Q型Hadamard全拟线性码秩和核
Hadamard Full Propelinear Codes of type Q. Rank and Kernel
摘要：
引入了Hadamard全推进线性码（HFP-码），并证明了它们与Hadamard群的等价性（另一方面，Hadamard群与群中的相对$(4n,2,4n,2n)$ -差集以及与上循环Hadamard矩阵的等价性也是已知的）.我们计算了Q型HFP码的核的秩和维数的可用值，并证明了核的维数总是1或$2$。我们还表明，当内核的维数是2，那么转置码的内核的维数是1（所以，这两个代码是不等价的）。最后，我们给出了一种构造方法，使得从长度为$4n$，核维数为$k=2$，最大秩为$r=2n$的HFP码得到长度为$8n$，核维数为$k=2$，最大秩为$r=4n$的HFP码.
英文摘要：
Hadamard full propelinear codes (HFP-codes) are introduced and their equivalence with Hadamard groups is proven (on the other hand, it is already known the equivalence of Hadamard groups with relative $(4n,2,4n,2n)$-difference sets in a group and also with cocyclic Hadamard matrices). We compute the available values for the rank and dimension of the kernel of HFP-codes of type Q and we show that the dimension of the kernel is always 1 or $2$. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP-code of length $4n$, dimension of the kernel $k=2$, and maximum rank $r=2n$, we obtain an HFP-code of double length $8n$, dimension of the kernel $k=2$, and maximum rank $r=4n$.
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