Two codes related to Hermitian forms (Q1868696)

Let \(\mathbb{F}_t\) denote the finite field with \(t\) elements and of characteristic \(p\). The classical Hermitian form theory on \(\mathbb{C}\) can be written on \(\mathbb{F}_{t^2}\) with the involution \(x\mapsto x^t\) instead of the application \(z\mapsto\overline z\). By considering the quadratic Hermitian forms on \(\mathbb{F}^N_{t^2}\) and on \(\mathbb{F}^{2N}_t\), the author constructs two linear codes using the same method as Reed-Muller codes. The aim of this paper is to consider such two trace-codes, to compute their parameters and to compare these to the classical Reed-Muller case.