Non-additive symmetric rank-metric codes

Abstract

We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd $q$, we introduce a quotient construction for ${\mathrm{Sym}}_3(q)$ in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on ${\mathbb{F}}_q^4$. This leads to a correspondence with partial ovoids of the elliptic quadric $Q^{-}(5,q)$. Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in ${\mathrm{Sym}}_3(q)$ of cardinality $q^4+2q^3$, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd $n\ge 3$ and every prime power $q>2$, we obtain \begin{equation*} A_q^{\rm sym}(n,2) \ge q^{(n^2-1)/2}+q^{(n^2-3)/2}+q^{(n^2-9)/2}, \end{equation*} which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension $n$ varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for $n=3$.