论文标题
使用简单复合物构建的某些二进制最小代码
Certain binary minimal codes constructed using simplicial complexes
论文作者
论文摘要
在此手稿中，我们在非链环$ \ Mathcal {r} = \ Mathbb {f} _2 [u]/\ langle u^3 -U \ u \ rangle $上工作。令$ m \ in \ mathbb {n} $，让$ l，m，n \ subseteq [m]：= \ {1，2，\ dots，m \} $。 For $X\subseteq [m]$, define $Δ_X:=\{v \in \mathbb{F}_2^m : \textnormal{Supp}(v)\subseteq X\}$ and $D:= (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3$, an ordered finite multiset consisting来自$ \ Mathcal {r}^m $的元素，其中$ d_1 \ in \ {Δ_l，δ_l^c \}，d_2 \ in \ in \ {Δ_m，Δ_m^c \}，d_m^c \}，d_3 \ in \ in \ in \ {Δ_n，δ_n，δ_n^c \} $。线性代码$ c_d $ over $ \ mathcal {r} $由$ \ {\ big（v \ cdot d \ big）_ {d \ in d}：v \ in \ in \ mathcal {r}^m \} $。此外，我们还考虑了上述工作中具有两个最大元素的简单复合物。我们研究了他们的二进制灰色图像和类似二进制子场的代码，对应于某个$ \ mathbb {f} _ {2} $ - $ \ mathcal {r} $的功能。这些二进制线性代码的足够条件是最小的，在每种情况下均可获得自我实施。此外，我们就Griesmer Bound产生了无限的最佳代码家族。在本手稿中获得的大多数代码都是重量代码。
In this manuscript, we work over the non-chain ring $\mathcal{R} = \mathbb{F}_2[u]/\langle u^3 - u\rangle $. Let $m\in \mathbb{N}$ and let $L, M, N \subseteq [m]:=\{1, 2, \dots, m\}$. For $X\subseteq [m]$, define $Δ_X:=\{v \in \mathbb{F}_2^m : \textnormal{Supp}(v)\subseteq X\}$ and $D:= (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3$, an ordered finite multiset consisting of elements from $\mathcal{R}^m$, where $D_1\in \{Δ_L, Δ_L^c\}, D_2\in \{Δ_M, Δ_M^c\}, D_3\in \{Δ_N, Δ_N^c\}$. The linear code $C_D$ over $\mathcal{R}$ defined by $\{\big(v\cdot d\big)_{d\in D} : v \in \mathcal{R}^m \}$ is studied for each $D$. Further, we also consider simplicial complexes with two maximal elements in the above work. We study their binary Gray images and the binary subfield-like codes corresponding to a certain $\mathbb{F}_{2}$-functional of $\mathcal{R}$. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.
