A construction of self-dual skew cyclic and negacyclic codes of length \(n\) over \(\mathbb{F}_{p^n}\) (Q2232214)

We say that a code \(C\) with generator matrix \(G\) is isodual if it is equivalent to its dual. A linear code \(C\) of length \(n\) and generator matrix \(G\) is a \((\theta, \nu)\)-isodual code if \(G\cdot D\) is a generator matrix of \(C^\bot\) where \(D\) is the \(n\times n\) diagonal matrix with diagonal coefficients \(\nu, \theta(\nu),\ldots, \theta^{n-1}(\nu)\) where \(\nu\in\mathbb{F}_q^\ast,\) \(\theta\in\text{Aut}(\mathbb{F}_q)\) and \(n\) is a natural number. The authors develop the notion of \((\theta, \nu)\)-isodual codes, where \(\theta\) is the Frobenius automorphism in \(\mathbb{F}_{q}\) and \(\nu\in\mathbb{F}_{q}^\ast.\) These codes form a subfamily of the family of isodual codes and the \((\theta, \nu)\)-isodual \(\theta\)-cyclic and \(\theta\)-negacyclic codes are characterised thanks to an equation satisfied by the skew check polynomials of the codes. The special case when \(q=p^n\) where \(p\) is a prime number and \(\theta\) is the Frobenius automorphism over \(\mathbb{F}_{p^n}\) is studied. A necessary and sufficient condition for the existence of \((\theta, \nu)\)-isodual as well as a construction and an enumeration formula for self-dual \(\theta\)-cyclic and \(\theta\)-negacyclic codes of length \(n\) over \(\mathbb{F}_{p^n}\) are derived. In the last section, a subclass of self-dual \(\theta\)-cyclic codes over \(\mathbb{F}_{p^n}\) which are self-dual Gabidulin codes are parametrize by a parameter which satisfies a polynomial system. For the entire collection see [Zbl 1470.11003].