### Some cyclic codes of length ${8p^n}$ over $GF(q)$, where order of $q$ modulo $8p^n$ is $\frac{\phi{(p^n)}}{2}$

#### 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.

**How to Cite this Article:**Jagbir Singh, Sonu Singh, Some cyclic codes of length ${8p^n}$ over $GF(q)$, where order of $q$ modulo $8p^n$ is $\frac{\phi{(p^n)}}{2}$, Journal of Mathematical and Computational Science, Vol 9, No 6 (2019), 654-677 Copyright © 2019 Jagbir Singh, Sonu Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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