A comment on "Systematic single asymmetric error-correcting Codes"

Bose and Al-Bassam designed quasi-systematic single asymmetric error-correcting codes that are able to encode k information bits in a systematic way, but cannot encode all 2^k information words. However, "Construction I" of their paper does not provide an efficient way for computing the number of codewords. In this correspondence, we derive a recursive relation based on generating functions in order to compute the maximum number of codewords in such codes. This recursion allows us to determine those numbers for large code lengths which may not be feasible otherwise.