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

arXiv:0802.2723v2 [cs.IT] 6 Oct 2008 On Strongly Controllable Group Codes and Mixing Group Shifts: Solvable Groups, Translation Nets, and Algorithms Kenneth M. Mackenthun Jr. October 5, 2008 Abstract The branch group of a strongly controllable group code is a shift group. We show that a shift group can be charac- terized 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 strongly controllable Latin group code. 1 Introduction Kitchens introduced the fundamental idea of a group shift and showed that a group shift is a shift of finite type [1]. A group shift is essentially a time invariant group code. Forney and Trott showed that a group code has a well defined state space and can be represented on a trellis, and a strongly controllable group code can be realized with a shift register [2]. In a following article, among other results, Loeliger and Mittelholzer gave an abstract characterization of the group which can appear as the branch group of a strongly controllable group code, which they call a group with a shift structure [3]. In this paper, we give a simple characterization of a group with a shift structure, or shift group. We show that a shift group G involves a normal chain {Xj} and a tower of isomorphisms using groups in the normal chain. In addition, there are two important normal subgroups X0 and Y0 of G which have normal chains which also characterize the shift group. These results are shown in Section 2. In Section 3, we use the theory of translation nets to show that if a group code is strongly controllable and is labeled with Latin squares, the shift group is solvable. We show that Latin squares which can appear in a Latin group code are isotopic to those constructed by the au- tomorphism method of Mann [19]. It is shown that if a group code is strongly controllable and if X0 ∩Y0 = 1, X0 ≃Y0, and X0 is elementary abelian, then a complete set of mutually orthogonal Latin squares can be used to label the group code (throughout the paper, we use 1 for the identity of a group). We show that the structure of a shift group is closely related to the structure of a Latin square as a translation net. In Section 4, we show that a shift group with X0∩Y0 = 1 can be represented as a subdirect product group. Then we give necessary and sufficient conditions for a subdirect product group to be a shift group. These conditions show that to find a shift group it is sufficient to construct the state group of a shift group. We give a characterization of the state group. Lastly in Section 5, we give an algorithm to find the state group of a shift group; this can be used to find a Latin group code. 2 Shift groups Let G be any graph with vertices V (also called states) and edges E; in shorthand we write G = (V, E). We say a graph G is l-controllable if for any ordered pair of states (s, s′) in G, there is a path of length l from s to s′ in G. A graph that is l-controllable for some integer l is said to be strongly controllable. The least integer l for which a strongly controllable graph G is l-controllable is denoted as ℓ, and we say G is ℓ-controllable. In this paper, we only study the case l = ℓ. The preceding definition uses the idea of controllability in systems theory and the theory of convolutional codes. There is a similar notion in the theory of symbolic dynam- ics, drawn from ergodic theory. A graph G is primitive if there is a positive integer M such that for any ordered pair of states (s, s′) in G and any m ≥M, there is a path of length m from s to s′ in G [11]. If a graph has an edge into each state, then an ℓ-controllable graph is primitive with M = ℓ. In this paper, we consider a particular graph con- structed using a group B, where the edges E form group B, and the vertices V form a quotient group in B. We de- note this graph as GB. We now discuss this construction in more detail. 1 Let B be a finite group which contains normal sub- groups B+ and B−such that B/B−is isomorphic to B/B+ via an isomorphism ψ : B/B−→B/B+. Let π+ be the (natural) map which sends each element of B to the coset of B+ that it belongs to; likewise for π−: B →B/B−. Let GB = (V, E) be the graph with vertices V = B/B+ and edges E = B, such that each edge e ∈E has initial state i(e) = π+(e) and terminal state t(e) = ψ ◦π−(e). (This discussion is taken from Problem 2.2.16 of [17], which is based on [2, 3].) It is know

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