The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank, which is, at most, a given even number. Furthermore, a conjecture for the minimum distance of codes from symmetric matrices with ranks bounded by an odd number was given. In this article, we continue the study of codes from symmetric matrices of bounded rank. A connection between the weights of the codewords of this code and Q-numbers of the association scheme of symmetric matrices is established. Consequently, we get a concrete formula for the weight distribution of these codes. Finally, we determine the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank at most a given number, both when this number is odd and when it is even.

翻译：作者在近期论文中首次研究了从有界秩对称矩阵构成的行列式簇导出的、定义在奇数阶有限域上的线性码。该研究确定了通过在所有秩不超过给定偶数的对称矩阵上评估齐次线性函数所得码的最小距离，并提出了关于秩不超过奇数的对称矩阵所生成码的最小距离猜想。本文继续研究有界秩对称矩阵生成的码，建立了该码字权重与对称矩阵结合方案的Q数之间的联系，从而推导出这些码权重分布的具体计算公式。最终，我们确定了在秩不超过给定数（包括奇数和偶数情形）的所有对称矩阵上评估齐次线性函数所得码的最小距离。