819 research outputs found
Tensor Network Models of Unitary Black Hole Evaporation
We introduce a general class of toy models to study the quantum
information-theoretic properties of black hole radiation. The models are
governed by a set of isometries that specify how microstates of the black hole
at a given energy evolve to entangled states of a tensor product
black-hole/radiation Hilbert space. The final state of the black hole radiation
is conveniently summarized by a tensor network built from these isometries. We
introduce a set of quantities generalizing the Renyi entropies that provide a
complete set of bipartite/multipartite entanglement measures, and give a
general formula for the average of these over initial black hole states in
terms of the isometries defining the model. For models where the dimension of
the final tensor product radiation Hilbert space is the same as that of the
space of initial black hole microstates, the entanglement structure is
universal, independent of the choice of isometries. In the more general case,
we find that models which best capture the "information-free" property of black
hole horizons are those whose isometries are tensors corresponding to states of
tripartite systems with maximally mixed subsystems.Comment: 22 pages, 4 figure
MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings
A finite ring R and a weight w on R satisfy the Extension Property if every
R-linear w-isometry between two R-linear codes in R^n extends to a monomial
transformation of R^n that preserves w. MacWilliams proved that finite fields
with the Hamming weight satisfy the Extension Property. It is known that finite
Frobenius rings with either the Hamming weight or the homogeneous weight
satisfy the Extension Property. Conversely, if a finite ring with the Hamming
or homogeneous weight satisfies the Extension Property, then the ring is
Frobenius.
This paper addresses the question of a characterization of all bi-invariant
weights on a finite ring that satisfy the Extension Property. Having solved
this question in previous papers for all direct products of finite chain rings
and for matrix rings, we have now arrived at a characterization of these
weights for finite principal ideal rings, which form a large subclass of the
finite Frobenius rings. We do not assume commutativity of the rings in
question.Comment: 12 page
Folding Alternant and Goppa Codes with Non-Trivial Automorphism Groups
The main practical limitation of the McEliece public-key encryption scheme is
probably the size of its key. A famous trend to overcome this issue is to focus
on subclasses of alternant/Goppa codes with a non trivial automorphism group.
Such codes display then symmetries allowing compact parity-check or generator
matrices. For instance, a key-reduction is obtained by taking quasi-cyclic (QC)
or quasi-dyadic (QD) alternant/Goppa codes. We show that the use of such
symmetric alternant/Goppa codes in cryptography introduces a fundamental
weakness. It is indeed possible to reduce the key-recovery on the original
symmetric public-code to the key-recovery on a (much) smaller code that has not
anymore symmetries. This result is obtained thanks to a new operation on codes
called folding that exploits the knowledge of the automorphism group. This
operation consists in adding the coordinates of codewords which belong to the
same orbit under the action of the automorphism group. The advantage is
twofold: the reduction factor can be as large as the size of the orbits, and it
preserves a fundamental property: folding the dual of an alternant (resp.
Goppa) code provides the dual of an alternant (resp. Goppa) code. A key point
is to show that all the existing constructions of alternant/Goppa codes with
symmetries follow a common principal of taking codes whose support is globally
invariant under the action of affine transformations (by building upon prior
works of T. Berger and A. D{\"{u}}r). This enables not only to present a
unified view but also to generalize the construction of QC, QD and even
quasi-monoidic (QM) Goppa codes. All in all, our results can be harnessed to
boost up any key-recovery attack on McEliece systems based on symmetric
alternant or Goppa codes, and in particular algebraic attacks.Comment: 19 page
Isometry and Automorphisms of Constant Dimension Codes
We define linear and semilinear isometry for general subspace codes, used for
random network coding. Furthermore, some results on isometry classes and
automorphism groups of known constant dimension code constructions are derived
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