Conventional Trellises Research Articles

Matrix theory for minimal trellises

Trellises provide a graphical representation for the row space of a matrix. The product construction of Kschischang and Sorokine builds minimal conventional trellises from matrices in minimal span form. Koetter and Vardy showed that minimal tail-biting trellises can be obtained by applying the product construction to submatrices of a characteristic matrix. We introduce the unique reduced minimal span form of a matrix and we obtain an expression for the unique reduced characteristic matrix. Among new properties of characteristic matrices we prove that characteristic matrices are in duality if and only if they have orthogonal column spaces, and that the transpose of a characteristic matrix is again a characteristic matrix if and only if the characteristic matrix is reduced. These properties have clear interpretations for the unwrapped unit memory convolutional code of a tail-biting trellis, they explain the duality for the class of Koetter and Vardy trellises, and they give a natural relation between the characteristic matrix based Koetter–Vardy construction and the displacement matrix based Nori–Shankar construction. For a pair of reduced characteristic matrices in duality, one is lexicographically first in a forward direction and the other is lexicographically first in the reverse direction. This confirms a conjecture by Koetter and Vardy after taking into account the different directions for the lexicographical ordering.

Rendimiento del frijol ayocote y maíz del agrosistema asociado en función del número de plantas por mata.

El frijol ayocote (Phaseolus coccineus L.) es una leguminosa nativa de México con potencial alimenticio. Para lograr mayor distribución en el espacio, captación de radiación solar y producción de materia seca (MS), el ayocote trepador necesita un soporte. Los soportes convencionales como postes, mallas, raf ia, elevan el costo de producción, sin embargo el ayocote puede sembrarse utilizando al maíz como soporte. Para incrementar el rendimiento de grano, se ha recurrido al manejo de la densidad de población y arreglo topológico. El objetivo del estudio conducido en Montecillo, México, en el ciclo primavera-verano de 2014, fue determinar si el número de plantas de frijol ayocote y maíz por mata en asociación, afectan la producción de biomasa, índice de cosecha, rendimiento de grano y sus componentes. Los tratamientos fueron nueve, resultado de la combinación de 1, 2 y 3 plantas de frijol ayocote y de 1, 2 y 3 de maíz por mata. Se registraron las etapas fenológicas, biomasa total (BT), índice de cosecha (IC), rendimiento de grano (RG) y sus componentes en ambos cultivos. El diseño experimental fue bloques completos al azar con cuatro repeticiones. Los días a ocurrencia de las fases fenológicas en frijol ayocote y maíz, fueron similares entre tratamientos. En la asociación maíz-frijol los cambios en el número de plantas por mata de ambos cultivos, afectaron la BT, IC, RG y sus componentes tanto en ayocote como en maíz. En el agrosistema ayocote-maíz, la BT, RG y sus componentes, presentaron cambios por el número de plantas de ayocote y de maíz por mata. La combinación para lograr la mayor BT, RG e ingreso neto se logró con tres plantas de ayocote y una de maíz.

On the Algebraic Structure of Linear Tail-Biting Trellises

Tail-biting trellises are important graphical representations of codes, which can be simpler to decode than conventional trellises. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. In this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions. In particular, we present a useful and powerful characterization of linear trellis isomorphy. We also obtain a new proof of the linear factorization theorem of Koetter/Vardy, and point out unnoticed problems for the group case. Next, we apply our framework to: 1) describe all the factorizations of linear trellises; 2) determine and classify all the minimal linear trellises for a given linear code; and 3) prove that nonmergeable, one-to-one, linear trellises are determined by the edge-label sequences of certain closed paths. We also provide new insight into mergeability and state how results on reduced linear trellises can be extended to nonreduced ones. Our classification results are relevant to iterative/linear programming decoding, as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure.

English

Cowpea (Vigna unguiculata (L.) Walp) is a considerable protein, carbohydrates, vitamins, minerals and fiber source. Indeterminate habit cultivars require trellis to achieve higher development. This practice increases production cost. Alternatively, crops like maize (Zea mays L.) and sunflower (Helianthus annuus L.) can be used to reduce expenses and provide physical support. Growth and yield of cowpea was evaluated using conventional and live trellis, under summer rain conditions in 2012. Experiments were setup in warm (Cocula, Guerrero, Mexico) and temperate climate (Montecillo, Edo. Mex.). One indeterminate growth variety of cowpea was used to measure yield and total biomass. Additionally, harvest index, water use efficiency and profitability for each trellis setup (conventional (CC), maize (CM) and sunflower (CS)). Maximum and minimum temperatures, rain quantity and evaporation were recorded. Warm climate grown cowpea had higher water use efficiency, which translated into higher grain yield (68 g m-2) and total biomass (253 g m-2). Cowpea grown with maize trellis had higher yield by 13% compared to the one grown with sunflower. Net income was higher with maize trellis, followed by sunflower and conventional trellises. Live trellis is an alternative for cowpea production. Key words: Vigna unguiculata, conventional trellis, economic analysis, live trellis.

A low-complexity maximum likelihood decoder for tail-biting trellis

Abstract Tail-biting trellises are defined on a circular time axis and can have a smaller number of states than equivalent conventional trellises. Existing circular Viterbi algorithms (CVAs) on the tail-biting trellis are non-convergent and sub-optimal. In this study, we show that the net path metric of each tail-biting path is lower-bounded during the decoding process of the CVA. This property can be applied to remove unnecessary iterations of the CVA and results in a convergent maximum likelihood (ML) decoder. Simulation results show that the proposed algorithm exhibits higher decoding efficiency compared with other existing ML decoders.

Characteristic Generators and Dualization for Tail-Biting Trellises

This paper focuses on dualizing tail-biting trellises, particularly KV trellises. These trellises are based on characteristic generators, as introduced by Koetter-Vardy (2003), and may be regarded as a natural generalization of minimal conventional trellises, even though they are not necessarily minimal. Two dualization techniques will be investigated: the local dualization, introduced by Forney (2001) for general normal graphs, and a linear-algebra-based dualization tailored to the specific class of tail-biting Bahl-Cocke-Jelinek-Raviv (BCJR) trellises, introduced by Nori-Shankar (2006). It turns out that, in general, the BCJR dual is a subtrellis of the local dual, while for KV trellises these two coincide. Furthermore, making use of both the BCJR construction and the local dualization, it will be shown that for each complete set of characteristic generators of a code there exists a complete set of characteristic generators of the dual code such that their resulting KV trellises are dual to each other if paired suitably. This proves a stronger version of a conjecture formulated by Koetter-Vardy.

Unifying Views of Tail-Biting Trellis Constructions for Linear Block Codes

In this paper, we present new ways of describing and constructing linear tail-biting trellises for block codes. We extend the well-known Bahl-Cocke-Jelinek-Raviv (BCJR) construction for conventional trellises to tail-biting trellises. The BCJR-like labeling scheme yields a simple specification for the tail-biting trellis for the dual code, with the dual trellis having the same state-complexity profile as that of the primal code . Finally, we show that the algebraic specification of Forney for state spaces of conventional trellises has a natural extension to tail-biting trellises

On viewing block codes as finite automata

Block codes are viewed from a formal language theoretic perspective. It is shown that properties of trellises for subclasses of block codes called rectangular codes follow naturally from the Myhill Nerode theorem. A technique termed subtrellis overlaying is introduced with the object of reducing decoder complexity. Necessary and sufficient conditions for trellis overlaying are derived from the representation of the block code as a group, partitioned into a subgroup and its cosets. The conditions turn out to be simple constraints on coset leaders. It is seen that overlayed trellises are tail-biting trellises for which decoding is generally more efficient than that for conventional trellises. Finally, a decoding algorithm for tail-biting trellises is described, and the results of some simulations are presented.

A new maximum likelihood decoding of high rate convolutional codes using a trellis

AbstractMaximum likelihood decoding of convolutional codes by Viterbi decoding method is an effective error correcting technique. However, little study on Viterbi decoding of high rate codes has been done since the number of paths entering each trellis state increases as the code rate increases. This paper presents a new maximum likelihood decoding of convolutional codes of rate (n ‐ 1)/n using a trellis. the trellis is drawn according to the state transitions of the syndrome former of the code, and has only one or two paths entering each state rather than 2n‐1 paths as in the case of conventional trellises. Thus, we can replace the 2n‐1‐ary comparisons at each state needed in the conventional Viterbi decoding for the code of rate (n ‐ 1)/n by binary comparisons. the implementation of maximum likelihood decoders for high rate convolutional codes is simplified greatly by this approach. Moreover, this decoding method shows the same performance as that of conventional Viterbi decoding method from the viewpoint of selecting the maximum likelihood code sequence.