Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum Hamming distance d. Deriving upper and lower bounds on A(n, d) have been a subject for extensive research in coding theory. In this paper, we examine upper and lower bounds on A(n, d) in the high-minimum distance regime, in particular, when $d = n/2 - \Theta(\sqrt{n})$. We will first provide a lower bound based on a cyclic construction for codes of length $n= 2^m -1$ and show that $A(n, d= n/2 - 2^{c-1}\sqrt{n}) \geq n^c$, where c is an integer with $1 \leq c \leq m/2-1$. With a Fourier-analytic view of Delsarte's linear program, novel upper bounds on $A(n, n/2 - \sqrt{n})$ and $A(n, n/2 - 2 \sqrt{n})$ are obtained, and, to the best of the authors' knowledge, are the first upper bounds scaling polynomially in n for the regime with $d = n/2 - \Theta(\sqrt{n})$.

翻译：A(n, d) 表示长度为 n 和最小 Hamming 距离的二进制代码中的最大编码字数。 在 A(n, d) 上下界对 A(n, d) 进行广泛的编码理论研究。 在本文中, 我们检查高最小距离系统中的 A(n, d) 上下界对 A(n, d) 值的最大编码数, 尤其是当$d = n/2 - \ Theta( sqrt{n) 美元时。 我们首先根据长度为 $n= 2 m - 1美元的周期构造提供较低的下界数。 显示 $(n, d= n/2 n) = n- nqqrt $, c是以 1\leq c\leq c\ leq m/2$ 的整数。 在 Delsarte 线性方案的四倍分析视图中, 在 $A(n, n/2 - srt{n) 和 $A(n, n/2) lefly) 的作者们的上端值为第一, N/2= 最高值。