On Perfect Binary Quadratic Forms

Perfect Forms over Totally Real Number Fields

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronöı and later generalized by Koecher to arbitrary number fields. One knows that up to a natural “change of variables” equivalence, there are only finitely many perfect f...

On the Number of Perfect Binary Quadratic Forms

Enumerating Perfect Forms

A positive definite quadratic form is called perfect, if it is uniquely determined by its arithmetical minimum and the integral vectors attaining it. In this self-contained survey we explain how to enumerate perfect forms in d variables up to arithmetical equivalence and scaling. We put an emphasis on practical issues concerning computer assisted enumerations. For the necessary theory of Vorono...

Binary quasi-perfect linear codes from APN quadratic functions

A mapping f from F2m to itself is almost perfect nonlinear (APN) if its directional derivatives in nonzero directions are all 2-to-1. Let Cf be the binary linear code of length 2 − 1, whose parity check matrix has its j-th column [ π f(π) ] , where π is a primitive element in F2m and j = 0, 1, · · · , 2 − 2. For m ≥ 3 and any quadratic APN function f(x) = ∑m−1 i,j=0 ai,jx 2+2 , ai,j ∈ F2m , it ...

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.