发表机构
Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文计算了Z2Z4-线性与Z2^s-线性Hadamard码的核块秩轮廓，证明其恒定，并利用下降定理与坐标置换统一论证，从而完成了Z2Z4Z8-线性Hadamard码的完全分类，仅一个无限族需二块细化区分。
AI 中文摘要
包含零字的二元码的核将二元坐标划分为若干块，当每个核字在两个坐标上取值相同时，这两个坐标位于同一块中，而\emph{核块秩轮廓}是其在这些块上截断的线性张成的维数的多重集。对于$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码族$H^{t_1,t_2,t_3}$，该不变量已知且给出了该族的完全分类。本文计算了与之比较的$\mathbb{Z}_2\mathbb{Z}_4$-线性和$\mathbb{Z}_{2^s}$-线性Hadamard族的不变量，并证明对于每个非线性$\mathbb{Z}_2\mathbb{Z}_4$-线性Hadamard码以及每个非线性$\mathbb{Z}_{2^s}$-线性Hadamard码$\bar H^{a_1,\dots,a_s}$（其中$s\geq2$），该不变量是常数。第二个结论的证明无需任何秩公式，而是通过展示保持码并对其核块可迁作用的坐标置换获得，这使得论证在$s$上统一。我们还证明了下降定理：若$\bar H^{a_1,\dots,a_s}$是非线性的且$a_1\geq2$，则在一个核块上截断的码是$\mathbb{Z}_{2^{s-1}}$-线性Hadamard码$\bar H^{a_1,\dots,a_{s-1}}$。因此，常数局部秩等于$\rank(\bar H^{a_1,\dots,a_{s-1}})$且至少为$t-\kappa+2$，其中$2^t$是长度，$\kappa$是核维数。这些结果将每个非线性$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码与每个同长度的$\mathbb{Z}_4$-线性、$\mathbb{Z}_2\mathbb{Z}_4$-线性和$\mathbb{Z}_{2^s}$-线性Hadamard码区分开来，除了单个无限族$H^{1,1,t-4}$和$\bar H^{2,0,t-5}$（其中$t\geq5$）。该无限族的成员在秩、核维数和核块秩轮廓上一致，但二块细化将它们区分开来。
英文摘要
The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on those blocks. For the family $H^{t_1,t_2,t_3}$ of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code and for every nonlinear $\mathbb{Z}_{2^s}$-linear Hadamard code $\bar H^{a_1,\dots,a_s}$ with $s\geq2$. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in $s$. We also prove a descent theorem: if $\bar H^{a_1,\dots,a_s}$ is nonlinear and $a_1\geq2$, then the code punctured on one kernel block is the $\mathbb{Z}_{2^{s-1}}$-linear Hadamard code $\bar H^{a_1,\dots,a_{s-1}}$. Consequently, the constant local rank equals $rank(\bar H^{a_1,\dots,a_{s-1}})$ and is at least $t-κ+2$, where $2^t$ is the length and $κ$ the kernel dimension. These results separate every nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code from every $\mathbb{Z}_4$-linear, $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard code of the same length, except for the single infinite family $H^{1,1,t-4}$ and $\bar H^{2,0,t-5}$ with $t\geq5$. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.
Comments34 pages
