I. Introduction
Let , where is an odd prime power. Let be the finite field with elements. An linear code over , denoted by , can be seen as a -dimensional subspace of with minimum distance . There are many tradeoffs between , and for a linear code. One of the most interesting tradeoffs is so-called the Singleton bound, which yields . When the equation is established, i.e., , is called an maximum distance separable (MDS) code. The Euclidean inner product of two vectors and in is defined by \mathbf {x}\cdot \mathbf {y}=\sum _{i=1}^{n} x_{i}y_{i}.
With respect to Euclidean inner product, we define the dual code of as
{\mathcal {C}}^{\bot }=\{\mathbf {x}\in {\mathbb {F}} _{q}^{n}: \mathbf {x}\cdot \mathbf {c}=0, \forall \mathbf {c}\in {\mathcal {C}} \}.
If , we call (Euclidean) self-orthogonal. Some constructions of MDS self-orthogonal codes were given and the reader may refer to [7] and [12]. If , we call (Euclidean) self-dual. In particular, if is MDS and (Euclidean) self-dual, is called an MDS (Euclidean) self-dual code. Clearly, the dimension and the minimal distance of an MDS self-dual code are uniquely determined by its length . Furthermore, the length of a self-dual code can only be even.