Linear exact repair is a method of recovering erased data in distributed storage systems. In 2017, Guruswami and Wootters introduced linear exact repair of Reed–Solomon codes, a family of error correcting codes defined by evaluating polynomials of bounded degree at elements of a finite field. This scheme considers codes with alphabets in a proper extension \(\mathbb {F}_{p^t}\) of a prime field \(\mathbb {F}_p\) and conducts repair using only elements of the prime field. It has been adapted to other families of evaluation codes, including algebraic geometry codes, Reed–Muller codes, and variants. In this paper, we consider codes that are not described as evaluation codes. We provide a linear exact repair for some families of codes whose duals contain some words that decompose in a particular way. We achieve exact recovery of missing data for these codes using only elements of a base field, while the code itself has a much larger alphabet by relying on the field trace along with dual bases for the field extension.
