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We study and propose schemes that map messages onto constant-weight codewordsusing variable-length prefixes. We provide polynomial-time computable formulasthat estimate the average number of redundant bits incurred by our schemes. Inaddition to the exact formulas, we also perform an asymptotic analysis anddemonstrate that our scheme uses $\frac12 \log n+O(1)$ redundant bits to encodemessages into length-$n$ words with weight $(n/2)+{\sf q}$ for constant ${\sfq}$. We also propose schemes that map messages into balanced codebooks witherror-correcting capabilities. For such schemes, we provide methods toenumerate the average number of redundant bits.

Average Redundancy of Variable-Length Balancing Schemes \`{a} la Knuth