Maximum-likelihood decoding of Reed-Solomon codes is NP-hard
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A polynomial algorithm for codes based on directed graphs
CATS '06 Proceedings of the 12th Computing: The Australasian Theroy Symposium - Volume 51
Algorithms for computing parameters of graph-based extensions of BCH codes
Journal of Discrete Algorithms
Note: Minimum light number of lit-only σ-game on a tree
Theoretical Computer Science
The general σ all-ones problem for trees
Discrete Applied Mathematics
An Algorithm for BCH Codes Extended with Finite State Automata
Fundamenta Informaticae - Workshop on Combinatorial Algorithms
Sort and Search: Exact algorithms for generalized domination
Information Processing Letters
Parameterized complexity of even/odd subgraph problems
Journal of Discrete Algorithms
An exact algorithm for connected red-blue dominating set
Journal of Discrete Algorithms
Parameterized complexity of generalized domination problems
Discrete Applied Mathematics
An exact algorithm for connected red-blue dominating set
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Parameterized complexity of even/odd subgraph problems
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
The birth and early years of parameterized complexity
The Multivariate Algorithmic Revolution and Beyond
Deciding first order properties of matroids
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
An Algorithm for BCH Codes Extended with Finite State Automata
Fundamenta Informaticae - Workshop on Combinatorial Algorithms
A polynomial algorithm for codes based on directed graphs
CATS '06 Proceedings of the Twelfth Computing: The Australasian Theory Symposium - Volume 51
Parameterized complexity of weak odd domination problems
FCT'13 Proceedings of the 19th international conference on Fundamentals of Computation Theory
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The parametrized complexity of a number of fundamental problems in the theory of linear codes and integer lattices is explored. Concerning codes, the main results are that MAXIMUM-LIKELIHOOD DECODING and WEIGHT DISTRUBUTION are hard for the parametrized complexity class W[1]. The NP-completeness of these two problems was established by Berlekamp, McEliece, and van Tilborg in 1978 using by means of a reduction from THREE-DIMENSIONAL MATCHING. On the other hand, our proof of hardness for W[1] is based on a parametric polynomial-time transformation from PERFECT CODE in graphs. An immediate consequence of our results is that bounded-distance decoding is likely to be hard for binary linear codes. Concerning lattices, we address the THETA SERIES problem of determining for an integer lattice $\L$ %given by a set of generators, and a positive integer k whether there is a vector $x \in \L$ of Euclidean norm k. We prove here for the first time that THETA SERIES is NP-complete and show that it is also hard for W[1]. Furthermore, we prove that the NEAREST VECTOR problem for integer lattices is hard for W[1]. These problems are the counterparts of WEIGHT DISTRUBUTION and MAXIMUM-LIKELIHOOD DECODING for lattices. Relations between all these problems and combinatorial problems in graphs are discussed.