102 research outputs found
Generalized weights: an anticode approach
In this paper we study generalized weights as an algebraic invariant of a
code. We first describe anticodes in the Hamming and in the rank metric,
proving in particular that optimal anticodes in the rank metric coincide with
Frobenius-closed spaces. Then we characterize both generalized Hamming and rank
weights of a code in terms of the intersection of the code with optimal
anticodes in the respective metrics. Inspired by this description, we propose a
new algebraic invariant, which we call "Delsarte generalized weights", for
Delsarte rank-metric codes based on optimal anticodes of matrices. We show that
our invariant refines the generalized rank weights for Gabidulin codes proposed
by Kurihara, Matsumoto and Uyematsu, and establish a series of properties of
Delsarte generalized weights. In particular, we characterize Delsarte optimal
codes and anticodes in terms of their generalized weights. We also present a
duality theory for the new algebraic invariant, proving that the Delsarte
generalized weights of a code completely determine the Delsarte generalized
weights of the dual code. Our results extend the theory of generalized rank
weights for Gabidulin codes. Finally, we prove the analogue for Gabidulin codes
of a theorem of Wei, proving that their generalized rank weights characterize
the worst-case security drops of a Gabidulin rank-metric code
Partial Spreads in Random Network Coding
Following the approach by R. K\"otter and F. R. Kschischang, we study network
codes as families of k-dimensional linear subspaces of a vector space F_q^n, q
being a prime power and F_q the finite field with q elements. In particular,
following an idea in finite projective geometry, we introduce a class of
network codes which we call "partial spread codes". Partial spread codes
naturally generalize spread codes. In this paper we provide an easy description
of such codes in terms of matrices, discuss their maximality, and provide an
efficient decoding algorithm
Projective normality of Artin-Schreier curves
In this paper we study the projective normality of certain Artin-Schreier
curves defined over a field \F of characteristic by the equations
, being a power of and f\in \F[x] being a polynomial in
of degree , with . Many curves are singular and so, to be
precise, here we study the projective normality of appropriate projective
models of their normalizations.Comment: Le Matematiche (to appear
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