K-Comma Codes and Their Generalizations

In this paper, we introduce the notion of k-comma codes a proper generalization of the notion of comma-free codes. For a given positive integer k, a k-comma code is a set L over an alphabet Σ with the property that LΣL ∩ ΣLΣ = ∅. Informally, in a k-comma code, no codeword can be a subword of the catenation of two other codewords separated by a “comma” of length k. A k-comma code is indeed a code, that is, any sequence of codewords is uniquely decipherable. We extend this notion to that of k-spacer codes, with commas of length less than or equal to a given k. We obtain several basic properties of k-comma codes and their generalizations, k-comma intercodes, and some relationships between the families of k-comma intercodes and other classical families of codes, such as infix codes and bifix codes. Moreover, we introduce the notion of n-k-comma intercodes, and obtain, for each k ≥ 0, several hierarchical relationships among the families of n-k-comma intercodes, as well as a characterization of the family of 1-k-comma intercodes. Address for correspondence: Bo Cui, Department of Computer Science, University of Western Ontario, London, Ontario, Canada, N6A 5B7 ∗This research was supported by The Natural Sciences and Engineering Council of Canada Discovery Grant R2824A01, and Canada Research Chair Award to L.K. 1002 Bo Cui, Lila Kari, and Shinnosuke Seki /K-Comma Codes and Their Generalizations