8 research outputs found
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
Restricting Dyck Paths and 312-avoiding Permutations
Dyck paths having height at most and without valleys at height are
combinatorially interpreted by means of 312-avoding permutations with some
restrictions on their \emph{left-to-right maxima}. The results are obtained by
analyzing a restriction of a well-known bijection between the sets of Dyck
paths and 312-avoding permutations. We also provide a recursive formula
enumerating these two structures using ECO method and the theory of production
matrices. As a further result we obtain a family of combinatorial identities
involving Catalan numbers
Comma-free Codes Over Finite Alphabets
Comma-free codes have been widely studied in the last sixty years, from points of view as diverse as biology, information theory and combinatorics. We develop new methods to study comma-free codes achieving the maximum size, given the cardinality of the alphabet and the length of the words. Specifically, we are interested in counting the number of such codes. We provide (two different proofs for) a closed-formula. The approach introduced is further developed to tackle well-known sub-families of comma-free codes, such as self-complementary and (generalisations of) non-overlapping codes. We also study codes that are not contained in strictly larger ones. For instance, we determine the maximal size of self-complementary comma-free codes and the number of codes reaching the bound. We provide a characterisation of-letter non-overlapping codes (over an alphabet of cardinality n), which allows us to devise the number of such codes that are not contained in any strictly larger one. Our approach mixes combinatorial and graph-theoretical arguments