Hulls Of Simple-root Cyclic Codes Over The Residue Class Ring Z_p~m

The hull of a linear code is the intersection of the code and its dual.The hulls of linear codes over finite fields,which were provided in 1990,have been widely used in the design of some algorithms in coding theory.Cyclic codes are one of the most important subclasses of linear codes.Such codes have good algebraic structures making them easier to encode and decode.In addition,they can reduce the error rate of various communication systems to improve communication quality.Thus,more and more researchers pay attention to the study of cyclic codes.Therefore,some researchers have studied properties of the hulls of cyclic codes over finite fields,especially the hull dimensions,and have achieved some results.In recent decades,interest has been shown in cyclic codes over finite rings.The finite commutative chain rings,as a class of the important finite rings,have been extensively studied.In general,if the characteristic of the finite commutative chain ring is relatively prime to the code length of a cyclic code,we call this code a simple-root cyclic code;otherwise it is called a repeated-root cyclic code.So far,many researchers have devoted themselves to the study of simple-root cyclic codes over finite commutative chain rings,and have obtained many influential results.In recent research,some researchers have studied the hulls of cyclic codes over some special finite commutative chain rings.In 2019,Jitman et al.studied the structures and related properties of the hulls of simple-root cyclic codes over the ring Z4.In this paper,we study the hulls of simple-root cyclic codes over the residue class ring Zpm,where p is an arbitrary prime.And as follows the details are given.Firstly,we give the generator polynomials and the sizes of the hulls of simple-root cyclic codes of length n over the ring Zpm,according to the generator polynomials and the sizes of these simple-root cyclic codes and their dual codes.Secondly,the p-dimension expression with parameters of the hull of a simple-root cyclic code of length n over the ring Zpm is given.At the same time,an algorithm for computing all possible values of this p-dimension is provided according to its expression and the parameter value ranges.Finally,let the p-dimension of the hull of a simple-root cyclic code of length n over the ring Zpm be the random variable.And a general formula for the average p-dimension of the hulls of simple-root cyclic codes of length n over the ring Zpm has been provided.Furthermore,the upper and lower bounds of this average p-dimension are given by the related conclusions of the Np-factorization of a positive integer.