Exploring Simplex Codes and Their Applications over $\mathbb{Z}_{3}\mathbb{Z}_{6}$

Abstract

This article delves into the investigation of simplex codes of types $\alpha$, $\beta$, and $\gamma$ over the ring $\mathbb{Z}_{3}\mathbb{Z}_{6}$. It examines the fundamental properties of these codes, including their covering radius, association schemes, and practical applications in multi-secret sharing schemes. The covering radius analysis sheds light on the error-correcting capabilities of $\mathbb{Z}_{3}\mathbb{Z}_{6}$-simplex codes, crucial for reliable communication systems. Additionally, association schemes for $\mathbb{Z}_{6}$-simplex codes provide insights into efficient encoding and decoding strategies, enhancing their performance in various applications. Furthermore, the development of a multi-secret sharing scheme based on these codes highlights their versatility beyond traditional error correction, offering promising avenues for secure multi-party communication and data storage. This exploration of simplex codes over $\mathbb{Z}_{3}\mathbb{Z}_{6}$ not only contributes to theoretical coding theory but also opens up new opportunities in practical cryptography, advancing the realm of secure information exchange protocols.