The parametrized complexity of some fundamental problems in coding theory

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 DISTRIBUTION 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 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 linear codes. Concerning lattices, we address the THETA SERIES problem of determining for an integer lattice Lambda and a positive integer k whether there is a vector x is an element of Lambda 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 DISTRIBUTION and MAXIMUM-LIKELIHOOD DECODING for lattices. Relations between all these problems and combinatorial problems in graphs are discussed.