On the Schneider-Vigneras functor for principal series

We study the Schneider-Vigneras functor attaching a module over the Iwasawa algebra $\Lambda(N_0)$ to a $B$-representation for irreducible modulo $\pi$ principal series of the group $\mathrm{GL}_n(F)$ for any finite field extension $F|\mathbb{Q}_p$.Comment: After major revision, 21 pages, to appear in Journal of Number Theor

Links between generalized Montr\'eal-functors

Let $o$ be the ring of integers in a finite extension $K/\mathbb{Q}_p$ and $G=\mathbf{G}(\mathbb{Q}_p)$ be the $\mathbb{Q}_p$-points of a $\mathbb{Q}_p$-split reductive group $\mathbf{G}$ defined over $\mathbb{Z}_p$ with connected centre and split Borel $\mathbf{B}=\mathbf{TN}$. We show that Breuil's pseudocompact $(\varphi,\Gamma)$-module $D^\vee_{\xi}(\pi)$ attached to a smooth $o$-torsion representation $\pi$ of $B=\mathbf{B}(\mathbb{Q}_p)$ is isomorphic to the pseudocompact completion of the basechange $\mathcal{O_E}\otimes_{\Lambda(N_0),\ell}\widetilde{D_{SV}}(\pi)$ to Fontaine's ring (via a Whittaker functional $\ell\colon N_0=\mathbf{N}(\mathbb{Z}_p)\to \mathbb{Z}_p$) of the \'etale hull $\widetilde{D_{SV}}(\pi)$ of $D_{SV}(\pi)$ defined by Schneider and Vigneras. Moreover, we construct a $G$-equivariant map from the Pontryagin dual $\pi^\vee$ to the global sections $\mathfrak{Y}(G/B)$ of the $G$-equivariant sheaf $\mathfrak{Y}$ on $G/B$ attached to a noncommutative multivariable version $D^\vee_{\xi,\ell,\infty}(\pi)$ of Breuil's $D^\vee_{\xi}(\pi)$ whenever $\pi$ comes as the restriction to $B$ of a smooth, admissible representation of $G$ of finite length.Comment: 50 pp, revise

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 $\mathrm{GL}_n$.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 $C$ be a linear code of length $n$ and dimension $k$ over the finite field $\mathbb{F}_{q^m}$. The trace code $\mathrm{Tr}(C)$ is a linear code of the same length $n$ over the subfield $\mathbb{F}_q$. The obvious upper bound for the dimension of the trace code over $\mathbb{F}_q$ is $mk$. If equality holds, then we say that $C$ 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 $C_{\mathbf{a}}$ denote the code obtained from $C$ and a multiplier vector $\mathbf{a}\in (\mathbb{F}_{q^m})^n$. In this paper, we give a lower bound for the probability that a random multiplier vector produces a code $C_{\mathbf{a}}$ 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 $n\geq m(k+h)$, where $h\geq 0$ is the Singleton defect of $C$. For the extremal case $n=m(h+k)$, 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 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