Automorphism groups of Gabidulin-like codes

Let K/k be a cyclic Galois extension of degree $${\ell}$$ℓ and $${\theta }$$θ a generator of Gal(K/k). For any $${v=(v_1, \ldots, v_m)\in K^{m}}$$v=(v1,…,vm)∈Km such that v is linearly independent over k, and any $${1\leq d < m }$$1≤d<m the Gabidulin-like code $${\mathcal{C}(v,\theta, d) \leq k^{\ell \times m }}$$C(v,θ,d)≤kℓ×m is a maximum rank distance code of dimension $${\ell d}$$ℓd over k. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.