Abstract
LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 n (n>1) be ϕ(2 n )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m−3, then the ring $$R_{2^n } = F\left[ x \right]/ $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 n generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n−1 primitive idempotents ofR 2 n are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n−1 idempotents are also obtained. The casen=2, 3 is dealt separately.
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