论文标题
小型重量代码单词的射影几何代码
Small Weight Code Words of Projective Geometric Codes
论文作者
论文摘要
我们研究了$ p $ -ary线性代码$ \ MATHCAL C_ {J，K}（n，Q）$的小重量代码单词，由$ k $ -spaces的发射矩阵和$ j $ -spaces PG $（n，q）$及其双重的$ j $ spaces生成的$及其双重的$ q $ q $ a priend and priend and priend and priend as priend and pripe a priend and pripe j <k <k <n $。首先，我们证明了$ \ MATHCAL C_ {J，K}（n，Q）的所有代码单词，重量$ \ weft（3 - \ Mathcal {o} \ left（\ frac 1 q \ frac 1 q \ right）\ right）\ right）\ genfrac）\ genfrac \ genfrac最多两个$ k $ - 空格的组合（即发射矩阵的两行）。至于双代码$ \ MATHCAL C_ {J，K}（N，Q）^\ PERP $，我们设法减少确定其最小重量（1）的问题，并将其最小重量代码单词（2）表征到Case $ \ Mathcal C_ {0,1}（N，Q）（N，Q）^\ perp $。这意味着解决问题（1）和（2）的解决方案，如果$ q $是主要的，并且解决问题（1）如果$ q $甚至是。
We investigate small weight code words of the $p$-ary linear code $\mathcal C_{j,k}(n,q)$ generated by the incidence matrix of $k$-spaces and $j$-spaces of PG$(n,q)$ and its dual, with $q$ a prime power and $0 \leq j < k < n$. Firstly, we prove that all code words of $\mathcal C_{j,k}(n,q)$ up to weight $\left(3 - \mathcal{O}\left(\frac 1 q \right) \right) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$ 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 code words (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.
