论文标题
最佳的可能是小尺寸的非线性3 PIR代码
Optimal possibly nonlinear 3-PIR codes of small size
论文作者
论文摘要
首先，我们指出对PIR代码的最小距离限制的概括。然后，我们使用包装设计描述了线性PIR代码的结构，并使用它来构建一些新的5-PIR代码。最后，我们表明，对于二进制$ r $ ther订单，没有编码器（线性或非线性）产生3-PIR代码，除非$ r = 2 $。我们使用这些结果来确定组合尺寸的二进制（可能是非线性）的最小长度长达〜6。长度为11和尺寸$ 2^7 $的二进制3-PIR代码必然是非线性的，我们构成了这样的代码，例如开放问题。
First, we state a generalization of the minimum-distance bound for PIR codes. Then we describe a construction for linear PIR codes using packing designs and use it to construct some new 5-PIR codes. Finally, we show that no encoder (linear or nonlinear) for the binary $r$-th order Hamming code produces a 3-PIR code except when $r=2$. We use these results to determine the smallest length of a binary (possibly nonlinear) 3-PIR code of combinatorial dimension up to~6. A binary 3-PIR code of length 11 and size $2^7$ is necessarily nonlinear, and we pose the existence of such a code as an open problem.
