Number Theoretical Locally Recoverable Codes

In this paper we give constructions for infinite sequences of finite non-linear locally recoverable codes $\mathcal{C}\subseteq {\prod }_{i=1}^N{\mathbb{F}}_{q_i}$ over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.