We investigate small weight codewords of the p-ary linear code ${\mathcal{C}}_{j,k}(n,q)$ generated by the incidence matrix of k-spaces and j-spaces of $\mathrm{\text{PG}}(n,q)$ and its dual, with q a prime power and $0⩽j<k<n$. Firstly, we prove that all codewords of ${\mathcal{C}}_{j,k}(n,q)$ up to weight $(3-\mathcal{O}\left(\frac{1}{q}\right)){[\begin{array}{c}k+1\\ j+1\end{array}]}_q$ are linear combinations of at most two k-spaces (i.e. two rows of the incidence matrix). As for the dual code ${\mathcal{C}}_{j,k}{(n,q)}^{\perp }$, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight codewords (2) to the case ${\mathcal{C}}_{0,1}{(n,q)}^{\perp }$. This implies the solution to both problem (1) and (2) if q is prime and the solution to problem (1) if q is even.

我们研究p -ary线性码的小权重码字${\mathcal{C}}_{Ĵ，ķ}（ñ，q）$由k的空间和j的空间的入射矩阵生成$\mathrm{\text{PG}}（ñ，q）$和它的对偶，q为素数，$0⩽Ĵ<ķ<ñ$。首先，我们证明所有的码字${\mathcal{C}}_{Ĵ，ķ}（ñ，q）$ 根据体重 $（3-\mathcal{Ø}（\frac{\mathrm{1个}}{q}））{[\begin{array}{c}ķ+\mathrm{1个}\\ Ĵ+\mathrm{1个}\end{array}]}_q$是最多两个k空间（即入射矩阵的两行）的线性组合。至于双码${\mathcal{C}}_{Ĵ，ķ}{（ñ，q）}^{\perp }$，我们设法减少了确定最小权重（1）和表征其最小权重码字（2）的问题 ${\mathcal{C}}_{0，\mathrm{1个}}{（ñ，q）}^{\perp }$。如果q是素数，则意味着问题（1）和（2）的解决方案；如果q是偶数，则意味着问题（1）的解决方案。