On the list decodability of Rank Metric codes [article]

Rocco Trombetti, Ferdinando Zullo
2020 arXiv   pre-print
Let k,n,m ∈Z^+ integers such that k≤ n ≤ m, let G_n,k∈F_q^m^n be a Delsarte-Gabidulin code. Wachter-Zeh proven that codes belonging to this family cannot be efficiently list decoded for any radius τ, providing τ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in F_q^m^n ∖G_n,k. In 2016, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when n| m. As a consequence, they were able to detect infinite families of
more » ... Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by the first author and Yue Zhou. By exploiting arguments suchlike those used in the above mentioned papers, when n| m, we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to d-1/2+1, where d is its minimum distance. Nonetheless, in all other examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to d-1/2+2, where d is its minimum distance. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove that any rank metric code of F_q^m^n of order q^kn with n dividing m, such that 4n-3 is a square in Z and containing G_n,2, is not efficiently list decodable at some values of the radius τ.
arXiv:1907.01289v2 fatcat:oab2kb3tnratffryrkwzzobzki