The Patterson–Wiedemann (PW) construction, which is defined for an odd number n of variables with n being the product of two distinct prime numbers p and q, can be interpreted as idempotent functions which are represented by the (d, r)-interleaved sequences formed by all-zero and all-one columns, where \(r=(2^p-1)(2^q-1)\) and \(d=\frac{(2^n-1)}{r}\). We here study a modified form of the PW construction, which only requires \(2^n-1\) \((= dr)\) be a composite number, by relaxing the constraint on the values of d and r. We first elaborate on the case \(n=15\) and consider the functions corresponding to the (217, 151)-interleaved sequences. Taking into account those satisfying \(f(\alpha ) = f(\alpha ^{2^k})\) for all \(\alpha \in \mathbb {F}_{2^{n}}\) in this scenario, where k is a fixed divisor of n, we obtain Boolean functions with nonlinearity 16268 exceeding the bent concatenation bound. Then we extend our study for the case \(n=11\) and obtain Boolean functions with nonlinearity 996 represented by the (89, 23)-interleaved sequences, which equals the best known nonlinearity result. In the process, we show that there is the possibility to exceed the best known nonlinearities using the functions corresponding to those interleaved sequences.

Patterson–Wiedemann (PW) 结构定义为奇数n个变量，n是两个不同质数p和q的乘积，可以解释为幂等函数，由 ( d ,  r )-由全零列和全一列形成的交错序列，其中\(r=(2^p-1)(2^q-1)\)和\(d=\frac{(2^n-1)} {r}\)。我们在这里研究 PW 构造的一种修改形式，它只需要\(2^n-1\) \((= dr)\)是一个合数，通过放宽对d和r值的约束. 我们首先详细说明\(n=15\) 的情况，并考虑对应于 (217, 151)-交错序列的函数。考虑到那些满足\(f(\alpha ) = f(\alpha ^{2^k})\)的所有\(\alpha \in \mathbb {F}_{2^{n}}\)在在这种情况下，其中k是n的固定除数，我们获得了非线性度为 16268 的布尔函数，超过了弯曲的串联界限。然后我们扩展我们对案例\(n=11\)的研究并获得非线性度为 996 的布尔函数，由 (89, 23)-交错序列表示，这等于最知名的非线性结果。在此过程中，我们表明有可能使用与那些交错序列对应的函数来超越最知名的非线性。