Primitive polynomial with three coefficients prescribed

Primitive Polynomials With Prescribed Second Coefficient

“…We string the 82 pairs (a, c), one example for each primitive quartic (2,14), (3,5), (4,35), (5,14), (6,24), (7,20), (8,34), (9,14), (10, 2), (11,2), (12,8), (13,15), (14,6), (15,14), (16,80), (17,6), (18,32), (19,19), (20,43), (21,15), (22, 32), (23,32), (24,8), (25,62) In summary, the above examples suffice to complete the proof (for odd q) of Theorem 1.2 for n = 4, m = 2 and a = 0.…”

“…The papers of Han [16] and Cohen and Mills [9] cover most cases with m = 2 and n ≥ 5 (although the situation when q is even and n = 5 or 6 is not altogether clear). For m = 3, the conjecture holds provided n ≥ 7 by [13], [14], [23] and [8]. It has to be said, however, that, when m = 2 or 3, some of these items dealt with the stronger requirement that the first m coefficients are prescribed and significant computer verification in a large (though finite) number of cases was necessary to resolve these questions, particularly when 5 ≤ n ≤ 7.…”

A construction of primitive polynomials over finite fields

“…Primitive polynomials over the binary field, F 2 , have received particular attention, due to their use in the generation of linear recurring sequences widely employed in testing, coding theory, cryptography, communication systems, and many other areas of electrical engineering [11,12,13]. There have appeared a number of recent results about primitive polynomials, for instace, dealing with the existence of primitive polynomials with prescribed coefficients [14,15,16].…”