Linear and group identifying codes in Hamming Graphs

Codes are crucial in many areas of applications. Different types of codes are designed to meet specific needs, which makes them more effective and useful. Linear codes are extensively used in data storage systems. Identifying codes are essential for locating malfunctioning processors. To combine these benefits, researchers have looked into a type of code called linear identifying codes. These codes blend the error-correction abilities of linear codes with the fault-finding capabilities of identifying codes. Group codes are also highly regarded for their strong properties and reliable decoding methods. In our work, we introduce a new type of identifying code called group Identifying codes. These codes aim to bring together the best features of both Identifying codes and group codes, offering enhanced performance in fault detection and system reliability. In this paper, we establish limits on the smallest size of a group identifying code when $G$ is an $n$-dimensional Hamming cube $K_{m_1}\mathrm{\square }{K}_{m_2}\mathrm{\square }\dots \mathrm{\square }{K}_{m_n}$. Additionally, we determine the smallest size of a linear identifying code in $K_p^n$ for a prime $p$ and $n\ge 2$. In [1], it was hypothesized that ${\gamma }^{ID}(K_m^3)=m^2$ for an integer $m\ge 2$. Although this conjecture was disproven in [2], we demonstrate that group identifying codes in $K_m^3$ for an integer $m\ge 2$ and linear identifying codes in $K_p^3$ for a prime $p$ indeed fulfill this conjecture.