718 research outputs found
Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions
We contribute to the knowledge of linear codes from special polynomials and
functions, which have been studied intensively in the past few years. Such
codes have several applications in secret sharing, authentication codes,
association schemes and strongly regular graphs.
This is the first work in which we study the dual codes in the framework of
the two generic constructions; in particular, we propose a Gram-Schmidt
(complexity of ) process to compute them explicitly. The
originality of this contribution is in the study of the existence or not of
defining sets , which can be used as ingredients to construct the dual code
for a given code in the context of the second
generic construction. We also determine a necessary condition expressed by
employing the Walsh transform for a codeword of to belong in the
dual. This achievement was done in general and when the involved functions are
weakly regularly bent. We shall give a novel description of the Hull code in
the framework of the two generic constructions. Our primary interest is
constructing linear codes of fixed Hull dimension and determining the (Hamming)
weight of the codewords in their duals
A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes
Some combinatorial designs, such as Hadamard matrices, have been extensively
researched and are familiar to readers across the spectrum of Science and
Engineering. They arise in diverse fields such as cryptography, communication
theory, and quantum computing. Objects like this also lend themselves to
compelling mathematics problems, such as the Hadamard conjecture. However,
complex generalized weighing matrices, which generalize Hadamard matrices, have
not received anything like the same level of scrutiny. Motivated by an
application to the construction of quantum error-correcting codes, which we
outline in the latter sections of this paper, we survey the existing literature
on complex generalized weighing matrices. We discuss and extend upon the known
existence conditions and constructions, and compile known existence results for
small parameters. Some interesting quantum codes are constructed to demonstrate
their value.Comment: 33 pages including appendi
On Fields of rationality for automorphic representations
This paper proves two results on the field of rationality \Q(\pi) for an
automorphic representation , which is the subfield of \C fixed under the
subgroup of \Aut(\C) stabilizing the isomorphism class of the finite part of
. For general linear groups and classical groups, our first main result is
the finiteness of the set of discrete automorphic representations such
that is unramified away from a fixed finite set of places,
has a fixed infinitesimal character, and [\Q(\pi):\Q] is bounded. The second
main result is that for classical groups, [\Q(\pi):\Q] grows to infinity in a
family of automorphic representations in level aspect whose infinite components
are discrete series in a fixed -packet under mild conditions
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