  This study focuses on efficient schemes for enumerative coding of
$\sigma$--ary sequences by mainly borrowing ideas from \"Oktem & Astola's
\cite{Oktem99} hierarchical enumerative coding and Schalkwijk's
\cite{Schalkwijk72} asymptotically optimal combinatorial code on binary
sequences. By observing that the number of distinct $\sigma$--dimensional
vectors having an inner sum of $n$, where the values in each dimension are in
range $[0...n]$ is $K(\sigma,n) = \sum_{i=0}^{\sigma-1} {{n-1} \choose
{\sigma-1-i}} {{\sigma} \choose {i}}$, we propose representing $C$ vector via
enumeration, and present necessary algorithms to perform this task. We prove
$\log K(\sigma,n)$ requires approximately $ (\sigma -1) \log (\sigma-1) $ less
bits than the naive $(\sigma-1)\lceil \log (n+1) \rceil$ representation for
relatively large $n$, and examine the results for varying alphabet sizes
experimentally. We extend the basic scheme for the enumerative coding of
$\sigma$--ary sequences by introducing a new method for large alphabets. We
experimentally show that the newly introduced technique is superior to the
basic scheme by providing experiments on DNA sequences.
