We consider linear codes over a finite field $\mathbb{F}_q$, for odd $q$, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a codeword is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upperbounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order $5$, are given.

翻译：我们考虑一个限定字段的线性代码 $mathbb{F ⁇ {F ⁇ q$, 奇数 q$, 奇数 $q$, 出自定数品种, 取自定数列的对称矩阵 。 计算出一个编码的重量公式 。 使用这个公式, 我们计算出与任何固定、 甚至级别所上浮的矩阵相对应的代码的最低距离 。 对于上界为奇数的情况, 提出了一种推测。 在文章结尾, 给出了这些编码重量的表格, 用于5美元的对称矩阵空间 。