Polyadic Codes of Prime Power Length

Abstract

Let \(m \geq 2\) be an integer and let \(G\) be a finite Abelian group of order \(p^n\), where \(p\) is an odd prime and \(n\) is a positive integer. In this paper, we derive necessary and sufficient conditions for the existence of an \(m\)-adic splitting of \(G\), and hence for the existence of polyadic codes (as ideals in an Abelian group algebra) of length \(p^n\). Additionally, we provide an algorithm to construct all \(m\)-adic splittings of \(G\). This work generalizes the results of Ling and Xing \([9]\) and Sharma, Bakshi, and Raka \([14]\).