Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most fundamental subjects in coding theory. In this paper, we explore new lower and upper bounds on $A(n, d)$ in the large-minimum distance regime, in particular, when $d = n/2 - \Omega(\sqrt{n})$. We first provide a new construction of cyclic codes, by carefully selecting specific roots in the binary extension field for the check polynomial, with length $n= 2^m -1$, distance $d \geq n/2 - 2^{c-1}\sqrt{n}$, and size $n^{c+1/2}$, for any $m\geq 4$ and any integer $c$ with $0 \leq c \leq m/2 - 1$. These code parameters are slightly worse than those of the Delsarte--Goethals (DG) codes that provide the previously known best lower bound in the large-minimum distance regime. However, using a similar and extended code construction technique we show a sequence of cyclic codes that improve upon DG codes and provide the best lower bound in a narrower range of the minimum distance $d$, in particular, when $d = n/2 - \Omega(n^{2/3})$. Furthermore, by leveraging a Fourier-analytic view of Delsarte's linear program, upper bounds on $A(n, n/2 - \rho\sqrt{n})$ with $\rho\in (0.5, 9.5)$ are obtained that scale polynomially in $n$. To the best of authors' knowledge, the upper bound due to Barg and Nogin \cite{barg2006spectral} is the only previously known upper bound that scale polynomially in $n$ in this regime. We numerically demonstrate that our upper bound improves upon the Barg-Nogin upper bound in the specified high-minimum distance regime.

翻译：设$A(n, d)$表示长度为$n$、最小汉明距离为$d$的二元码的最大尺寸。研究$A(n, d)$，包括确定其值以及推导对于大$n$的$A(n, d)$的界，是编码理论中最基础的课题之一。本文探讨了在大最小距离情形下$A(n, d)$的新的下界和上界，特别是当$d = n/2 - \Omega(\sqrt{n})$时。首先，通过仔细选择二元扩域中校验多项式的特定根，我们构造了一类新的循环码，其长度为$n= 2^m -1$，距离$d \geq n/2 - 2^{c-1}\sqrt{n}$，尺寸为$n^{c+1/2}$，适用于任意$m\geq 4$和满足$0 \leq c \leq m/2 - 1$的整数$c$。这些码参数略逊于之前在大最小距离情形下提供已知最佳下界的Delsarte--Goethals (DG)码。然而，利用类似且扩展的码构造技术，我们展示了一系列循环码，这些码改进了DG码，并在更窄的最小距离$d$范围内（特别是当$d = n/2 - \Omega(n^{2/3})$时）提供了最佳下界。此外，通过利用Delsarte线性规划的傅里叶分析视角，我们获得了关于$A(n, n/2 - \rho\sqrt{n})$（其中$\rho\in (0.5, 9.5)$）的上界，这些上界在$n$上呈多项式规模。据作者所知，Barg和Nogin \cite{barg2006spectral}给出的上界是该情形下唯一已知的呈多项式规模的上界。我们通过数值计算证明，在指定的大最小距离情形下，我们的上界优于Barg-Nogin上界。