Explicit lifts of quintic Jacobi sums and period polynomials for $\mathbf {F}_{ {q}}$

Hamming weights in irreducible cyclic codes

“…Gaussian periods of order 5, 6, 8 and 12 are computed in [16] and [14] respectively. So the weight distribution of the code C(r, N) in (1) can be computed by these Gaussian periods and (11).…”

On period polynomials of degree 2 and weight distributions of certain irreducible cyclic codes

“…Gurak [7] obtained similar results for e ∈ {6, 8, 12, 24}; see also [6] for the case s = 2, e ∈ {6, 8, 12}. Hoshi [8] considered the case e = 5. Note that if −1 is a power of p modulo e, then the period polynomials can also be easily obtained.…”

“…Finally, if 4 s, then P * 16 (X) has complex conjugate roots q 8 i, and the other roots are integers (see Table 3). Taking into account the multiplicities given in Table 3, we obtain the desired factorization.…”

Ideal access structures based on a class of minimal linear codes

“…At present, they can be determined for N ∈ {3, 4} and little progress on larger N has been made. According to Lemma 10 in [22] and Theorem 3.2 in [23], we perform extensive computation and get the following lemmas. A c c e p t e d M a n u s c r i p t (ii) If p ≡ 2(mod 5) or p ≡ 3(mod 5), then…”