431 research outputs found
Elusive Codes in Hamming Graphs
We consider a code to be a subset of the vertex set of a Hamming graph. We
examine elusive pairs, code-group pairs where the code is not determined by
knowledge of its set of neighbours. We construct a new infinite family of
elusive pairs, where the group in question acts transitively on the set of
neighbours of the code. In our examples, we find that the alphabet size always
divides the length of the code, and prove that there is no elusive pair for the
smallest set of parameters for which this is not the case. We also pose several
questions regarding elusive pairs
Feynman graphs and the large dimensional limit of multipartite entanglement
We are interested in the properties of multipartite entanglement of a system
composed by -level parties (qudits).
Focussing our attention on pure states we want to tackle the problem of the
maximization of the entanglement for such systems. In particular we effort the
problem trying to minimize the purity of the system. It has been shown that not
for all systems this function can reach its lower bound, however it can be
proved that for all values of a can always be found such that the lower
bound can be reached.
In this paper we examine the high-temperature expansion of the distribution
function of the bipartite purity over all balanced bipartition considering its
optimization problem as a problem of statistical mechanics. In particular we
prove that the series characterizing the expansion converges and we analyze the
behavior of each term of the series as .Comment: 29 pages, 11 figure
On palimpsests in neural memory: an information theory viewpoint
The finite capacity of neural memory and the
reconsolidation phenomenon suggest it is important to be able
to update stored information as in a palimpsest, where new
information overwrites old information. Moreover, changing
information in memory is metabolically costly. In this paper, we
suggest that information-theoretic approaches may inform the
fundamental limits in constructing such a memory system. In
particular, we define malleable coding, that considers not only
representation length but also ease of representation update,
thereby encouraging some form of recycling to convert an old
codeword into a new one. Malleability cost is the difficulty of
synchronizing compressed versions, and malleable codes are of
particular interest when representing information and modifying
the representation are both expensive. We examine the tradeoff
between compression efficiency and malleability cost, under a
malleability metric defined with respect to a string edit distance.
This introduces a metric topology to the compressed domain. We
characterize the exact set of achievable rates and malleability as
the solution of a subgraph isomorphism problem. This is all done
within the optimization approach to biology framework.Accepted manuscrip
Maximum Weight Spectrum Codes
In the recent work \cite{shi18}, a combinatorial problem concerning linear
codes over a finite field \F_q was introduced. In that work the authors
studied the weight set of an linear code, that is the set of non-zero
distinct Hamming weights, showing that its cardinality is upper bounded by
. They showed that this bound was sharp in the case ,
and in the case . They conjectured that the bound is sharp for every
prime power and every positive integer . In this work quickly
establish the truth of this conjecture. We provide two proofs, each employing
different construction techniques. The first relies on the geometric view of
linear codes as systems of projective points. The second approach is purely
algebraic. We establish some lower bounds on the length of codes that satisfy
the conjecture, and the length of the new codes constructed here are discussed.Comment: 19 page
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
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