Linear and group identifying codes in Hamming Graphs

TL;DR

This paper introduces and analyzes group and linear identifying codes in Hamming graphs, establishing bounds and confirming conjectures about their minimal sizes for fault detection and system reliability.

Contribution

It defines group identifying codes, establishes bounds on their sizes in Hamming graphs, and confirms conjectures for linear and group codes in specific Hamming cube cases.

Findings

Established bounds on smallest group identifying codes in Hamming graphs.

Determined minimal sizes of linear identifying codes in prime power Hamming graphs.

Confirmed conjecture for group and linear codes in specific Hamming cube cases.

Abstract

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 \( K_{m_1} \square K_{m_2} \square \dots \square K_{m_n} \). Additionally, we determine the smallest size of a linear identifying code in \( K_p^n \) for a prime \( p \) and \( n \geq 2 \). In [1], it was hypothesized that \( \gamma^{ID}(K_m^3) = m^2 \) for an integer \( m \geq 2 \). Although this conjecture was disproven in [2], we demonstrate that group identifying codes in \( K_m^3 \) for an integer \( m \geq 2 \) and linear identifying codes in \( K_p^3 \) for a prime \( p \) indeed fulfill this conjecture.