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

TL;DR

This paper introduces new bounds on the maximum size of binary codes with large minimum distances, providing improved constructions and upper bounds in the regime where the minimum distance is close to half the code length.

Contribution

It presents novel cyclic code constructions and improved upper bounds for binary codes with large minimum distances, advancing understanding in the high-distance regime.

Findings

New cyclic code constructions with improved parameters

Enhanced lower bounds surpassing previous codes in certain regimes

Polynomially scaling upper bounds that improve upon existing bounds

Abstract

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.