# Coding theory and complete problems

Coding theory is an useful topic in theoretical computer science. There are known examples of problems coming from coding theory which turn out to be NP complete.

My questions are the following:

$(1)$ Are there natural good examples of coding theory problems that lie in different levels of the polynomial hierarchy above NP and are complete for those levels?

$(2)$ Are there natural good examples of coding theory problems that are PSPACE complete? (Note that a coding theory approach is used to show IP=PSPACE in http://www.math.ias.edu/~ormeir/papers/ip_pspace.pdf)

In subsection "Coding and cryptology" (see p. 24), two $\Pi_2^p$-complete problems appear: checking an upper bound on the covering radius of a linear code, and deciding whether a linear code is $r$-identifying ($r$ part of the input). A certain $\Sigma_3^p$-complete problem called "Minimum block decoder" can be found on p. 28 (see also $\Sigma_2^p$-complete "Minimal block encoder and decoder", p. 25).