Narrow Results

In this paper, we discuss the relationship between the generation polynomial of a linear sequence and its decimation sequence over the finite field 𝔽₂. Then we propose an upper bound of the number of terms in the generation polynomial of a decimation sequence of a linear sequence whose generation polynomial is trinomial. Finally, we suggest a method to calculate the terms of a decimation polynomial and their number directly, which can be used in the construction of irreducible polynomials with a controlled number of terms, when the source trinomial and the decimation distance satisfy a certain condition.

We use the Smith normal form of the augmented degree matrix to estimate the number of rational points on a toric hypersurface over a finite field. This is the continuation of a previous work by Cao in 2009.

Let 𝔽_q be the finite field of q elements and f be a nonzero polynomial over 𝔽_q. For each b ϵ 𝔽_q, let N_q(f = b) denote the number of 𝔽_q-rational points on the affine hypersurface f = b. We obtain the formula of N_q(f = b) for a class of hypersurfaces over 𝔽_q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.