On The Orthogonality And Hull Of Cyclic Codes

The intersection of a linear codeand its duality is called the Hull of.A linear code is called complementary dual linear code(abbreviated LCD code)or self-orthogonal code,if the Hull of the linear code is{0}or.LCD linear codes have important applications in side-channel attacks,and have been widely studied by scholars in recent years.As a kind of very important linear codes,cyclic codes have a very special structure,that is,any code word in the cyclic code is still in this code after cyclic shifting.Cyclic codes have a rich algebraic structure and efficient decoding algorithms.They have been widely used in information commu-nication and storage,and have also been studied in depth in theory.In recent years,the work on LCD codes has focused on constructing LCD linear codes and optimal LCD linear codes(i.e.,LCD codes that reach the Singleton bound),but the minimum distance of ordinary LCD linear codes is rarely considered.This article consists of two parts:first,a class of LCD cyclic codes and self-orthogonal cyclic codes with a clear minimum distance are constructed using the corre-lation properties of the partial repair cyclic codes and BCH codes.Secondly,we study the Hull of a cyclic code of a specific length.Specifically,the Hull of all possible dimensions of a cyclic code of length q~m-1(where8)is a power of primes)and the number of cyclic codes with this dimension are determined.This gives all possi-ble dimensions and corresponding numbers of the Hull of the LCD cyclic code and self-orthogonal cyclic code of length q~m-1(where8)is a power of primes).