On the Zeros of the Askey–Wilson Polynomials, with Applications to Coding Theory

In a symmetric association scheme that is (P and Q)-polynomial, the P and Q eigenmatrices are given by balanced ${}_4 \phi _3 $ Askey–Wilson polynomials. In this paper, the parameters of the Askey–Wilson polynomial are classified so that its zeros are not contained in its spectrum. These results, together with theorems of Biggs and Delsarte, imply the nonexistence of perfect codes and tight designs in the classical association schemes of type $A_N $, $B_N $, $C_N $, $D_N $ and the afl'lne matrix schemes.