16 research outputs found
On the Schneider-Vigneras functor for principal series
We study the Schneider-Vigneras functor attaching a module over the Iwasawa
algebra to a -representation for irreducible modulo
principal series of the group for any finite field extension
.Comment: After major revision, 21 pages, to appear in Journal of Number Theor
Links between generalized Montr\'eal-functors
Let be the ring of integers in a finite extension and
be the -points of a
-split reductive group defined over
with connected centre and split Borel . We show that
Breuil's pseudocompact -module attached
to a smooth -torsion representation of is
isomorphic to the pseudocompact completion of the basechange
to Fontaine's
ring (via a Whittaker functional ) of the \'etale hull of
defined by Schneider and Vigneras. Moreover, we construct a -equivariant map
from the Pontryagin dual to the global sections
of the -equivariant sheaf on attached to a
noncommutative multivariable version of
Breuil's whenever comes as the restriction to of
a smooth, admissible representation of of finite length.Comment: 50 pp, revise
The distribution and density of cyclic groups of the reductions of an elliptic curve over a function field
Matrix Kloosterman sums modulo prime powers
We give optimal bounds for matrix Kloosterman sums modulo prime powers
extending earlier work of the first two authors on the case of prime moduli.
These exponential sums arise in the theory of the horocyclic flow on
.Comment: 17 pages, comments welcom
On the Schneider-Vigneras functor for principal series
We study the Schneider-Vigneras functor attaching a module over the Iwasawa algebra Λ(N0) to a B-representation for irreducible modulo π principal series of the group GLn(F) for any finite field extension F|Qp. © 2015 Elsevier Inc
On Linear Codes with Random Multiplier Vectors and the Maximum Trace Dimension Property
Let be a linear code of length and dimension over the finite
field . The trace code is a linear code of
the same length over the subfield . The obvious upper bound
for the dimension of the trace code over is . If equality
holds, then we say that has maximum trace dimension. The problem of finding
the true dimension of trace codes and their duals is relevant for the size of
the public key of various code-based cryptographic protocols. Let
denote the code obtained from and a multiplier vector
. In this paper, we give a lower bound for
the probability that a random multiplier vector produces a code
of maximum trace dimension. We give an interpretation of the
bound for the class of algebraic geometry codes in terms of the degree of the
defining divisor. The bound explains the experimental fact that random
alternant codes have minimal dimension. Our bound holds whenever , where is the Singleton defect of . For the extremal case
, numerical experiments reveal a closed connection between the
probability of having maximum trace dimension and the probability that a random
matrix has full rank
On linear codes with random multiplier vectors and the maximum trace dimension property
Let C C be a linear code of length n n and dimension k k over the finite field F q m {{\mathbb{F}}}_{{q}^{m}} . The trace code Tr ( C ) {\rm{Tr}}\left(C) is a linear code of the same length n n over the subfield F q {{\mathbb{F}}}_{q} . The obvious upper bound for the dimension of the trace code over F q {{\mathbb{F}}}_{q} is m k mk . If equality holds, then we say that C C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a {C}_{{\boldsymbol{a}}} denote the code obtained from C C and a multiplier vector a ∈ ( F q m ) n {\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n} . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a {C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m ( k + h ) n\ge m\left(k+h) , where h ≥ 0 h\ge 0 is the Singleton defect of C C . For the extremal case n = m ( h + k ) n=m\left(h+k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank
On linear codes with random multiplier vectors and the maximum trace dimension property
Let CC be a linear code of length nn and dimension kk over the finite field Fqm{{\mathbb{F}}}_{{q}^{m}}. The trace code Tr(C){\rm{Tr}}\left(C) is a linear code of the same length nn over the subfield Fq{{\mathbb{F}}}_{q}. The obvious upper bound for the dimension of the trace code over Fq{{\mathbb{F}}}_{q} is mkmk. If equality holds, then we say that CC has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let Ca{C}_{{\boldsymbol{a}}} denote the code obtained from CC and a multiplier vector a∈(Fqm)n{\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n}. In this study, we give a lower bound for the probability that a random multiplier vector produces a code Ca{C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n≥m(k+h)n\ge m\left(k+h), where h≥0h\ge 0 is the Singleton defect of CC. For the extremal case n=m(h+k)n=m\left(h+k), numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank