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
本文引入核块秩轮廓不变量，完全分类$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码，证明同长码等价当且仅当类型相同，并给出不等价码计数公式。
AI 中文摘要
$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-加性码是$\mathbb{Z}_2^{\alpha_1}\times\mathbb{Z}_4^{\alpha_2}\times\mathbb{Z}_8^{\alpha_3}$的子群，而$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码是此类码的Gray映射像。已知$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-加性Hadamard码$\mathcal H^{t_1,t_2,t_3}$（其中所有$\alpha_i\neq0$，$t_1\geq1$，$t_2\geq0$，$t_3\geq1$）存在递归构造，且相应码$H^{t_1,t_2,t_3}$（长度为$2^t$，其中$t+1=3t_1+2t_2+t_3$）的线性性、核维数和秩已知。然而，这些不变量并不能完全分类该族。两对无限的不同类型具有相同的长度、秩和核维数，对于已分类的长度$2^t$（$3\leq t\leq11$），此类对仅通过计算机等价性测试区分。本文引入一个等价不变量来解决这些情况。核将二元坐标划分为块，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；\emph{核块秩轮廓}是在这些块上截断的线性张成空间的维数多重集。与全局性的秩和核维数不同，该不变量记录每个块上张成空间的存活程度。我们计算了整个族：共有$2^{t_1+t_2+t_3-1}$个块，每个块大小为$2^{2t_1+t_2}$，轮廓至多取两个值$t_2+\binom{t_1+2}{2}$和$t_2+2+\binom{t_1+1}{2}$，其差为$t_1-1$；当且仅当$t_1=1$时轮廓为常数。因此，轮廓恢复$t_1$，长度和核维数恢复$t_2$和$t_3$。该族中两个长度相同的码等价当且仅当它们的类型一致，且长度为$2^t$的两两不等价码的数量对于每个$t\geq3$为$\lfloor(t^2+6)/12\rfloor$。
英文摘要
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1}\times\mathbb{Z}_4^{α_2}\times\mathbb{Z}_8^{α_3}$, and a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $α_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity, kernel dimension, and rank of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$, where $t+1=3t_1+2t_2+t_3$. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths $2^t$, $3\leq t\leq11$, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the \emph{kernel-block rank profile} is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are $2^{t_1+t_2+t_3-1}$ blocks, all of size $2^{2t_1+t_2}$, and the profile takes at most two values $t_2+\binom{t_1+2}{2}$ and $t_2+2+\binom{t_1+1}{2}$, whose difference is $t_1-1$; it is constant precisely when $t_1=1$. Hence, the profile recovers $t_1$, and the length and kernel dimension recover $t_2$ and $t_3$. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for every $t\geq3$.
Comments25 pages; a related paper will be released soon
