Matrix theory for minimal trellises

  title={Matrix theory for minimal trellises},
  author={Iwan M. Duursma},
  journal={Designs, Codes and Cryptography},
  • I. Duursma
  • Published 2019
  • Computer Science, Mathematics
  • Designs, Codes and Cryptography
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… Expand


OF DISSERTATION MINIMALITY AND DUALITY OF TAIL-BITING TRELLISES FOR LINEAR CODES Codes can be represented by edge-labeled directed graphs called trellises, which are used in decoding with the ViterbiExpand
Characteristic Generators and Dualization for Tail-Biting Trellises
It will be shown that for each complete set of characteristic generators of a code there exists a completeSet of characteristic generator of the dual code such that their resulting KV trellises are dual to each other if paired suitably. Expand
On the Algebraic Structure of Linear Tail-Biting Trellises
A new algebraic framework for a systematic analysis of linear trellises is developed which enables it to address open foundational questions and provides new insight into mergeability and state how results on reduced linear Trellises can be extended to nonreduced ones. Expand
Unifying Views Of Tail-Biting Trellises For Linear Block Codes
This thesis presents new techniques for the construction and specification of linear tail-biting trellises. Tail-biting trellises for linear block codes are combinatorial descriptions in the form ofExpand
On the BCJR trellis for linear block codes
  • R. McEliece
  • Mathematics, Computer Science
  • IEEE Trans. Inf. Theory
  • 1996
It is shown that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf, Massey, and Forney, uniquely minimizes the edge count. Expand
On the trellis structure of block codes
The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete and the number of distinct minimal linear block code trellises is a Stirling number of the second kind. Expand
Minimal tail-biting trellises: The Golay code and more
A 16-state 12-section structurally invariant tail-biting trellis is constructed for the (24, 12, 8) binary Golay code, which has attractive performance/complexity properties and simultaneously minimizes all conceivable measures of state complexity. Expand
Linear Tail-Biting Trellises: Characteristic Generators and the BCJR-Construction
This paper investigates the constructions of tail-biting trellises for linear block codes as introduced by Koetter and Vardy (2003) and Nori and Shankar (2006) and shows that each KV-trellis is a span-based BCJR-Trellis and that the latter are always nonmergeable. Expand
The structure of tail-biting trellises: minimality and basic principl
This work observes that a trellis - either tail-biting or conventional - is linear if and only if it factors into a product of elementary trellises, and devise a linear-programming algorithm that starts with the characteristic matrix and produces a linear tail- biting trell is for /spl Copf/; which minimizes the maximum state-space size. Expand
Bruhat decomposition and solution of sparse linear algebraic systems
Gauss-type decompositions commonly used for solving linear algebraic systems of equations with sparse coefficient matrices do not make it possible to take into consideration more subtle matrixExpand