Invariants of trace forms over finite fields of characteristic 2
Let \(F_2\) be the 2-element field and let \(K=F_{2^k}\) be its extension of degree \(k\). For \(R(x)=\sum _{i=0}^n\,\varepsilon_ix^{2^i}, \epsilon_i\in K,\) the number of zeros \(\text{N}(Q_R^K)\) (in \(K\)) of the quadratic form \(Q_R^K(x)=\text{tr}_{K/F}(xR(x))\) has appeared in many applications. It turns out that \[ \text{N}(Q_R^K)=\tfrac 12(2^k+\Lambda(Q_R^K)\sqrt{2^{k+w}}), \] where \(w\) is the dimension of the radical of \(Q_R^K\) and \(\Lambda(Q_R^K)\in \{-1,0,1\}\) depends on the classification of \(Q_R^K\). However, it is not easy to find \(w\) and \(\Lambda(Q_R^K).\) \textit{A. Klapper} [Des. Codes Cryptography 3, No. 4, 347--377 (1993; Zbl 0780.94006)] did it for \(R\) with a single term. The aim of the paper under review is to do it for \(R(x)=x^{2^a}+x^{2^b}.\) The computation depends on a general reduction result which holds for any \(R(x)\) with all \(\varepsilon_i\in\{0,1\}.\) The author applies the obtained formula to finding cross-correlation of certain binary sequences and to finding the size of the intersection of two conics.
scientific article; zbMATH DE number 5541694