Subspaces and polynomial factorizations over finite fields
Let \(f(x)\in \mathrm{GF}(q) [x]\) be square-free of degree \(n\) over the finite field \(\mathrm{GF}(q)\) and define the Berlekamp subspace as \[ B=\{ b(x): b^q(x) \equiv b(x) \bmod f(x)\}.\] For \(q=p\) a prime, define the Niederreiter subspace as follows. Consider the differential equation \(y^{(p-1)}+ y^p=0\) where \(y^{(n)}\) denotes the \(n\)-th derivative for \(y\in K[x]\). Then for \(f(x)\in K[x]\), \(K= \mathrm{GF}(p)\) square-free, monic, of degree at least one, define the Niederreiter subspace with respect to \(f(x)\) as \[ N= \{ h(x)\in K[x]: \frac{h}{f} \text{ satisfies the equation }y^{(p-1)}+ y^p= 0\}. \] For general field order, the subspace is defined similarly using the notion of a Hasse-Teichmüller derivative. For a given polynomial and field order, the possibilities of intersection of the Berlekamp and Niederreiter subspaces is considered. The notion of a subspace that factors a polynomial \(f(x)\) completely and the notion of ``equivalent'' subspaces is introduced. The Niederreiter and Berlekamp subspaces are shown to be equivalent under a certain group action. All subspaces equivalent to the Niederreiter and Berlekamp subspaces are characterized. The orbit \(M\) under this group action which contains \(B\) and \(N\) is described.
scientific article; zbMATH DE number 750797