Some Properties of Extractable Codes and Insertable Codes

This paper deals with insertability and mainly extractablity of codes. A code C is called insertable (or extractable) if the free submonoid C^* generated by C satisfies if z, $xy\in C^{*}$ implies $x\mathrm{zy}\in C^{*}$ (or z, $x\mathrm{zy}\in C^{*}$ implies $xy\in C^{*}$). We show that a finite insertable code is a full uniform code. On the other hand there are many finite extractable codes which are not full uniform codes. We cannot still characterize the structures of infinite extractable codes. Here we give some results on the class of infix extractable codes. First, we consider a necessary and sufficient condition whether a given infix code C is extractable or not by using the syntactic graph of the code. Secondly, we investigate the extractability for the families of other related bifix codes. We newly define the bifix codes, called e(m)-codes and $\overline{e}$-codes, and refer to the extractability of them.