  The Learning Parity with Noise (LPN) problem underlines several classic
cryptographic primitives. Researchers have endeavored to demonstrate the
algorithmic difficulty of this problem by attempting to find a reduction from
the decoding problem of linear codes, for which several hardness results exist.
Earlier studies used code smoothing as a technical tool to achieve such
reductions, showing that they are possible for codes with vanishing rate. This
has left open the question of attaining a reduction with positive-rate codes.
Addressing this case, we characterize the efficiency of the reduction in terms
of the parameters of the decoding and LPN problems. As a conclusion, we isolate
the parameter regimes for which a meaningful reduction is possible and the
regimes for which its existence is unlikely.
