  New lower and upper bounds on the reliability function of typewriter channels
are given. Our lower bounds improve upon the (multiletter) expurgated bound of
Gallager, furnishing a new and simple counterexample to a conjecture made in
1967 by Shannon, Gallager and Berlekamp on its tightness. The only other known
counterexample is due to Katsman, Tsfasman and Vl\u{a}du\c{t} who used
algebraic-geometric codes on a $q$-ary symmetric channels, $q\geq 49$. Here we
prove, by introducing dependence between codewords of a random ensemble, that
the conjecture is false even for a typewriter channel with $q=4$ inputs. In the
process, we also demonstrate that Lov\'asz's proof of the capacity of the
pentagon was implicitly contained (but unnoticed!) in the works of Jelinek and
Gallager on the expurgated bound done at least ten years before Lov\'asz. In
the opposite direction, new upper bounds on the reliability function are
derived for channels with an odd number of inputs by using an adaptation of
Delsarte's linear programming bound. First we derive a bound based on the
minimum distance, which combines Lov\'asz's construction for bounding the graph
capacity with the McEliece-Rodemich-Rumsey-Welch construction for bounding the
minimum distance of codes in the Hamming space. Then, for the particular case
of cross-over probability $1/2$, we derive an improved bound by also using the
method of Kalai and Linial to study the spectrum distribution of codes.
