I. Introduction
Let be the finite field with elements, where is a prime power. Let be the multiplicative group of nonzero elements of . An linear code over is a -dimensional subspace of with minimum distance . The Singleton bound states that . If , then is called a maximum distance separable (MDS) code. MDS codes achieve optimal parameters that allow correction of maximal number of errors. For two vectors , define the Euclidean inner product . The dual code of is defined by \begin{equation*} \mathcal {C}^{\perp }=\left \{{\boldsymbol {x} \in \mathbb {F}_{q}^{n}:(\boldsymbol {x}, \boldsymbol {y})=0, \forall \boldsymbol {y} \in \mathcal {C}}\right \}.\end{equation*} The code is called self-dual if . In particular, if is MDS and self-dual, is called an MDS self-dual code.