构造$\gf_{2^n}$上二次伪平面函数的新方法
A New Approach to Constructing Quadratic Pseudo-Planar Functions over $\gf_{2^n}$
摘要：
有限域上的平面函数产生有限射影平面。它们也被用于构造类似DES的迭代密码、纠错码和码本。它们最初只定义在具有奇特征的有限域中，但最近周引入了具有偶特征的伪平面函数，这产生了类似的应用。所有已知的伪平面函数都是二次的，因此它们给出了presemifields。本文给出了一种构造二次伪平面函数的新方法。然后构造了五个显式的伪平面函数族，其中一个是二项式，两个是三项式，另外两个是四项式。所有已知的伪平面函数的重新审视，其中一些是广义的。这些函数不仅可以导出投影平面、相对差集和预半域，而且可以给出满足Levenstein界的最优码本、互无偏基完备集（MUB）和低相干性的压缩感知矩阵。
英文摘要：
Planar functions over finite fields give rise to finite projective planes. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and codebooks. They were originally defined only in finite fields with odd characteristic, but recently Zhou introduced pesudo-planar functions in even characteristic which yields similar applications. All known pesudo-planar functions are quadratic and hence they give presemifields. In this paper, a new approach to constructing quadratic pseudo-planar functions is given. Then five explicit families of pseudo-planar functions are constructed, one of which is a binomial, two of which are trinomials, and the other two are quadrinomials. All known pesudo-planar functions are revisited, some of which are generalized. These functions not only lead to projective planes, relative difference sets and presemifields, but also give optimal codebooks meeting the Levenstein bound, complete sets of mutually unbiased bases (MUB) and compressed sensing matrices with low coherence.
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