New Bounds on the Size of Binary Codes With Large Minimum Distance - Pang, James Chin-Jen; Mahdavifar, Hessam; Pradhan, S. Sandeep (IEEE journal on selected areas in information theory, 2023)
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 - Ω (\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 ≥ n/2 - 2^{c-1}\sqrt {n} , and size n^{c+1/2} , for any m≥ 4 and any integer c with 0 ≤ c ≤ 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 - Ω (n^{2/3}) . Furthermore, by leveraging a Fourier-analytic view of Delsarte's linear program, upper bounds on A(n, \left \lceil{ n/2 - ρ \sqrt {n}\, }\right \rceil) with ρ \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.