Bounds on the Asymptotic Coding Gain of Long Binary Block Codes

For some time it has been known that, for fixed code length n , binary BCH codes appear to be most efficient when the number of information bits k is between 1/4 n and 3/4 n [1, p. 443], [2, p. 219]. In this correspondence the efficiency of block codes on an binary-quantized additive white Gaussian noise channel is analyzed as a function of the code rate r = k/n for hard decision decoding. A closed form analytical expression for the upper and lower bounds on block code performance is derived for large code lengths n . They show that, for best codes, a relatively broad maximum occurs for rates of approximately 0.4. The performance of the BCH codes is also compared with the bounds.