The problem studied in this work is to determine the higher weight spectra of the Projective Reed-Muller codes associated to the Veronese $3$-fold $\mathcal V$ in $PG(9,q)$, which is the image of the quadratic Veronese embedding of $PG(3,q)$ in $PG(9,q)$. We reduce the problem to the following combinatorial problem in finite geometry: For each subset $S$ of $\mathcal V$, determine the dimension of the linear subspace of $PG(9,q)$ generated by $S$. We develop a systematic method to solve the latter problem. We implement the method for $q=3$, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field $\mathbb F_q$ will be treated in a future work.

翻译：本文研究的问题是与Veronese三维簇$\mathcal V$相关联的射影Reed-Muller码的高权谱，其中$\mathcal V$是$PG(3,q)$在$PG(9,q)$中的二次Veronese嵌入像。我们将该问题归结为有限几何中的如下组合问题：对于$\mathcal V$的每个子集$S$，确定由$S$生成的$PG(9,q)$中线性子空间的维数。我们发展了一种系统的方法来解决后一个问题。针对$q=3$的情况实现了该方法，并利用它获得了相关码的高权谱。一般有限域$\mathbb F_q$的情况将在后续工作中处理。