Additionally, we extend some known results on $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets.
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Key Findings & Data:We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting.
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Conclusion & Impact:Additionally, we extend some known results on $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Composite $G$-codes · Group rings · Finite chain rings
Abstract
In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting. We extend the known results on projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings to composite $G$-codes. Additionally, we extend some known results on $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.
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