On Cyclic Codes with Length 2 p e over Finite Fields

Cyclic codes are an important class of linear codes. Cyclic codes are applied in data storage, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. We utilize the cyclotomy of order 2 to study the cyclic codes with length n = 2 p e and dimension k = n /2 = p e . The cyclic codes from our construction are either optimal or almost optimal among all cyclic codes with the same length and dimension. We get the enumeration of these cyclic codes and study the hull of cyclic codes of length n over F q . We obtain the range of l = dim(Hull( C )). Finally, we construct and enumerate cyclic codes of length n having hull of given dimension.