Construction of linear codes over $\mathfrak{R}=\sum_{s=0}^{4} v_{5}^{s}\mathcal{A}_{4}$

The aim of this paper is to propose a new family of codes.  We define this family over the ring $\mathfrak{R}=\sum_{s=0}^{4} v_{5}^{s}\mathcal{A}_{4}$, with $v_{5}^{5}=v_{5}$.  We derive its properties, a generator matrix and Gray images.  This new family of codes is illustrated by three applications.

codes over the rings
idempotents
Gray map
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