Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = Fleft[ x right]/< x^{2^n } - 1 > $$ 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.