Some constructions and existence conditions for Hermitian self-dual skew codes (Q6564088)

For a finite field \(\mathbb{F}_q,\) and \(\theta\) its automorphism, a ring structure on the set \(R=\mathbb{F}_q[X;\theta]\) can be defined. Let \(a\in\mathbb{F}_q^\ast\) and \(n,\) \(k\) are integers such that \(0 \leq k \leq n.\) A \((\theta, a)\)-constacyclic code or skew constacyclic code \(C\) of length \(n\) is a left \(R\)-submodule \(Rg/R(X^n-a)\subset R/R(X^n-a)\) in the basis \(1, X, \ldots, X^{n-1}\) where \(g\) is a monic skew polynomial dividing \(X^n-a\) on the right in \(R\) with degree \(n-k.\) When \(a=-1\) the code is called \(\theta\)-negacyclic.\N\NHermitian self-dual \(\theta\)-cyclic and \(\theta\)-negacyclic codes over \(\mathbb{F}_{p^2}\) when \(\theta\) is the Frobenius automorphism are studied. Using a system of homogeneous polynomial equations of degree \(p+1\) these codes are described. By delving into the factorization of skew polynomials, the authors prove that there exists no Hermitian self-dual \(\theta\)-cyclic code of any dimension over \(\mathbb{F}_{p^2},\) and a construction as well as an exact formula for the number of Hermitian self-dual \(\theta\)-negacyclic codes is given. A sufficient number of examples are shown, including a quaternary \([68, 34, 18]\) Hermitian self-dual code improving the best-known such codes.\N\NIn this paper, it's also proven that there is no Hermitian self-dual \(\theta\)-cyclic code over any finite field \(\mathbb{F}_e\) with \(e\) even and \(p\) odd and necessary and sufficient conditions for the existence of Hermitian self-dual \(\theta\)-negacyclic codes over \(\mathbb{F}_e\) with \(e>2\) are derived.