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

Let <inline-formula> <tex-math notation="LaTeX">$A(n, d)$ </tex-math></inline-formula> denote the maximum size of a binary code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and minimum Hamming distance <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>. Studying <inline-formula> <tex-math notation="LaTeX">$A(n, d)$ </tex-math></inline-formula>, including efforts to determine it as well to derive bounds on <inline-formula> <tex-math notation="LaTeX">$A(n, d)$ </tex-math></inline-formula> for large <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>’s, is one of the most fundamental subjects in coding theory. In this paper, we explore new lower and upper bounds on <inline-formula> <tex-math notation="LaTeX">$A(n, d)$ </tex-math></inline-formula> in the large-minimum distance regime, in particular, when <inline-formula> <tex-math notation="LaTeX">$d = n/2 - \Omega (\sqrt {n})$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$n= 2^{m} -1$ </tex-math></inline-formula>, distance <inline-formula> <tex-math notation="LaTeX">$d \geq n/2 - 2^{c-1}\sqrt {n}$ </tex-math></inline-formula>, and size <inline-formula> <tex-math notation="LaTeX">$n^{c+1/2}$ </tex-math></inline-formula>, for any <inline-formula> <tex-math notation="LaTeX">$m\geq 4$ </tex-math></inline-formula> and any integer <inline-formula> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$0 \leq c \leq m/2 - 1$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>, in particular, when <inline-formula> <tex-math notation="LaTeX">$d = n/2 - \Omega (n^{2/3})$ </tex-math></inline-formula>. Furthermore, by leveraging a Fourier-analytic view of Delsarte’s linear program, upper bounds on <inline-formula> <tex-math notation="LaTeX">$A(n, \left \lceil{ n/2 - \rho \sqrt {n}\, }\right \rceil)$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$\rho \in (0.5, 9.5)$ </tex-math></inline-formula> are obtained that scale polynomially in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> in this regime. We numerically demonstrate that our upper bound improves upon the Barg-Nogin upper bound in the specified high-minimum distance regime.