Linear Codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$ and their Applications

In the paper, we explore the simplex and MacDonald codes over the finite ring $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$. Our investigation focuses on the unique properties of these codes, with the particular attention to their weight distributions and Gray images. The weight distribution is a crucial aspect as it provides insights into the error-detection and error-correction capabilities of the codes. Gray images play a significant role in understanding the structure and behavior of these codes. By examining the dual Gray images of simplex and MacDonald codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$, we aim to develop efficient secret sharing schemes. These schemes benefit from the inherent properties of the codes, such as minimal weight and redundancy, which are essential for secure and reliable information sharing. Understanding the access structure of these schemes is vital, as it determines which subsets of participants can reconstruct the secret. Our study draws on various properties to elucidate this access structure, ensuring that the schemes are secure and efficient. Through this comprehensive analysis, we contribute to the field of coding theory by demonstrating how simplex and MacDonald codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$ can be effectively utilized in cryptographic applications, particularly in designing robust and reliable secret sharing mechanisms.