On r-primitive k-normal polynomials with two prescribed coefficients

This article investigates the existence of an r-primitive k-normal polynomial, defined as the minimal polynomial of an r-primitive k-normal element in Fₐ䂞, with a specified degree n and two given coefficients over the finite field Fₐ. Here, q represents an odd prime power, and n is an integer. The article establishes a sufficient condition to ensure the existence of such a polynomial. Using this condition, it is demonstrated that a 2-primitive 2-normal polynomial of degree n always exists over Fₐ when both q 11 and n 15. However, for the range 10 n 14, uncertainty remains regarding the existence of such a polynomial for 71 specific pairs of (q, n). Moreover, when q<11, the number of uncertain pairs reduces to 16. Furthermore, for the case of n=9, extensive computational power is employed using SageMath software, and it is found that the count of such uncertain pairs is reduced to 3988.