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We use a Diamond diagram attached to a $2$-dimensional reducible split mod$p$ Galois representation of$\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p^{2}})$ to construct anon-admissible smooth irreducible mod $p$ representation of$GL_{2}(\mathbb{Q}_{p^{2}})$ following the approach of Daniel Le.

On non-admissible irreducible modulo $p$ representations of $GL_{2}(\mathbb{Q}_{p^{2}})$