论文标题
在准循环代码的非零重量的数量上狭窄的上限
A tight upper bound on the number of non-zero weights of a quasi-cyclic code
论文作者
论文摘要
令$ \ mathcal {c} $为索引$ l（l \ geq2）$的准环境代码。令$ g $为$ρ^l $生成的$ \ mathcal {c} $的自动形态组的子组，而$ \ mathcal {c} $的标量乘数，其中$ρ$表示标准的环形偏移。在本文中，我们在$ \ Mathcal {c} \ setMinus \ {\ Mathbf {0} \} $上找到了$ g $的轨道的明确公式。因此，立即得出了对$ \ Mathcal {C} $的非零权重的数量的明确上限，并且显示了满足限制的代码的必要条件。如果$ \ Mathcal {c} $是单一的准循环代码，则可以通过考虑较大的自动形态亚组来获得$ \ MATHCAL {C} $的非零权重的数量，这是由Multiplier，$ρ^l $和$ρ^l $和$ salcar乘积生成的较大的自动形态亚组。特别是，我们列出了一些示例以显示界限很紧。我们的主要结果改善并概括了\ cite {m2}中的一些结果。
Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $ρ^l$ and the scalar multiplications of $\mathcal{C}$, where $ρ$ denotes the standard cyclic shift. In this paper, we find an explicit formula of orbits of $G$ on $\mathcal{C}\setminus \{\mathbf{0}\}$. Consequently, an explicit upper bound on the number of nonzero weights of $\mathcal{C}$ is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. If $\mathcal{C}$ is a one-generator quasi-cyclic code, a tighter upper bound on the number of nonzero weights of $\mathcal{C}$ is obtained by considering a larger automorphism subgroup which is generated by the multiplier, $ρ^l$ and the scalar multiplications of $\mathcal{C}$. In particular, we list some examples to show the bounds are tight. Our main result improves and generalizes some of the results in \cite{M2}.
