11 research outputs found

    On Codes of Bounded Trellis Complexity

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    In this paper, we initiate a structure theory of linear codes with bounded trellis complexity. The theory is based on the observation that the family of linear codes over Fq, some permutation of which has trellis state-complexity at most w, is a minor-closed family. It then follows from a deep result of matroid theory that such codes are characterized by finitely many excluded minors. We provide the complete list of excluded minors for w = 1, and give a partial list for w = 2. We also give a polynomial-time algorithm for determining whether or nor a given code has a permutation with state-complexity at most 1

    Finding branch-decompositions of matroids, hypergraphs, and more

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    Given nn subspaces of a finite-dimensional vector space over a fixed finite field F\mathcal F, we wish to find a "branch-decomposition" of these subspaces of width at most kk, that is a subcubic tree TT with nn leaves mapped bijectively to the subspaces such that for every edge ee of TT, the sum of subspaces associated with leaves in one component of Tāˆ’eT-e and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most kk. This problem includes the problems of computing branch-width of F\mathcal F-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most kk, if it exists, for input subspaces of a finite-dimensional vector space over F\mathcal F. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of F\mathcal F-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time O(n3)O(n^3) where nn is the number of elements of the input F\mathcal F-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on F\mathcal F-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed kk.Comment: 73 pages, 10 figure

    Typical Sequences Revisited ā€” Computing Width Parameters of Graphs

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    In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in polynomial time

    The Telecommunications and Data Acquisition Report

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    This quarterly publication provides archival reports on developments in programs managed by JPL's Telecommunications and Mission Operations Directorate (TMOD), which now includes the former Telecommunications and Data Acquisition (TDA) Office. In space communications, radio navigation, radio science, and ground-based radio and radar astronomy, it reports on activities of the Deep Space Network (DSN) in planning, supporting research and technology, implementation, and operations. Also included are standards activity at JPL for space data and information systems and reimbursable DSN work performed for other space agencies through NASA. The preceding work is all performed for NASA's Office of Space Communications (OSC)

    Measurement based fault tolerant error correcting quantum codes on foliated cluster states

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    The "Art of Trellis Decoding" is NP-Hard

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    Given a linear code C, the fundamental problem of trellis decoding is to find a coordinate permutation of C that yields a code C ā€² whose minimal trellis has the least state-complexity among all codes obtainable by permuting the coordinates of C. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of finding such a coordinate permutation is NP-hard, thus settling a long-standing conjecture
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