| Any cyclic code is a direct sum of the minimal ideals, generated by the primitive idempotents. Thus the problem of determining the primitive idempotents is very important.The primitive idempotent is different from the generating polynomial for a cyclic code, but it can also generate the cyclic code,and the idempotent itself contains much information about the cyclic code. Sometimes,it is easier to find the idempotent than the generating polynomial. Construction of binary idempotents from the cyclotomic cosets is easy. In general, however, we do not have much information about the codes generated. Only in special situations do we know the dimension.In this thesis,suppose than p and q are different odd primes,n ≥ 1 is a positive integer,and q are both a primitive root mod pN and 4.In this case, there are exactly An + 3 different cyclotomic cosets mod 4pn. We have derived all the idempotents in the ring GF(q)[x]/ (x4pn — 1). When we determine the primitive idempotents,we have considered two cases,one is C1 = -C1,the other is C1 = -Ca.We have also discussed the dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length 4pn. |
