Combinatorial properties of codes with degree n

The classification of the very thin codes depends on their degrees. We call the very thin codes of degree 1 the synchronizing codes. By using the synchronizing codes, we are able to localize the position of a coded message through which the decoding of the message passes. Upon this position, the message can be divided into two parts and we can always decode these two parts separately. In this paper, we study the very thin codes of degree n. The combinatoric properties of the synchronizing codes are extended to the codes of degree n⩾2, and in particular, we study the properties of codes with degree 2.