Enumeration of sequences with large alphabets

Abstract

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.