二元码的半速率线性规划界为$\frac12-\frac1π$
The half-rate linear programming bound for binary codes is $\frac12-\frac1π$
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中文总结 AI 辅助
本研究证明了渐近二元Delsarte线性规划的半速率点为$1/2-1/\pi$，解决了二元编码领域对应于球体填充线性规划指数猜想的对应问题，确定了两个Krawtchouk符号不确定问题的渐近行为，构造了各层级相容二元证书并在极限下达到上界。
中文摘要 AI 辅助
在关于球体填充与共形自举的研究中，Afkhami-Jeddi、Cohn、Hartman、de Laat和Tajdini提出了关于Cohn-Elkies球体填充线性规划的精确高维指数猜想。OpenAI的Chapter 1随后通过证明两个傅里叶符号不确定半径均为$(1/\pi+o(1))\sqrt d$，证实了该猜想。我们证明了二元编码领域的对应结论：渐近二元Delsarte线性规划的半速率点为$1/2-1/\pi$；等价地，\\[ R_D\\!\left(\frac12-\frac1\pi\right)=\frac12. \\]我们还提出了两个Krawtchouk符号不确定问题，并确定了二者的渐近行为。若$A^{\mathrm K}_{\pm}(n)$表示原点处为零的Krawtchouk $(\pm1)$本征函数能保持非负的第一个径向层的位置，则\\[ \frac{A^{\mathrm K}_{\pm}(n)}n\longrightarrow \frac12-\frac1\pi. \\]其中公共下界是汉明空间中对应于Chapter 1证明里质量集中原理的对应结论。上界则有不同的来源，它是OpenAI的Chapter 2中最终球面码构造的二元码对应版本。Gay、Jeronimo和Liu改进了所得的二元界，并提出了本文使用的泛函$\Phi$，但仅显式计算了对应层级结构的少数低阶层级。我们构造并评估了每一层级的相容二元证书，在极限情况下达到了该上界。该构造使用了Alrabiah和Guruswami纯态信道的$N$量子比特推广形式。
英文摘要
In their work on sphere packing and the modular bootstrap, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini conjectured the exact high-dimensional exponent of the Cohn-Elkies sphere-packing linear program. OpenAI's Chapter 1 subsequently proved their conjecture by establishing that both Fourier sign-uncertainty radii are $(1/π+o(1))\sqrt d$. We prove the binary coding analogue: \[ R_D\!\left(\frac12-\frac1π\right)=\frac12. \] We also formulate the two Krawtchouk sign-uncertainty problems and determine both of their asymptotics. If $A^{\mathrm K}_{\pm}(n)$ denotes the smallest radius $r$ for which a nonzero Krawtchouk $(\pm1)$-eigenfunction $f$ exists with $f(0) = 0$ and $f(x) \ge 0$ for all $|x| \ge r$, then \[ \frac{A^{\mathrm K}_{\pm}(n)}n\longrightarrow \frac12-\frac1π. \] The lower bound proves a mass-concentration principle similar to OpenAI's Chapter 1 for Hamming space. The upper bound, on the other hand, follows the approach of the spherical-code construction in OpenAI's Chapter 2. Gay, Jeronimo, and Liu formulated a hierarchy for binary codes analogous to the spherical-code construction and improved the best known binary coding rate bounds by evaluating the first level of the corresponding hierarchy. We prove this hierarchy bounds the Delsarte program and give a construction at arbitrarily deep levels of the hierarchy, attaining the upper bound in the limit. The construction uses an $N$-qubit generalization of the pure-state channel of Alrabiah and Guruswami.
