The Parameterized Complexity of Some Fundamental Problems of Linear Codes and Integer Lattices

The parameterized complexity of a number of fundamental problems of linear codes and integer lattices is explored. Concerning codes, the main results are that Maximum Likelihood Decoding and Weight Distribution are hard for W 1] and NP by parametric polynomial-time transformations from the Perfect Code problem in graphs. Concerning lattices, we prove a similar result for 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. This problem is closely related to the Shortest Vector problem that has assumed great importance due to the results of Ajtai on the generation of hard instances. From the W 1]-hardness of Maximum Likelihood Decoding, Weight Distribution and Theta Series, it is concluded by a theorem of Cai and Chen that these problems do not have fully polynomial-time approximation schemes unless the parameterized complexity hierarchy collapses.