The random coding bound is tight for the average linear code

In 1973, Gallager published a correspondence, consisting a proof that the random coding bound is exponentially tight for the random code ensemble at all rates, even below the expurgation rate. This result explained that the random coding upper bound does not achieve the expurgation bound due to the properties of the random ensemble, irrespective of the utilized bounding technique. It has been conjectured that this same behavior holds true, also for a random linear ensemble. This conjecture is proved in this paper. Additionally, it is shown that exponential tightness for both ensembles can be achieved by considering only the triple-wise error events.