Some cyclic codes of length 8p^n over GF(q), where order of q modulo 8p^n is φ(p^n)/2

Abstract

Let $G$ be a finite group and $F$ be finite field of prime power order $q$ (of type $8k+5$) and order of $q$ modulo $8p^n$ is $\frac{\phi(p^n)}{2}$. If $p$ is prime of type $4k+1$, then the semi-simple ring $R_{8p^n}\equiv \frac{GF(q)[x]}{}$ has $16n+6$ primitive idempotents and for $p$ of type $4k+3$, then $R_{8p^n}$ has $12n+6$ primitive idempotents. The explicit expression for these idempotents are obtained, the generating polynomials and minimum distance bounds for cyclic codes are also completely described.