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
本文给出了Z2Z4Z8-线性Hadamard码的秩的闭式公式，并结合核维数对码族进行分类，确定了唯一共享两个不变量的型对及与Z4、Z2Z4、Z8线性码的重合情况。
AI 中文摘要
Z2Z4Z8-加性码是Z2^α1×Z4^α2×Z8^α3的子群。Z2Z4Z8-线性Hadamard码是此类码的Gray映射像。已知Z2Z4Z8-加性Hadamard码H^{t1,t2,t3}（其中所有α_i≠0，t1≥1，t2≥0，t3≥1）的递归构造，以及相应长度为2^t的码H^{t1,t2,t3}的线性度和核。然而，其秩的闭式形式未知。本文确定了每个允许三元组的rank(H^{t1,t2,t3})，并得到显式闭式公式。证明计算了码的二元坐标函数（视为消息二进制位的布尔函数）所张成空间的维数。然后，我们利用秩公式和已知的核维数，对族进行分类，尽可能利用这两个不变量。唯一具有相同长度且共享这两个不变量的不同型对是t≥7时的(H^{1,2,t-6},H^{2,0,t-5})和t≥10时的(H^{2,2,t-9},H^{3,0,t-8})。因此，长度为2^t的码中不同对(r,k)的数量为：3≤t≤6时为⌊(t^2+6)/12⌋，7≤t≤9时为⌊(t^2+6)/12⌋-1，t≥10时为⌊(t^2+6)/12⌋-2。因此，当且仅当3≤t≤6时，秩和核维数完全分类该族。我们还确定了与同长度的Z4-、Z2Z4-和Z8-线性Hadamard码的所有重合：与前两者恰好有一个无限族，即t≥10时的H^{2,1,t-7}和H^{5,t-9}，与Z8-线性码恰好有五个族。
英文摘要
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1}\times\mathbb{Z}_4^{α_2}\times\mathbb{Z}_8^{α_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which 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 and kernel of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$. Their rank, however, was not known in closed form. In this paper, we determine $rank(H^{t_1,t_2,t_3})$ for every admissible triple and obtain an explicit closed formula. The proof computes the dimension of the space spanned by the binary coordinate functions of the code, viewed as Boolean functions of the binary digits of the message. We then use the rank formula, together with the known kernel dimension, to classify the family as far as these two invariants allow. The only pairs of distinct types of the same length sharing both invariants are $\bigl(H^{1,2,t-6},H^{2,0,t-5}\bigr)$ for $t\geq7$ and $\bigl(H^{2,2,t-9},H^{3,0,t-8}\bigr)$ for $t\geq10$. Consequently, the number of distinct pairs $(r,k)$ among the codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for $3\leq t\leq6$, $\lfloor(t^2+6)/12\rfloor-1$ for $7\leq t\leq9$, and $\lfloor(t^2+6)/12\rfloor-2$ for $t\geq10$. Thus, rank and kernel dimension classify the family completely if and only if $3\leq t\leq6$. We also determine all coincidences with the $\mathbb{Z}_4$-, $\mathbb{Z}_2\mathbb{Z}_4$-, and $\mathbb{Z}_8$-linear Hadamard codes of the same length: there is exactly one infinite family with the first two, namely $H^{2,1,t-7}$ and $H^{5,t-9}$ for $t\geq10$, and exactly five families with the $\mathbb{Z}_8$-linear codes.
Comments33 pages; a related paper will be released soon
