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We propose a variation of Construction A of lattices from linear codes defined using the quotient $\Lambda/\mathfrak p\Lambda$ of some order $\Lambda$ inside a cyclic division $F$-algebra, for $\mathfrak p$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $\Lambda/\mathfrak p\Lambda$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A., Comment: 15 pages

We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes and characterize $[n,k]$ cyclic codes of rank metric 1 when (1) $k=1$, (2) $n$ and $q$ are coprime, and (3) the characteristic $char(\mathbb F_q)$ divides $n$. Finally, for $n$ and $q$ coprime, cyclic codes of minimal $r$-rank are characterized, and a refinement of the Singleton bound for the rank weight is derived.

We consider linear codes over some fixed finite field extension over an arbitrary finite field. Gabidulin introduced rank metric codes, by endowing linear codes over the extension field with a rank weight over the base field and studied their basic properties in analogy with linear codes and the classical Hamming distance. Inspired by the characterization of wiretap II codes in terms of generalized Hamming weights by Wei, Kurihara et al. defined some generalized rank weights and showed their relevance for secure network coding. In this paper, we derive a statement for generalized rank weights of the dual code, completely analogous to Wei's one for generalized Hamming weights and we characterize the equality case of the r-generalized Singleton bound for the generalized rank weights, in terms of the rank weight of the dual code., Comment: 8 pages

In this paper, we generalize Serre's splitting theorem for cohomological invariants of the symmetric group to finite Coxeter groups, provided that the ground field has characteristic zero. We then use this principle to determine all the cohomological invariants of Weyl groups of classical type with coefficients modulo 2., Comment: arXiv admin note: substantial text overlap with arXiv:1112.6290

In a previous paper, we showed that all the cohomological invariants of Weyl groups are completely determined by their restrictions to the abelian subgroups generated by reflections. Using this principle, we describe all the cohomological invariants of the Weyl groups of classical type modulo 2., Comment: The content of this article has been added to the article "Cohomological invariants of finite Coxeter groups