# Linked Coupling Faults Detection by Multirun March Tests
## 6 Conclusions  
A formal mathematical model is proposed to describe linked coupling faults, based on the behavior of cells involved in the fault. The cell behavior scenario is determined by the address sequence of a one-run march test. The necessary and sufficient conditions for detecting linked coupling faults are provided. Theoretically and experimentally, undetectable _LCF_s are effectively detected by multirun tests using random address sequences \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{CFin}(\mathbf{a_{i}},\mathbf{a_{j}})\)} & \multirow{2}{*}{\(\mathbf{CFin}(\mathbf{a_{j}},\mathbf{a_{k}})\)} & \multicolumn{5}{c}{\(\mathbf{FC_{Test}}(f)\)[\%]} \\  & & \(\mathbf{f=1}\) & \(\mathbf{f=2}\) & \(\mathbf{f=4}\) & \(\mathbf{f=8}\) & \(\mathbf{f=10}\) & \(\mathbf{f=12}\) \\ \hline \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & 49.97 & 75.26 & 93.70 & 100.00 & 100.00 & 100.00 \\ \hline \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & 50.04 & 74.25 & 94.65 & 99.51 & 100.00 & 100.00 \\ \hline \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & 49.99 & 75.03 & 94.11 & 100.00 & 100.00 & 100.00 \\ \hline \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & 50.15 & 75.24 & 95.27 & 99.71 & 100.00 & 100.00 \\ \hline \multicolumn{1}{c}{TOTAL \(FC_{Test}(f)\)} & 50.04 & 74.95 & 94.43 & 99.80 & 100.00 & 100.00 \\ \hline \end{tabular}
\end{table} 
Table 4: Fault coverage \(\text{FC}_{Test}(f)\) of the _LCF_ for the multirun March C- test (address sequences: counting sequences, and background: random). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{CFin}(\mathbf{a_{i}},\mathbf{a_{j}})\)} & \multirow{2}{*}{\(\mathbf{CFin}(\mathbf{a_{j}},\mathbf{a_{k}})\)} & \multicolumn{5}{c}{\(\mathbf{FC_{Test}}(f)\)[\%]} \\  & & \(\mathbf{f=1}\) & \(\mathbf{f=2}\) & \(\mathbf{f=4}\) & \(\mathbf{f=8}\) & \(\mathbf{f=10}\) & \(\mathbf{f=12}\) \\ \hline \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & 33.53 & 55.93 & 80.51 & 96.25 & 98.53 & 99.31 \\ \hline \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & 66.35 & 88.59 & 98.64 & 99.99 & 100.00 & 100.00 \\ \hline \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & 66.47 & 88.83 & 98.62 & 99.98 & 100.00 & 100.00 \\ \hline \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\downarrow\),\(\downarrow\)\(>\) & 33.52 & 56.06 & 80.49 & 96.23 & 98.53 & 99.34 \\ \hline \multicolumn{1}{c}{TOTAL \(FC_{Test}(f)\)} & 49.97 & 72.35 & 89.57 & 98.12 & 99.27 & 99.66 \\ \hline \end{tabular}
\end{table} 
Table 2: Fault coverage \(FC_{Test}(f)\) of the _LCF_ for the multirun MATS+ test (address sequences: random, and background: zeros). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{CFin}(\mathbf{a_{i}},\mathbf{a_{j}})\)} & \multirow{2}{*}{\(\mathbf{CFin}(\mathbf{a_{j}},\mathbf{a_{k}})\)} & \multicolumn{5}{c}{\(\mathbf{FC_{Test}}(f)\)[\%]} \\  & & \(\mathbf{f=1}\) & \(\mathbf{f=2}\) & \(\mathbf{f=4}\) & \(\mathbf{f=8}\) & \(\mathbf{f=10}\) & \(\mathbf{f=12}\) \\ \hline \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & \(\wedge\)\(<\)\(\uparrow\),\(\downarrow\)\(>\) & 49.98 & 74.42 & 93.57 &and variable initial memory states. The experiments demonstrate the high efficiency of detecting \(LCF1\)s using multirun memory tests. For instance, an eight-run test with a random background technique achieves 100% coverage of \(LCF1\)s, even for simple tests, like the \"one march element\" ( \(\{\Uparrow(r0,w1)\}\)). The proposed mathematical model for linked coupling faults appears promising for their identification and diagnostic localization.