Character sums over Galois rings and primitive polynomials over finite fields

Existence of primitive normal pairs with one prescribed trace over finite fields

“…It is easy to observe that an element in F * q m is (q m − 1)-free if and only if it is primitive. A special case of [17,Lemma 10], provides an interesting result.…”

“…(5, 6), (5, 7), (5,9), (5,11), (5,12), (5,13), (5,14), (5,17), (5,18), (5,19), (5, 21), (5,22), (5,27), (5,30), (5,36), (5 2 , 7), (5 2 , 9), (5 2 , 11), (5 3 , 6), (5 5 , 6).…”

“…36 holds for all (q, m) except (5,24). Next, we refer to Table 2 to note that Theorem 4.1 holds for the pairs (5,11), (5,13), (5,14), (5,15), (5,16), (5,17), (5,18), (5,19), (5, 21), (5,22), (5, 24), (5, 27), (5, 30), (5, 36), (5, 40), (5 2 , 7), (5 2 , 9), (5 2 , 11), (5 3 , 6), (5 3 , 8), (5 5 , 6). Thus, the possible exceptions in the case m q − 1 are only (5, 6),(5, 7), (5,8), (5,9), and (5, 12).…”

Additive cyclic codes over finite commutative chain rings

“…It can be shown directly that the Euclidean orthogonal code of a code is a linear code and that in the case of linear codes over Galois rings it coincides with the dual code, see [21].…”

“…It is because, some chain rings are not free over their coefficient rings and hence we can not define a trace function for some elements over the base ring (Lemma 5.1). Hence one can not make use of the equivalence of Euclidean orthogonality and duality via the trace map (see [21,Lemma 6]). Thus we use the general idea of duality, constructed via annihilators of characters (see [24]).…”