A Simple Proof of the Main Inequalities for Fundamental Parameters of Codes in Polynomial Association Schemes

$P$- and $Q$-polynomial association schemes are considered. A simple proof of seven main inequalities for such code parameters as the minimum distance, the dual distance, the number of distances, the external distance, and the covering radius is given. This proof is based on using the annihilating and dual-annihilating polynomials for codes and orthogonality conditions for polynomial systems which are adjacent to systems $P$ and $Q$. All of these inequalities, some of which are new, are attained in some cases.