A New Class Of Cyclic Codes Related To Reed-Muller Codes

With the popularization of intelligent devices,the rapid growth of the network bandwidth and the rapid development of the new network technology.The network communication has become the main body of the data communication system,and the increasing of data also makes us step into the era of big data.As a channel coding,Reed-Muller(RM)codes is mostly used in wireless communication applications,especially in deep space communication.Because of its good theoretical and mathematical properties,it has also been widely studied in theoretical computer science.The famous Barnes-Wall(BW)lattices can be obtained by applying Construction D to a chain of RM codes.By applying Construction D(cyc)to a chain of extended cyclic codes sandwiched between RM codes,Hu and Nebe(J.London Math.Soc.(2)101(2020)10681089)constructed new series of universally strongly perfect lattices sandwiched between BW lattices.This thesis will study the cyclic codes sandwiched between RM codes.Hu and Nebe obtained a new code between the RM codes by dividing the zeros of the RM codes more finely.The lower bound of the minimum distance of the new codes can be obtained by applying the BCH bound.The main research purpose of this thesis is to determine the minimum distance of the codewords of the new codes and all the minimum weight codewords of the new codes for some special cases.We mainly use some of the methods in Charpin(Handbook of coding theory,1998,1:963-1063),and we adopt her methods to completely determine the minimum weight codewords of the new codes.The main tools we use are group algebra,Newton identitity,affine polynomial and affine invariant codes.Recently,Kudekar et al.showed that a sequence of affine-invariant codes of increasing length,rates converging to r∈(0,1),achieve capacity on the BEC(binary erasure channel)under bit-MAP decoding.The new code offers a new capacity-achieving sequence of cyclic codes over the erasure channel.