On equivalency of zero-divisor codes via classifying their idempotent generator

Abstract

In 2009, Ted and Paul Hurley proposed a code construction method using group rings. These codes with single generator are termed group ring codes and in particular zero-divisor codes when using zero-divisors as generators. In this paper, we mainly study the equivalency of zero-divisor codes in $F_2G$ having generator from I(G), the set of all idempotents in $F_2G$. For abelian G, our previous notion of generated idempotents completely classified I(G) by serving as its basis. Here, we first extend the notion of generated idempotents to study and classify some elements in I(G) for non-abelian G. Later, the study is generally done on equivalency of zero-divisor codes in $F_2G$, then concentrating on those with idempotent generator. In particular, we affirm the conjecture “Every group ring code in $F_2D_{2n}$ is equivalent to some in $F_2C_{2n}$” in the cases where the generators are our classified idempotents. We also show that the equivalency of zero-divisor codes in $F_2C_n$ with generated idempotent as generators can be established sufficiently on the generator property.