arXiv subjects

Kenneth M. Mackenthun Jr

Publications and source records attributed to Kenneth M. Mackenthun Jr.

5 recordsLinked to original sources

arXiv2022

Any strongly controllable group system or group shift or any linear block code is isomorphic to a generator group

Consider any sequence of finite groups $A^t$, where $t$ takes values in an integer index set $\mathbf{Z}$. A group system $A$ is a set of sequences with components in $A^t$ that forms a group under componentwise addition in $A^t$, for each $t\in\mathbf{Z}$. As shown previously, any strongly controllable complete group system $A$ can be decomposed into generators. We study permutations of the generators when sequences in the group system are multiplied. We show that any strongly controllable complete group system $A$ is isomorphic to a generator group $({\mathcal{U}},\circ)$. The set ${\mathcal{U}}$ is a set of tensors, a double Cartesian product space of sets $G_k^t$, with indices $k$, for $0\le k\le\ell$, and time $t$, for $t\in\mathbf{Z}$. $G_k^t$ is a set of unique generator labels for the generators in $A$ with nontrivial span for the time interval $[t,t+k]$. We show the generator group contains a unique elementary system, an infinite collection of elementary groups, one for each $k$ and $t$, defined on small subsets of ${\mathcal{U}}$, in the shape of triangles, which form a tile like structure over ${\mathcal{U}}$. There is a homomorphism from each elementary group to any elementary group defined on smaller tiles of the former group. The group system $A$ may be constructed from either the generator group or elementary system. These results have application to linear block codes, any algebraic system that contains a linear block code, group shifts, and harmonic theory in mathematics, and systems theory, coding theory, control theory, and related fields in engineering.

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arXiv2021

Any strongly controllable group system or group shift or any linear block code is a linear system whose input is a generator group

Consider any sequence of finite groups $A^t$, where $t$ takes values in an integer index set $\mathbf{Z}$. A group system $A$ is a set of sequences with components in $A^t$ that forms a group under componentwise addition in $A^t$, for each $t\in\mathbf{Z}$. In the setting of group systems, a natural definition of a linear system is a homomorphism from a group of inputs to an output group system $A$. We show that any group can be the input group of a linear system and some group system. In general the kernel of the homomorphism is nontrivial. We show that any $\ell$-controllable complete group system $A$ is a linear system whose input group is a generator group $({\mathcal{U}},\circ)$, deduced from $A$, and then the kernel is always trivial. The input set ${\mathcal{U}}$ is a set of tensors, a double Cartesian product space of sets $R_{0,k}^t$, with indices $k$, for $0\le k\le\ell$, and time $t$, for $t\in\mathbf{Z}$. $R_{0,k}^t$ is a set of unique generator labels for the generators in $A$ with nontrivial span for the time interval $[t,t+k]$. We show the generator group contains an elementary system, an infinite collection of elementary groups, one for each $k$ and $t$, defined on small subsets of ${\mathcal{U}}$, in the shape of triangles, which form a tile like structure over ${\mathcal{U}}$. There is a homomorphism from each elementary group to any elementary group defined on smaller tiles of the former group. Any elementary system is sufficient to define a unique generator group up to isomorphism, and therefore is sufficient to construct a linear system and group system as well. Any linear block code is a strongly controllable group system. Then we can obtain new results on the structure of block codes using the generator group. There is a harmonic theory of group systems which we study using the generator group.

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arXiv2017

Time and harmonic study of strongly controllable group systems, group shifts, and group codes

In this paper we give a complementary view of some of the results on group systems by Forney and Trott. We find an encoder of a group system which has the form of a time convolution. We consider this to be a time domain encoder while the encoder of Forney and Trott is a spectral domain encoder. We study the outputs of time and spectral domain encoders when the inputs are the same, and also study outputs when the same input is used but time runs forward and backward. In an abelian group system, all four cases give the same output for the same input, but this may not be true for a nonabelian system. Moreover, time symmetry and harmonic symmetry are broken for the same reason. We use a canonic form, a set of tensors, to show how the outputs are related. These results show there is a time and harmonic theory of group systems.

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arXiv2011

Group Codes and the Schreier matrix form

In a group trellis, the sequence of branches that split from the identity path and merge to the identity path form two normal chains. The Schreier refinement theorem can be applied to these two normal chains. The refinement of the two normal chains can be written in the form of a matrix, called the Schreier matrix form, with rows and columns determined by the two normal chains. Based on the Schreier matrix form, we give an encoder structure for a group code which is an estimator. The encoder uses the important idea of shortest length generator sequences previously explained by Forney and Trott. In this encoder the generator sequences are shown to have an additional property: the components of the generators are coset representatives in a chain coset decomposition of the branch group B of the code. Therefore this encoder appears to be a natural form for a group code encoder. The encoder has a register implementation which is somewhat different from the classical shift register structure. This form of the encoder can be extended. We find a composition chain of the branch group B and give an encoder which uses coset representatives in the composition chain of B. When B is solvable, the generators are constructed using coset representatives taken from prime cyclic groups.

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arXiv2008

On strongly controllable group codes and mixing group shifts: solvable groups, translation nets, and algorithms

The branch group of a strongly controllable group code is a shift group. We show that a shift group can be characterized in a very simple way. In addition it is shown that if a strongly controllable group code is labeled with Latin squares, a strongly controllable Latin group code, then the shift group is solvable. Moreover the mathematical structure of a Latin square (as a translation net) and the shift group of a strongly controllable Latin group code are closely related. Thus a strongly controllable Latin group code can be viewed as a natural extension of a Latin square to a sequence space. Lastly we construct shift groups. We show that it is sufficient to construct a simpler group, the state group of a shift group. We give an algorithm to find the state group, and from this it is easy to construct a stronlgy controllable Latin group code.

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