# Linked Coupling Faults Detection by Multirun March Tests
## 3 Necessary and Sufficient Conditions for \(LCF\) Detection  
The problem of detecting single and multiple \(CF\)s has long been the primary stimulus for developing new match tests [3]. The widely known March C and March C- tests, developed to detect all kinds of single \(CFid\) faults, are very effective in detecting faulty memory states and their low time complexity [3,7]. Within the framework of one-run march tests, the March Mest was developed, in which the conditions for detecting coupling faults were implemented for the first time, including double coupling faults of various configurations [13]. More efficient tests were proposed by ascertaining the presence of undetectable \(LCF\)s using one-run march tests, such as the March LR and March LA [13,14]. These tests provided conditions for detecting realistic \(LCF\)s within the considered models of linked faults and determining the technological and design features of the memory under testing [13,14]. Various modifications of the March LR and March LA tests are effectively used to detect dynamic unlinked faults [15], test dynamic memory (DRAM) [16], and implement memory built-in self-tests (BISTs) [17,18]. Specific march tests, such as March SS, March BLC-Opt, March SR, March MSS, and March m-MSS, are applied to cover bit-line coupling faults [19,20,21] and their diagnoses by applying March Cd [22]. The required sets of march elements and their sequence in the testing were determined as conditions for detecting various types of \(LCF\)s, ensuring the detection of all faults of the considered type [3]. For example, the condition for detecting all single \(CFid\)s and all linked double \(LCFid\)s is the presence in the test of the following marching elements \(\Uparrow\{ra,w\overline{a},\ldots,wa,\ldots\};\Uparrow\{r\overline{a},wa, \ldots,w\overline{a},\ldots\};\Downarrow\{ra,w\overline{a},\ldots,wa,\ldots\}\). In turn, the conditions for the presence of the necessary march elements in the test were formulated based on the conditions for activating a particular fault and the conditions for its detection. Similar conditions apply to the memory cells directly involved in the fault and describe the sequence of actions necessary to detect the fault. For the simplest \(CF\) of the form \(\wedge\langle<\uparrow\rangle\partial>\), where cell \(a_{i}\) is the aggressor (\(\uparrow\)) and cell \(a_{j}\) is the victim (0), the following conditions must be fulfilled sequentially. First, the initial setting of both cells to the initial state is necessary: cell \(a_{i}\) to State 0 and cell \(a_{j}\) to State 1. Then, the fault activation condition is executed to perform a 0 to 1 state change in cell \(a_{i}\). Finally, the condition for detecting the fault results from reading the content (0) of cell \(a_{j}\) and comparing it with the reference (previously set in cell \(a_{j}\)) value equal to 1. To obtain and formulate the conditions for detecting complex, numerous, and diverse configurations of linked coupling faults \(LCF(a_{i},a_{j},\mu_{x},\ldots,a_{z})\) including \(r\) cells \(a_{i},a_{j},a_{k},\ldots,a_{z}\) with an ordered relation of addresses \(i<j<k<\ldots<z\), we introduce their formal description. The basis of such a description is the role of each cell of the considered \(LCF\)s concerning all other cells included in the fault description. Similarly, as in the simplest case of a \(CF\), two cells, \(a_{i}\) and \(a_{j}\), where \(i\
eq j\), can influence each other. One can be a victim (\(V\)) and the other an aggressor (\(A\)). In the examples of fault models \(LCF1\), \(LCF2\), \(LCF3\), \(LCF4\), and \(LCF5\), the roles of each cell are indicated by links and arrows. For the general case,within the framework of linked faults that combine two single _CF_s that include the same two cells, \(a_{i}\) and \(a_{j}\), we introduce the third role (both, \(B\)) of cell \(a_{i}\), which means two roles, \(A\) and \(V\), for cell \(a_{j}\). Accordingly, cell \(a_{j}\) in relation to \(a_{i}\) also has two roles, \(V\) and \(A\), denoted by the same symbol \(B\).