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

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 $A(n, \left \lceil{ n/2 - \rho \sqrt {n}\, }\right \rceil)$ 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 (2006) 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.