The \(p\)-ary linear code \(\mathcal {C}_{k}\!\left( n,q\right) \) is defined as the row space of the incidence matrix \(A\) of \(k\)-spaces and points of \(\textrm{PG}\!\left( n,q\right) \). It is known that if \(q\) is square, a codeword of weight \(q^k\sqrt{q}+\mathcal {O}\!\left( q^{k-1}\right) \) exists that cannot be written as a linear combination of at most \(\sqrt{q}\) rows of \(A\). Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if \(q\geqslant 32\) is a composite prime power, every codeword of \(\mathcal {C}_{k}\!\left( n,q\right) \) up to weight \(\mathcal {O}\!\left( q^k\sqrt{q}\right) \) is a linear combination of at most \(\sqrt{q}\) rows of \(A\). We also generalise this result to the codes \(\mathcal {C}_{j,k}\!\left( n,q\right) \), which are defined as the \(p\)-ary row span of the incidence matrix of k-spaces and j-spaces, \(j < k\).

\(p\)元线性代码\ (\mathcal {C}_{k}\!\left( n,q\right) \)定义为关联矩阵\(A\)的行空间\(k\) - \(\textrm{PG}\!\left( n,q\right) \)的空间和点。已知如果\(q\)是方阵，则权重为\(q^k\sqrt{q}+\mathcal {O}\!\left(q^{k-1}\right)\)的码字存在不能写成最多\(\sqrt{q}\)行\(A\)的线性组合。在过去的几十年里，研究人员投入了大量的精力来证明任何较小权重的码字确实满足这一特性。我们证明，如果\(q\geqslant 32\)是复合素数幂，则 \(\mathcal {C}_{k}\!\left( n,q\right) \)的每个码字最多可达权重\( \mathcal {O}\!\left( q^k\sqrt{q}\right) \)是\(A\)至多\(\sqrt{q}\)行的线性组合。我们还将这个结果推广到代码\(\mathcal {C}_{j,k}\!\left( n,q\right) \)，它们被定义为\(p\) -ary 行跨度k空间和j空间的关联矩阵，\(j < k\)。