Decoding Trombetti-Zhou codes: a new syndrome-based decoding approach

In 2019, Trombetti and Zhou introduced a new family of $\mathbb{F}_{q^n}$-linear Maximum Rank Distance (MRD) codes over $\mathbb{F}_{q^{2n}}$. For such codes we propose a new syndrome-based decoding algorithm. It is well known that a syndrome-based decoding approach relies heavily on a parity-check matrix of a linear code. Nonetheless, Trombetti-Zhou codes are not linear over the entire field $\mathbb{F}_{q^{2n}}$, but only over its subfield $\mathbb{F}_{q^{n}}$. Due to this lack of linearity, we introduce the notions of $\mathbb{F}_{q^{n}}$-generator matrix and $\mathbb{F}_{q^{n}}$-parity-check matrix for a generic $\mathbb{F}_{q^{n}}$-linear rank-metric code over $\mathbb{F}_{q^{rn}}$ in analogy with the roles that generator and parity-check matrices play in the context of linear codes. Accordingly, we present an $\mathbb{F}_{q^n}$-generator matrix and $\mathbb{F}_{q^n}$-parity-check matrix for Trombetti-Zhou codes as evaluation codes over an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^{2n}}$. This relies on the choice of a particular basis called \emph{trace almost dual basis}. Subsequently, denoting by $d$ the minimum distance of the code, we show that if the rank weight $t$ of the error vector is strictly smaller than $\frac{d-1}{2}$, the syndrome-based decoding of Trombetti-Zhou codes can be converted to the decoding of Gabidulin codes of dimension one larger. On the other hand, when $t=\frac{d-1}{2}$, we reduce the decoding to determining the rank of a certain matrix. The complexity of the proposed decoding for Trombetti-Zhou codes is also discussed.