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 Fqⁿ, with a specified degree n and two given coefficients over the finite field Fq. 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 Fq 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 $3988$.