  In this work, we consider adaptive linear programming (ALP) decoding of
linear codes over the finite field $\mathbb{F}_p$ of size $p$ where $p$ is a
prime. In particular, we provide a general construction of valid inequalities
for the codeword polytope of the so-called constant-weight embedding of a
single parity-check (SPC) code over any prime field. The construction is based
on classes of building blocks that are assembled to form the left-hand side of
an inequality according to several rules. In the case of almost
doubly-symmetric valid classes we prove that the resulting inequalities are all
facet-defining, while we conjecture this to be true if and only if the class is
valid and symmetric. For $p=3$, there is only a single valid symmetric class
and we prove that the resulting inequalities together with the so-called
simplex constraints give a completely and irredundant description of the
codeword polytope of the embedded SPC code. For $p>5$, we show that there are
additional facets beyond those from the proposed construction. We use these
inequalities to develop an efficient (relaxed) ALP decoder for general
(non-SPC) linear codes over prime fields. The key ingredient is an efficient
separation algorithm based on the principle of dynamic programming.
Furthermore, we construct a decoder for linear codes over arbitrary fields
$\mathbb{F}_q$ with $q=p^m$ and $m>1$ by a factor graph representation that
reduces to several instances of the case $m=1$, which results, in general, in a
relaxation of the original decoding polytope. Finally, we present an efficient
cut-generating algorithm to search for redundant parity-checks to further
improve the performance towards maximum-likelihood decoding for short-to-medium
block lengths. Numerical experiments confirm that our new decoder is very
efficient compared to a static LP decoder for various field sizes, check-node
degrees, and block lengths.
