Quadratic Residue Codes Over Z_2^K

Many typical nonlinear binary codes,such as Nordstrom-Robinson,Kerdock,Preparata,Goethals and Delsarte-Goethals,have ex-cellent error-correcting capabilities .These codes can be mapped intolinear codes over Z4 by Gray map,which generates a lot of work on Z4-cyclic codes.Quadratic residue(QR) codes are a class of cyclic codeswell defined by their idempotent generators,and the minimum dis-tances of binary QR codes are quite high for the codes'length,anddecoding is comparatively easy.Although some properties of cycliccodes and QR codes over Z4 or Z8 are studied in some reference litera-tures,QR codes over general ring Z_2^k have not been mentioned mainlybecause of adding of a parameter k.In this paper,we define QR codesover Z_2^k,and show the existence of Z_2^k-QR codes.Then we show thatthey have many good properties which are analogous in many respectsto properties of QR codes over a field,and get an important automor-phism group of extended QR codes,and generalize some results con-tained in the reference literatures about QR codes over Z4 and Z8.Firstly,by judging the solution of congruence equations,we showthere are only 4 Z_2^k-QR codes,and making use of their relationship,wealso show that they have properties analogous to those of QR codesover a field.Secondly,according to the properties of QR codes,it is shown thatthe extended QR codes with the same length are equivalent with eachother,and an important automorphism group is given.