It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called "quadratic Fourier analysis", we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L.
On error linear complexity of new generalized cyclotomic binary sequences of period $p^2$
本文研究了Z. Xiao等人，他计算了序列的线性复杂度（Designs，Codes and Cryptography，2017，https：//doi.org/10.1007/s10623-017-0408-7）。更确切地说，我们根据费马行列式理论，确定了几乎$k>0$的$\mathbb{F}_2$上的$k$ -误差线性复杂度的值。结果表明，该序列具有良好的稳定性。
