Minimal cyclic codes of length $$8p^{n}$$ 8 p n over $$GF(q)$$ G F ( q ) , where $$q$$ q is prime power of the form $$8k+5$$ 8 k + 5

Abstract

The explicit expression for the $$8n+6$$ primitive idempotents in $$FG$$ (the group algebra of the cyclic group $$G$$ of order $$8p^{n}$$ , where $$p$$ is an odd prime, $$n\ge 1)$$ over the finite field $$F$$ of prime power order $$q$$ , where $$q$$ is of the form $$8k+5$$ and is a primitive root modulo $$p^{n}$$ are obtained. The minimum distances, dimensions and the generating polynomials of the minimal cyclic codes generated by these primitive idempotents are also obtained.