On the Error Coefficients of Asymptotic Frame Error Rate Optimal Binary Linear Codes

A binary linear code is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">asymptotic frame error rate (AFER)-optimal</i> if it achieves the maximum possible value of the minimum distance while having the smallest value of the corresponding error coefficient. Over the additive white Gaussian noise channel and under maximum-likelihood decoding, AFER-optimal codes attain the best possible asymptotic frame error rate at high signal-to-noise ratio. In this paper, we present several bounds on the smallest error coefficients of binary linear codes and give several constructions of AFER-optimal binary linear codes. Many examples confirm that our bounds are sharp on numerous occasions. In addition, we give two families of AFER-optimal codes that respectively attain the proposed bounds with equality.