# Linked Coupling Faults Detection by Multirun March Tests
## 1 Introduction
Thus, each cell of a single \(C\)_F_ has only one role: the aggressor (\(A\)) or victim (\(V\)). Due to the tremendous capacity of modern memory devices, tests are currently widely used. They are still being developed with a time complexity that depends linearly on the memory capacity. Such tests are called march tests [1; 3; 5; 6; 7; 8; 9] and are used for bit-oriented memory, where each cell stores one bit. A march test consists of a finite number of march elements [3]. Each march element contains a symbol that determines the order of formation of the address sequence of the cells of the memory, determining the order in which the cells are accessed. The symbol \(\Uparrow\) indicates a sequential enumeration of memory addresses in increasing order ( forward), and the symbol \(\Downarrow\) indicates decreasing order ( in reverse order). March elements contain a sequence of read (r) and write (\(w\)) operations, enclosed in parentheses and separated by semicolons. The transition to the next cell, according to the address sequence, is conducted only after performing all the operations with the current cell according to the march element [1; 3; 6; 7]. For example, the simplest march element used in many tests has the form {\(\Uparrow(r0,w1)\)}. In accordance with this element and the address sequence, 0 is read from each cell, and 1 is written. The read operation compares the read value with the expected value (0). A flowchart of this march element is presented in Figure 1.  
The detection of faults in memory, as a result of the march test, is based on receiving incorrect values read from its cells that are different from the reference ones. Thus, a faulty memory state is fixed, and a diagnostic procedure is performed if necessary [3; 8; 9]. Detecting single and multiple unlinked coupling faults within the march tests framework is considered solved [3]. Effective tests have proven to detect them in practice [3; 6; 7; 11]. March tests that detect coupling faults assume a one-run memory test with a standard initial state of storage cells, usually zero, and a standard counter address sequence [3]. The problem of detecting linked coupling faults, _LCFs_, using one-run march tests remains practically open, requiring further research due to the variety and complexity of the manifestation mechanisms of such faults [6; 7; 8; 9]. This article proposes a formal model of linked coupling faults, describing the roles of all cells involved in the fault and the scenario of their manifestation regarding the march test and its address sequence. The necessary and sufficient conditions for detecting such faults within the framework of one-run tests are determined, and the efficiency of their detection using multiple tests is evaluated. In the era of rapidly advancing computing technologies and the ever-increasing importance of data integrity, our research plays an important role in ensuring the reliability of memory devices. The formal model and conditions we propose for detecting complex linked coupling faults have significant implications in various computing domains, including big data analytics, where memory errors can have cascading effects on data processing and reliability. Moreover, the ability to detect and address such faults is of paramount importance in networked systems, cloud computing infrastructure, and data storage, where data integrity and availability are critical. Our results not only contribute to the field of memory testing but also contribute to the overall robustness and reliability of computing systems. One well-known march test is the MATS+ written as:  
\[\{\hat{\Updownarrow}(w0);\quad\hat{\Updownarrow}(r0,w1);\quad\Downarrow(r1,w0)\}. \tag{1}\]  
\(M0\)\(M1\)\(M2\)  
It has three march elements \(M0:\hat{\Updownarrow}(w0)\), \(M1:\hat{\Updownarrow}(r0,w1)\), and \(M2:\Downarrow(r1,w0)\). Algorithm 1 presents the interpretation of this notation in pseudocode.