324 research outputs found
Semianalytical calculation of the zonal-flow oscillation frequency in stellarators
Due to their capability to reduce turbulent transport in magnetized plasmas,
understanding the dynamics of zonal flows is an important problem in the fusion
programme. Since the pioneering work by Rosenbluth and Hinton in axisymmetric
tokamaks, it is known that studying the linear and collisionless relaxation of
zonal flow perturbations gives valuable information and physical insight.
Recently, the problem has been investigated in stellarators and it has been
found that in these devices the relaxation process exhibits a characteristic
feature: a damped oscillation. The frequency of this oscillation might be a
relevant parameter in the regulation of turbulent transport, and therefore its
efficient and accurate calculation is important. Although an analytical
expression can be derived for the frequency, its numerical evaluation is not
simple and has not been exploited systematically so far. Here, a numerical
method for its evaluation is considered, and the results are compared with
those obtained by calculating the frequency from gyrokinetic simulations. This
"semianalytical" approach for the determination of the zonal-flow frequency
reveals accurate and faster than the one based on gyrokinetic simulations.Comment: 30 pages, 14 figure
On the Kernel of -Linear Hadamard Codes
The -additive codes are subgroups of ,
and can be seen as a generalization of linear codes over and
. A -linear Hadamard code is a binary Hadamard
code which is the Gray map image of a -additive code. It is
known that the dimension of the kernel can be used to give a complete
classification of the -linear Hadamard codes. In this paper, the
kernel of -linear Hadamard codes and its dimension are
established for . Moreover, we prove that this invariant only provides a
complete classification for some values of and . The exact amount of
nonequivalent such codes are given up to for any , by using
also the rank and, in some cases, further computations
On Z8-linear Hadamard codes : rank and classification
The Z2s -additive codes are subgroups of ℤZn2s, and can be seen as a generalization of linear codes over ℤ2 and ℤ4. A Zs-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ℤs -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the ℤ4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of ℤ2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s=3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established
Linearity and Classification of -Linear Hadamard Codes
The -additive codes are subgroups of
. A -linear
Hadamard code is a Hadamard code which is the Gray map image of a
-additive code. A recursive construction
of -additive Hadamard codes of type
with , , , , , and is
known. In this paper, we generalize some known results for
-linear Hadamard codes to
-linear Hadamard codes with , , and . First, we show for which
types the corresponding -linear Hadamard
codes of length are nonlinear. For these codes, we compute the kernel and
its dimension, which allows us to give a partial classification of these codes.
Moreover, for , we give a complete classification by
providing the exact amount of nonequivalent such codes. We also prove the
existence of several families of infinite such nonlinear
-linear Hadamard codes, which are not
equivalent to any other constructed
-linear Hadamard code, nor to any
-linear Hadamard code, nor to any previously
constructed -linear Hadamard code with , with the
same length .Comment: arXiv admin note: text overlap with arXiv:2301.0940
On recursive constructions of Z2Z4Z8-linear Hadamard codes
The Z2Z4Z8-additive codes are subgroups of Z α1 2 × Z α2 4 × Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ̸= 0, α2 ̸= 0, and α3 ̸= 0. First, we give a recursive construction of Z2Z4Z8- additive Hadamard codes of type (α1, α2, α3;t1, t2, t3) with t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1. It is known that each Z4-linear Hadamard code is equivalent to a Z2Z4-linear Hadamard code with α1 ̸= 0 and α2 ̸= 0. Unlike Z2Z4-linear Hadamard codes, in general, this family of Z2Z4Z8-linear Hadamard codes does not include the family of Z4-linear or Z8-linear Hadamard codes. We show that, for example, for length 211, the constructed nonlinear Z2Z4Z8-linear Hadamard codes are not equivalent to each other, nor to any Z2Z4-linear Hadamard, nor to any previously constructed Z2s -Hadamard code, with s ≥ 2. Finally, we also present other recursive constructions of Z2Z4Z8-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones
-Additive Hadamard Codes
The -additive codes are subgroups of
, and can be seen as linear codes over
when , -additive or -additive
codes when or , respectively, or
-additive codes when . A
-linear Hadamard code is a Hadamard code
which is the Gray map image of a
-additive code. In this paper, we
generalize some known results for -linear Hadamard
codes to -linear Hadamard codes with
, , and . First, we give a
recursive construction of -additive
Hadamard codes of type with
, , and . Then, we show that in general the
-linear, -linear and
-linear Hadamard codes are not included in the family
of -linear Hadamard codes with , , and . Actually, we point out that
none of these nonlinear -linear Hadamard
codes of length is equivalent to a
-linear Hadamard code of any other type,
a -linear Hadamard code, or a
-linear Hadamard code, with , of the same length
- …