Regular Representations of Codes and Duality

Let K⁄F be finite fields with [K:F]=m. Let {u 1,…,um } be a basis of this extension. By the regular representation with this basis, every element of K corresponds to an m×m matrix over F. Applying this representation, every linear code over K induces a linear code over F. Another code may be induced by another basis. We study the condition of the basis which preserves duality, and also we show the existence of such a basis for any extension.