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On subfields of the Hermitian function fields involving the involution automorphism
A function field over a finite field is called maximal if it achieves the
Hasse-Weil bound. Finding possible genera that maximal function fields achieve
has both theoretical interest and practical applications to coding theory and
other topics. As a subfield of a maximal function field is also maximal, one
way to find maximal function fields is to find all subfields of a maximal
function field. Due to the large automorphism group of the Hermitian function
field, it is natural to find as many subfields of the Hermitian function field
as possible. In literature, most of papers studied subfields fixed by subgroups
of the decomposition group at one point (usually the point at infinity). This
is because it becomes much more complicated to study the subfield fixed by a
subgroup that is not contained in the decomposition group at one point.
In this paper, we study subfields of the Hermitian function field fixed by
subgroups that are not contained in the decomposition group of any point except
the cyclic subgroups. It turns out that some new maximal function fields are
found
Matrix-Monotonic Optimization for MIMO Systems
For MIMO systems, due to the deployment of multiple antennas at both the
transmitter and the receiver, the design variables e.g., precoders, equalizers,
training sequences, etc. are usually matrices. It is well known that matrix
operations are usually more complicated compared to their vector counterparts.
In order to overcome the high complexity resulting from matrix variables, in
this paper we investigate a class of elegant multi-objective optimization
problems, namely matrix-monotonic optimization problems (MMOPs). In our work,
various representative MIMO optimization problems are unified into a framework
of matrix-monotonic optimization, which includes linear transceiver design,
nonlinear transceiver design, training sequence design, radar waveform
optimization, the corresponding robust design and so on as its special cases.
Then exploiting the framework of matrix-monotonic optimization the optimal
structures of the considered matrix variables can be derived first. Based on
the optimal structure, the matrix-variate optimization problems can be greatly
simplified into the ones with only vector variables. In particular, the
dimension of the new vector variable is equal to the minimum number of columns
and rows of the original matrix variable. Finally, we also extend our work to
some more general cases with multiple matrix variables.Comment: 37 Pages, 5 figures, IEEE Transactions on Signal Processing, Final
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