New Bounds on the Size of Binary Codes with Large Minimum Distance

Let A(n, d) denote the maximum number of code-words in a binary code of length n and minimum Hamming distance d. Deriving upper and lower bounds on A(n, d) has 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 $A\left({n,d = n/2 - {2^{c - 1}}\sqrt n }\right) \geq {n^c}$, where c is an integer with 1 ⩽ c ⩽ m/2 − 1. With a Fourier-analytic view of Delsarte’s linear program, novel upper bounds on $A(n,n/2 - \sqrt n ){\text{ and }}A(n,n/2 - 2\sqrt n )$ are obtained, and, to the best of the authors’ knowledge, are the first upper bounds scaling polynomially in n for the regime with $d = n/2 - \Theta (\sqrt n )$.