AI 中文总结
研究离散 \(q\) - 系统秩的两个界何时相等及相关性质，通过构造极值准最大 \((m - 3)\) - 离散系统，保持初始码广义秩权重下界，满足条件时产生对偶准 - MRD 码对，还对扩展进行参数化并给出不等价输出数量下界及分离结果。
AI 中文摘要
两个一般估计控制着 \(h\) - 离散 \(q\) - 系统的秩：离散上界和包含关系下最大系统的下界。我们确定了它们何时相等，除 \(m = h + 1\) 和 \(m = h + 2\) 外，当 \(h\) 为偶数，\(m = h + 3\) 且 \(k\equiv h/2\pmod{h + 1}\) 时相等。我们还证明了前两种情况下的刚性。对于 \(h = m - 3\) 和特定的 \(k\)，两个界相差一。从任意最大离散 \(q\) - 系统出发，构造了极值准最大 \((m - 3)\) - 离散系统，其保持初始码广义秩权重的下界，满足特定条件时还产生非退化对偶准 - MRD 码对。通过商空间子空间对这些扩展进行参数化并得出不等价输出数量的统一下界。已知二阶族在特定数值条件下与每个非平凡直和分离，高阶族与指定一阶直和类分离。最后给出了一个完全明确的二进制示例。
英文摘要
Two general estimates govern the rank of an $h$-scattered $q$-system: the scattered upper bound and the lower bound for systems maximal under inclusion. We determine exactly when they coincide. Besides $m=h+1$ and $m=h+2$, equality occurs precisely when $h$ is even, $m=h+3$, and $k\equiv h/2\pmod{h+1}$. We also prove rigidity in the first two cases: every extremal system is equivalent to either $\mathbb{F}_q^k$ or a direct sum of elementary Gabidulin systems and $\mathbb{F}_q$-directions. For $h=m-3$ and $k=\ell(m-2)+s$, with $1\leq s<(m-2)/2$, the two bounds differ by one, apart from one boundary case. Starting from an arbitrary maximum scattered $q$-system, we construct extremal quasi-maximum $(m-3)$-scattered systems of rank $\ell m+s$. Every such extension preserves lower bounds on the generalized rank weights of the initial code. When an explicit independence condition holds, the construction also produces a nondegenerate dual pair of quasi-MRD codes. We parametrize these extensions by subspaces of a quotient space and derive a uniform lower bound on the number of inequivalent outputs. Known order-two families yield codes separated from every nontrivial direct sum under an explicit numerical condition, while higher-order families are separated from a specified order-one direct-sum class. We conclude with a fully explicit binary example.
