Cohomological invariants of finite Coxeter groups

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:1112.629

Rank weight hierarchy of some classes of cyclic codes

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

Generalized rank weights : a duality statement

We consider linear codes over some fixed finite field extension Fq m/Fq, where Fq is an arbitrary finite field. In [1], Gabidulin introduced rank metric codes, by endowing linear codes over Fq m with a rank weight over Fq and studied their basic properties in analogy with linear codes and the classical Hamming distance. Inspired by the characterization of the security in wiretap II codes in terms of generalized Hamming weights by Wei [8], Kurihara et al. defined in [3] 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 rth-generalized Singleton bound for the generalized rank weights, in terms of the rank weight of the dual code.Accepted versio

An analysis of small dimensional fading wiretap lattice codes

We consider sums of inverse of algebraic norms as a code design criterion for fading wiretap channels. We study their behavior for small dimensional lattices built over the ring of integers of a number field, where the lattice points are taken from finite constellations, whose shaping is either cubic or spheric. Our analysis shows that unimodular lattices whose underlying number field has a small discriminant give the best performance so far.Accepted versio

Rank weight hierarchy of some classes of polynomial codes

We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field \FF_{q^m}, $q$ a prime power, $m \geq 2$. We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of $[n,n-1]$ constacyclic codes. We characterize polynomial codes of $r$th rank weight $r$, and in particular of irst rank or minimum rank distance 1. Finally, we provide a refinement of the Singleton bound, from which we show that cyclic codes cannot be MRD (maximum rank distance) codes, but constacyclic codes can be.National Research Foundation (NRF)Submitted/Accepted versionThe early stage of this research by J. Ducoat and F. Oggier was supported by the Singapore National Research Foundation under Research Grant NRF-RF2009-07

Lattice Encoding of Cyclic Codes from Skew-polynomial Rings

We propose a construction of lattices from cyclic codes from skew-polynomial rings. This construction may be seen as a variation of Construction A of lattices from linear codes, obtained from quotients of orders in cyclic division algebras. An application is coset encoding of wiretap space-time codes.Accepted versio