Linear codes with arbitrary dimensional hull and pure LCD code

In this paper, we introduce a general construction of linear codes with small dimension hull from any non LCD codes. Furthermore, we show that for any linear code $\text{Co}$ over ${\text{F}}_q$ ($q>3$) with $dim(Hull(\text{Co}))=h$ there exist an equivalent codes ${\text{Co}}_j$ with $dim(Hull({\text{Co}}_j))=j$ for any integer $0\le j\le h$. We also introduce the notion of pure LCD code; an LCD code and all its equivalent are LCD; and construct an infinite family of pure LCD codes. In addition, we introduce a general construction of linear codes with one dimension hull.