Based on recent studies regarding high-temperature (high-$T_c$) La-Y ternary hydrides (e.g., $P{bar{1}}$-La$_2$YH$_{12}$, $Pm{bar{3}}m$-LaYH$_{12}$, and $Pm{bar{3}}m$-(La,Y)H$_{10}$ with a maximum $T_c sim 253$ K), we examined the phase and structural stabilities of the (LaH$_6$)(YH$_6$)$_y$ series as high-$T_c$ ternary hydride compositions using a genetic algorithm and $it ab$ $it initio$ calculations. Our evaluation showed that the $Pmbar{3}m$-LaYH$_{12}$ reported in the previous study was unstable during decomposition into $Rbar{3}c$-LaH$_{6}$ + $Imbar{3}m$-YH$_{6}$. We also discovered new crystal structures, namely $Cmmm$-LaYH$_{12}$ ($y=1$), $Rbar{3}c$-LaYH$_{12}$ ($y=1$), $Cmmm$-LaY$_3$H$_{24}$ ($y=3$), and $Rbar{3}$-LaY$_3$H$_{24}$ ($y=3$), showing stability against such decomposition. While $Rbar{3}c$ ($y=1$) and $Rbar{3}$ ($y=3$) did not exhibit superconductivity owing to the extremely low density of states at the Fermi level, $Cmmm$ phases exhibited a $T_{c}$ of approximately 140~K at around 200~GPa owing to the extremely high electron--phonon coupling constant ($lambda$ = 1.876 for LaYH$_{12}$). By the twice longer stacking for $Cmmm$-LaY$_3$H$_{24}$, the coupling constant increased owing to the chemical pressure of Y, leading to a slightly increased $T_{c}$.
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