The connection between quadratic bent-negabent functions and the Kerdock code

In this paper we prove that all bent functions in the Kerdock code, except for the coset of the symmetric quadratic bent function, are bent–negabent. In this direction, we characterize the set of quadratic bent–negabent functions and show some results connecting quadratic bent–negabent functions and the Kerdock code. Further, we note that there are bent–negabent preserving nonsingular transformations outside the well known class of orthogonal ones that might provide additional functions in the bent– negabent set. This is the first time we could identify non-orthogonal (nonsingular) linear transformations that preserve bent–negabent property for a special subset

Semi-Regularity of Pairs of Boolean Polynomials

Semi-regular sequences over $\mathbb{F}_2$ are sequences of homogeneous elements of the algebra $B^{(n)}=\mathbb{F}_2[X_1,...,X_n]/(X_1^2,...,X_n^2)$, which have a given Hilbert series and can be thought of as having as few relations between them as possible. It is believed that most such systems are semi-regular and this property has important consequences for understanding the complexity of Grobner basis algorithms such as F4 and F5 for solving such systems. We investigate the case where the sequence has length two and give an almost complete description of the number of semi-regular sequences for each $n$

Pairs of Quadratic Forms over p-Adic Fields

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$