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The dynamics of group codes: State spaces, trellis diagrams, and canonical encoders

The dynamics of group codes: State spaces, trellis diagrams, and canonical encoders,10.1109/18.259635,IEEE Transactions on Information Theory,G. David

The dynamics of group codes: State spaces, trellis diagrams, and canonical encoders
A group code C over a group G is a set of sequences of group elements that itself forms a group under a component-wise group operation. A group code has a well-defined state space Σk at each time k. Each code sequence passes through a well-defined state sequence. The set of all state sequences is also a group code, the state code of C. The state code defines an essentially unique minimal realization of C. The trellis diagram of C is defined by the state code of C and by labels associated with each state transition. The set of all label sequences forms a group code, the label code of C, which is isomorphic to the state code of C. If C is complete and strongly controllable, then a minimal encoder in controller canonical (feedbackfree) form may be constructed from certain sets of shortest possible code sequences, called granules. The size of the state space Σk is equal to the size of the state space of this canonical encoder, which is given by a decomposition of the input groups of C at each time k. If C is time-invariant and ν-controllable, then |Σk|=Π1&les;j&les;v|Fj/F j-1|j, where F0 ⊆···⊆ Fν is a normal series, the input chain of C. A group code C has a well-defined trellis section corresponding to any finite interval, regardless of whether it is complete. For a linear time-invariant convolutional code over a field G, these results reduce to known results; however, they depend only on elementary group properties, not on the multiplicative structure of G. Moreover, time-invariance is not required. These results hold for arbitrary groups, and apply to block codes, lattices, time-varying convolutional codes, trellis codes, geometrically uniform codes and discrete-time linear systems
Journal: IEEE Transactions on Information Theory - TIT , vol. 39, no. 5, pp. 1491-1513, 1993
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Citation Context (54)

• ...Approaching information theoretic performance limits of communication systems using structured codes has been an area of great interest in recent years [2], [5], [6], [8], [11], [14]...
• ...1) Group Codes: Given a group G, a group code C over G with block length n is any subgroup of G n [4], [8]...
• ...It has been shown in [8] that P[n;n](C) is a subgroup of G. Set H = P[n;n](C) to conclude the first part of the claim...

Aria Ghasemian Sahebi, et al. On the Capacity of Abelian Group Codes Over Discrete Memoryless Channe...

• ... will be represented as triples and thus the edge set decomposes into where .N otice that we compute modulo on the time axis . Referring to the familiar dynamical interpretation, we call the state spaceof the trellis at time ,and its elements are thestatesat that time.Theedgesreflectthepresent-statetonext-statetransitions, and therefore the sets will be called transition spaces .T hese spaces have also been called trellissections [3], [8] ...

Heide Gluesing-Luerssen, et al. Characteristic Generators and Dualization for Tail-Biting Trellises

• ...Formally, a group code over a group is defined as the set of sequences of group elements that itself forms a group under componentwise group operation [16]...

Sudharshan Srinivasan, et al. Decoding of high rate convolutional codes using the dual trellis

• ...In addition, if the subsequences form linear subspaces generated by the same polynomial, the DC­ substituted rows of D2n x I form a subset of a quasi-cyclic code whose algebraic properties are well known [12], [16]...

Sang Hyun Lee, et al. Boolean functions over nano-fabrics: Improving resilience through codi...

• ...Most of the literature on convolutional codes over rings adopts an approach in which code sequences are semi-infinite Laurent series [1], [2], [9]‐[14]...
• ...For example (see [1], [10], [15]) the convolutionalcodeover withencoder does not admit a noncatastrophic encoder...
• ...Formally, we define a trellis section as a three-tuple , where is the trellis state set and is the set of branches which is a subset of , see also [1], [2]...
• ... is called a trellis representation for a convolutional code if . A trellis representation for a convolutional code is called minimal if the size of its trellis state set is minimal among all trellis representations of . It is well known how to construct a minimal trellis representation in terms of the code sequences of . In fact, the theory of canonical trellis representations from the field case carries through to the ring case, see [1], ...
• ...Thus a difference between a -encoder and the encoding matrix of (1), is that the inputs of take their values in rather than in . Note that the idea of using a -adic expansion for the input sequence is already present in the 1993 paper[1].Itwasnotuntil1996thatthecrucialnotionof -generator sequenceappeared in [20], butonly forconstant vectors—it was extended to polynomial vectorsin [21]...
• ...of Example 1 is clearly minimal, so that its corresponding trellis is minimal with two states which concurs with [1]...
• ...In this Appendix, we recall the construction of a minimal trellis for a convolutional code as a so-called two-sided realization of , see [1], [2], [10], [11], [14], and [24]...

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