Research On The Construction Of MDS Self-dual Codes And Subfields Codes

Since error-correcting codes can effectively reduce the error rate of information in the transmission process,improving the reliability of the communication system,error-correcting codes has an important position in coding theory.At present,with the continuous in-depth research on the theory of error-correcting codes,error-correcting codes in finite fields have been widely used in military,economic,political and other fields.MDS self-dual codes are a kind of optimal linear codes,which are closely related to the fields of cryptography,quantum communication and combinatorial design.Therefore,constructing MDS self-dual codes with optimal parameters is a key research content of current coding theory.At present,the main mathematical tools for constructing MDS self-dual code applications include elliptic curve,combination design,coding theory,etc.This thesis mainly uses the cyclotomic number to construct MDS self-dual codes.The weight distribution of the subfield codes can reflect the error correction and error detection capabilities of the codes,so the weight distribution has important research value.But the calculation of the weight distribution is closely related to the exponential sum on the finite fields.Generally speaking,it is very difficult to give the exact value of the exponential sum in finite fields,this thesis calculates the weight distribution of two types of subfield codes.In this thesis,we mainly focus on the constructions of MDS self-dual codes and the weight distribution of subfield codes.The specific content of this thesis as follows:(1)Based on the generalized Reed-Solomon codes and extended generalized Reed-Solomon codes over the finite fields,using 6th and 8th order cyclotomic numbers as tools,the conditions for the existence of MDS self-dual codes are given.(2)By calculating the exact values of the sum of several types of exponents,the weight distributions of the two types of subfield codes on the permutation polynomials f₁(x)=x⁴ and f₂(x)=x⁶+x⁴+x² are determined,and the dual codes of them are proved to be the optimal with respect to the sphere-packing bound.