On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric
Daniele Bartoli,Leo Storme +1 more
TLDR
Here, the functional codes arising from the intersections of the algebraic hypersurfaces of small degree $h$ with a given non-singular quadric $\mathcal{Q}_N$ in PG$(N,q)$.Abstract:
We discuss the functional codes $C_h(\mathcal{Q}_N)$, for small $h\geq 3$, $q>9$, and for $N\geq 6$. This continues the study of different classes of functional codes, performed on functional codes arising from quadrics and Hermitian varieties. Here, we consider the functional codes arising from the intersections of the algebraic hypersurfaces of small degree $h$ with a given non-singular quadric $\mathcal{Q}_N$ in PG$(N,q)$.read more
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Journal ArticleDOI
Galois geometries and coding theory
Tuvi Etzion,Leo Storme +1 more
TL;DR: Galois geometries and coding theory are two research areas which have been interacting with each other for many decades as mentioned in this paper, from the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed---Muller codes and even further to LDPC codes, random network codes, and distributed storage.
Book Chapter
Galois geometries and coding theory
Ivan Landjev,Leo Storme +1 more
TL;DR: The known links are reviewed, new results and open problems are presented to stimulate the research on Galois geometries, coding theory, and on their continuously developing and increasing interactions.
Journal ArticleDOI
Linear codes from Denniston maximal arcs
TL;DR: This paper constructs functional codes from Denniston maximal arcs and finds a number of linear codes which appear to have larger minimum distance with respect to the known codes with same length and dimension.
Posted Content
Linear codes from Denniston maximal arcs
TL;DR: In this article, the authors constructed functional codes from Denniston maximal arcs with parameters $[((sqrt{q}-1)(q+1),5,d]_q$ where
References
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BookDOI
General Galois geometries
TL;DR: In this paper, the authors define Hermitian varieties, Grassmann varieties, Veronese and Segre varieties, and embedded geometries for finite projective spaces of three dimensions.
Journal ArticleDOI
Improved explicit estimates on the number of solutions of equations over a finite field
Antonio Cafure,Guillermo Matera +1 more
TL;DR: In this article, the Bertini theorem is used to estimate the number of q-ratinoal points of an F"q-definable affine affine absolutely irreducible variety of [email protected]?"q^n.
Journal ArticleDOI
Functional codes arising from quadric intersections with Hermitian varieties
Anja Hallez,Leo Storme +1 more
TL;DR: This paper will answer the question about the minimum distance in general dimension N, with N.
Journal ArticleDOI
The small weight codewords of the functional codes associated to non-singular Hermitian varieties
TL;DR: The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q2) consisting of q + 1 hyperplanes through a common (N − 2)-dimensional space Π, forming a Baer subline in the quotient space of Π.
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Functional codes arising from quadric intersections with Hermitian varieties
Anja Hallez,Leo Storme +1 more