Abstract
The error coefficient of a linear code, defined as the number of its minimum weight codewords, is a key performance metric to evaluate codes with a given length, dimension, and minimum distance. In this paper, we propose novel approaches, different from existing methods, to produce three new families of binary optimal linear codes with the smallest possible error coefficients. These codes are known as asymptotic frame error rate (AFER)-optimal codes, achieving the best known performance in the additive white Gaussian noise channel and under maximum-likelihood decoding. In particular, we solve a conjecture originally proposed by Li et al. (2025).