关于自对偶量子码、图和布尔函数
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
摘要：
简要介绍了量子纠错，指出零维量子码可以表示为GF（4）上的自对偶加法码，也可以表示为图.我们表明，图表示几个这样的代码与高的最小距离可以描述为嵌套的正则图具有最小的正则顶点度和包含长周期。两个图对应于等价的量子码，如果它们通过一个局部互补序列相关联。我们使用这个操作来生成图的轨道，从而对GF（4）上长度为12的所有不等价自对偶加法码进行分类，而以前只有长度为9的所有码是已知的。我们表明，这些代码可以被解释为二次布尔函数，我们定义了非二次量子码，对应于更高的程度的布尔函数。我们将研究布尔函数的各种密码学性质，特别是传播准则。然后定义了新的非周期传播准则（APC）和APC距离。我们证明了零维量子码的距离等于相应布尔函数的APC距离。生成布尔函数关于{I，H，N}^n变换集的轨道。研究了二次布尔函数关于{I，H，N}^n变换集的峰均功率平均功率比（PAR_IHN），证明了二次布尔函数的PAR_IHN与图的轨道上的最大独立集的大小有关.提出了一种低PAR_IHN的非二次布尔函数的构造方法。最后指出PAR_IHN和APC距离都可以解释为部分纠缠度量。
英文摘要：
A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.
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