Applications Of Gr(?)bner Bases In Coding Theory

This paper focuses on the application of the theory of Gr(?)bner bases to the minimum distance of linear codes.The minimum distance reflects the ability of error detection and error correction of linear codes,and it is an important parameter of linear codes.In Chapter 3,a more efficient method to solve the minimum distance of linear codes is proposed based on the algebraic coding theory and the Gr(?)bner bases theory.The original method is to find the minimum distance of the linear code by calculating the ideal zero set generated by the code word.However,in the case of large code length,this method is more complex and time-consuming.The new method transforms the ideal set of zeros in the original method into solving its Gr(?)bner bases,which solves some complicated problems in the calculation process of the original method,and the calculation speed is faster than that of the original method.In Chapter 4 and Chapter 5 of this paper,cyclic codes and quasi-cyclic codes in linear codes are introduced respectively,and the minimum distance can also be determined by linear codes.In these two chapters,other methods to determine the minimum distance of cyclic codes and quasi-cyclic codes based on Gr(?)bner bases are also given.