26 research outputs found
Cross-Bifix-Free Codes Within a Constant Factor of Optimality
A cross-bifix-free code is a set of words in which no prefix of any length of
any word is the suffix of any word in the set. Cross-bifix-free codes arise in
the study of distributed sequences for frame synchronization. We provide a new
construction of cross-bifix-free codes which generalizes the construction in
Bajic (2007) to longer code lengths and to any alphabet size. The codes are
shown to be nearly optimal in size. We also establish new results on Fibonacci
sequences, that are used in estimating the size of the cross-bifix-free codes
Non-Overlapping Codes
We say that a -ary length code is \emph{non-overlapping} if the set of
non-trivial prefixes of codewords and the set of non-trivial suffices of
codewords are disjoint. These codes were first studied by Levenshtein in 1964,
motivated by applications in synchronisation. More recently these codes were
independently invented (under the name \emph{cross-bifix-free} codes) by
Baji\'c and Stojanovi\'c.
We provide a simple construction for a class of non-overlapping codes which
has optimal cardinality whenever divides . Moreover, for all parameters
and we show that a code from this class is close to optimal, in the
sense that it has cardinality within a constant factor of an upper bound due to
Levenshtein from 1970. Previous constructions have cardinality within a
constant factor of the upper bound only when is fixed.
Chee, Kiah, Purkayastha and Wang showed that a -ary length
non-overlapping code contains at most codewords; this bound is
weaker than the Levenshtein bound. Their proof appealed to the application in
synchronisation: we provide a direct combinatorial argument to establish the
bound of Chee \emph{et al}.
We also consider codes of short length, finding the leading term of the
maximal cardinality of a non-overlapping code when is fixed and
. The largest cardinality of non-overlapping codes of
lengths or less is determined exactly.Comment: 14 pages. Extra explanations added at some points, and an extra
citation. To appear in IEEE Trans Information Theor