论文标题
Delsarte线性程序的独特优势
Unique Optima of the Delsarte Linear Program
论文作者
论文摘要
delsarte线性程序用于限制代码的大小，因为它们的块长度$ n $和最小的距离$ d $，通过从代码到准码的线性放松来限制代码。我们研究了$（n，d）$的哪个值此线性程序具有独特的最佳限制：虽然我们证明它并不总是具有唯一的最佳效果，但我们证明它在$ d> n/2 $或$ d \ d \ d \ leq 2 $中。引入了Quasicode的Krawtchouk分解，我们证明了$（n，2e）$和$（n-1,2e-1）$线性程序的最佳状态，这些程序基本上具有相同的Krawtchouk分解，揭示了Delsarte Lineareal程序中的平等现象。我们概括了将代码扩展和刺穿代码的概念，从中我们可以从中看到这种奇偶校关系是通过扩展/刺穿给出的。我们进一步表征了这些Optima对，特别是证明它们表现出对称性能，有效地减少了决策变量的数量。
The Delsarte linear program is used to bound the size of codes given their block length $n$ and minimal distance $d$ by taking a linear relaxation from codes to quasicodes. We study for which values of $(n,d)$ this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if $d>n/2$ or if $d \leq 2$. Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the $(n,2e)$ and $(n-1,2e-1)$ linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.
