460 research outputs found
Erasure Techniques in MRD codes
This book is organized into six chapters. The first chapter introduces the
basic algebraic structures essential to make this book a self contained one.
Algebraic linear codes and their basic properties are discussed in chapter two.
In chapter three the authors study the basic properties of erasure decoding in
maximum rank distance codes. Some decoding techniques about MRD codes are
described and discussed in chapter four of this book. Rank distance codes with
complementary duals and MRD codes with complementary duals are introduced and
their applications are discussed. Chapter five introduces the notion of integer
rank distance codes. The final chapter introduces some concatenation
techniques.Comment: 162 pages; Published by Zip publishing in 201
Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries
In this paper we construct infinite families of non-linear maximum rank
distance codes by using the setting of bilinear forms of a finite vector space.
We also give a geometric description of such codes by using the cyclic model
for the field reduction of finite geometries and we show that these families
contain the non-linear maximum rank distance codes recently provided by
Cossidente, Marino and Pavese.Comment: submitted; 22 page
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
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