246,731 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
In search of maximum non-overlapping codes
Non-overlapping codes are block codes that have arisen in diverse contexts of
computer science and biology. Applications typically require finding
non-overlapping codes with large cardinalities, but the maximum size of
non-overlapping codes has been determined only for cases where the codeword
length divides the size of the alphabet, and for codes with codewords of length
two or three. For all other alphabet sizes and codeword lengths no
computationally feasible way to identify non-overlapping codes that attain the
maximum size has been found to date. Herein we characterize maximal
non-overlapping codes. We formulate the maximum non-overlapping code problem as
an integer optimization problem and determine necessary conditions for
optimality of a non-overlapping code. Moreover, we solve several instances of
the optimization problem to show that the hitherto known constructions do not
generate the optimal codes for many alphabet sizes and codeword lengths. We
also evaluate the number of distinct maximum non-overlapping codes
The disjointness of stabilizer codes and limitations on fault-tolerant logical gates
Stabilizer codes are a simple and successful class of quantum
error-correcting codes. Yet this success comes in spite of some harsh
limitations on the ability of these codes to fault-tolerantly compute. Here we
introduce a new metric for these codes, the disjointness, which, roughly
speaking, is the number of mostly non-overlapping representatives of any given
non-trivial logical Pauli operator. We use the disjointness to prove that
transversal gates on error-detecting stabilizer codes are necessarily in a
finite level of the Clifford hierarchy. We also apply our techniques to
topological code families to find similar bounds on the level of the hierarchy
attainable by constant depth circuits, regardless of their geometric locality.
For instance, we can show that symmetric 2D surface codes cannot have non-local
constant depth circuits for non-Clifford gates.Comment: 8+3 pages, 2 figures. Comments welcom
Constructions and bounds for codes with restricted overlaps
Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which over-laps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein [9] in 1964
Constructions and bounds for codes with restricted overlaps
Non-overlapping codes have been studied for almost 60 years. In such a code,
no proper, non-empty prefix of any codeword is a suffix of any codeword. In
this paper, we study codes in which overlaps of certain specified sizes are
forbidden. We prove some general bounds and we give several constructions in
the case of binary codes. Our techniques also allow us to provide an
alternative, elementary proof of a lower bound on non-overlapping codes due to
Levenshtein in 1964.Comment: 25 pages. Extra citations, typos corrected and explanations expande
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