In this paper, we present some new constructions of the binary linear block codes (BLBCs) that achieve the largest possible minimum Hamming distance and the lowest possible number of codewords of this Hamming weight (also known as error coefficient), and they are said to be [d, A d ]-optimal (n, k) linear codes. These (n, k) BLBCs give the best possible frame error rate (FER) in the asymptotic regime under maximum-likelihood decoding over the additive white Gaussian noise channel. Specifically, for all positive integers k and m, and 0 ≤ l ≤ k − 1, we give the constructions of ((2 k − 1)m + l, k), ((2 k − 1) + k, k + 1), (15m + 4, 4), (15m + 6, 4), (12, 5), and (33, 5) BLBCs. Many of these BLBCs have A d = 1, and some achieve the lower bound on A d, which asserts their optimality. Our constructions show the asymptotic E b /N o gain of up to 1.24 dB over their d-optimal counterpart at a FER of 10 −7 .