Bounds on the Asymptotic Coding Gain of Long Binary Block Codes

For some time it has been known that, for fixed code lengthn, binary BCH codes appear to be most efficient when the number of information bitskis between1/4 nand3/4 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 rater = k/nfor hard decision decoding. A closed form analytical expression for the upper and lower bounds on block code performance is derived for large code lengthsn. 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.