Characterization And Classification Of Permutation Groups Of Finite Maximal Prefix Codes

In 1996,Shyr.H.J.proved that each irreducible finite maximal prefix code partially generated an irreducible maximal prefix code and a non-s-prime finite maximal prefix code.In 2007,For every admissible infinite length distribution one can construct a maximal prefix codes whose code words satisfy this length distribution.In 2004,Long Fengshan and Long Fang proved that the set of all maximal prefix codes on alphabet A is a free monoid.In 2010,He Yong and Tang Mei Xia proved that each prefix code is the intersection of two maximal prefix codes.In 2019,Liu Yun and Qu Bingjie prove the relationship between the number of prefix codes and the number of maximal prefix codes.In this paper,X is used to represent a finite maximal prefix code on the alphabet A.On the premise that the calculation formula of the number of finite maximal prefix codes and that it can be represented by a full binary tree,this paper determines whether there is an asynchronous code for the finite maximal prefix code with n=1,…,16,In the case of n ≤6,the maximal prefix codes corresponding to them are synchronization codes.Base on that,we design an algorithm that automatically identify the full binary tree corresponding to the maximal code and calculate its transition state monoid,so that when n>6,the corresponding maximal prefix code has both synchronous code and non-synchronous code.Except for the case of n=11,12,14,the maximal prefix codes corresponding to these three cases are synchronization codes.In addition,we prove the following theorem about the finite maximal prefix code:Let G be a finite permutation group,G(X)(?)G,then if G=Zn,there are |X|≥2n;if G=S3,there are|X|≥9.