding a new family of -ary
subspace codes
Dynamics for holographic codes
We describe how to introduce dynamics for the holographic states and codes
introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the
definition of a continuous limit of the kinematical Hilbert space which we
argue may be achieved via the semicontinuous limit of Jones. Dynamics is then
introduced by building a unitary representation of a group known as Thompson's
group T, which is closely related to the conformal group in 1+1 dimensions. The
bulk Hilbert space is realised as a special subspace of the semicontinuous
limit Hilbert space spanned by a class of distinguished states which can be
assigned a discrete bulk geometry. The analogue of the group of large bulk
diffeomorphisms is given by a unitary representation of the Ptolemy group Pt,
on the bulk Hilbert space thus realising a toy model of the AdS/CFT
correspondence which we call the Pt/T correspondence.Comment: 40 pages (revised version submitted to journal). See video of related
talk: https://www.youtube.com/watch?v=xc2KIa2LDF
Orbits of rational n-sets of projective spaces under the action of the linear group
For a fixed dimension we compute the generating function of the numbers
(respectively ) of -orbits of rational
-sets (respectively rational -multisets) of the projective space
\mathb{P}^N over a finite field . For these results
provide concrete formulas for and as a polynomial in
with integer coefficients
Tetrads of lines spanning PG(7,2)
Our starting point is a very simple one, namely that of a set L_4 of four
mutually skew lines in PG(7,2): Under the natural action of the stabilizer
group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1,
omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that
the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic
quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to
have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to
(Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and
each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a
triplet of 27-sets. We show in particular that the constituents of precisely 8
of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with
the recent finding that each Segre S = S_3(2) in PG(7,2) determines a
distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S"
of S.Comment: Some typos correcte
Tetrads of lines spanning PG(7,2)
Our starting point is a very simple one, namely that of a set L₄ of four mutually skew lines in PG(7,2). Under the natural action of the stabilizer group G(L₄)<GL(8,2) the 255 points of PG(7,2) fall into four orbits ω₁,ω₂,ω₃,ω4, of respective lengths 12,54,108,81. We show that the 135 points ∈ω₂∪ω₄ are the internal points of a hyperbolic quadric H7 determined by L₄, and that the 81-set ω₄ (which is shown to have a sextic equation) is an orbit of a normal subgroup G₈₁≅(Z₃)4 of G(L4). There are 40 subgroups ≅(Z₃)3 of G₈₁, and each such subgroup H<G₈₁ gives rise to a decomposition of ω4 into a triplet {RH,R′H,R′′H} of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S₃(2) in PG(7,2). This ties in with the recent finding 225-239 --- that each S=S₃(2) in PG(7,2) determines a distinguished Z₃ subgroup of GL(8,2) which generates two sibling copies S′,S′′ of S
Enumeration of Nonsingular Buekenhout Unitals
The only known enumeration of Buekenhout unitals occurs in the Desarguesian plane . In this paper we develop general techniques for enumerating the nonsingular Buekenhoutunitals embedded in any two-dimensional translation plane, and apply these techniques to obtain such an enumeration in the regular nearfield planes, the odd-order Hall planes, and the flag-transitive affine planes. We also provide some computer data for small-order André planes of index two and give partial results toward an enumeration in this case
Deformation classification of real non-singular cubic threefolds with a marked line
We prove that the space of pairs formed by a real non-singular cubic hypersurface with a real line has 18 connected components and give for them several quite explicit interpretations. The first one relates these components to the orbits of the monodromy action on the set of connected components of the Fano surface formed by real lines on . For another interpretation we associate with each of the 18 components a well defined real deformation class of real non-singular plane quintic curves and show that this deformation class together with the real deformation class of characterizes completely the component
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