On a family of circulant matrices for quasi-cyclic low-density generator matrix codes

We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors. © 2011 IEEE

Rigidity for rigid analytic motives

In this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of \ue9tale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to-coefficients

Invariant States on Noncommutative Tori

For any number h such that h := h/2 pi is irrational and any skew-symmetric, non-degenerate bilinear form sigma : Z(2g) x Z(2g) -> Z, let be A(g,sigma)(h) be the twisted group *-algebra C[Z(2g)] and consider the ergodic group of *-automorphisms of A(g,sigma)(h) induced by the action of the symplectic group Sp (Z(2g), sigma). We show that the only Sp (Z(2g),sigma)-invariant state on A(g,sigma)(h) is the trace state tau

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry

Analytic geometry over F1 and the Fargues-Fontaine curve

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toën-Vaquié theory of schemes over F1, i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (e.g. polydisks) are recovered as a base change of analytic spaces over F1. We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry