129 research outputs found
One-adhesive polymatroids
Adhesive polymatroids were defined by F. Mat\'u\v{s} motivated by entropy
functions. Two polymatroids are adhesive if they can be glued together along
their joint part in a modular way; and are one-adhesive, if one of them has a
single point outside their intersection. It is shown that two polymatroids are
one-adhesive if and only if two closely related polymatroids have any
extension. Using this result, adhesive polymatroid pairs on a five-element set
are characterized
Polar Codes for the m-User MAC
In this paper, polar codes for the -user multiple access channel (MAC)
with binary inputs are constructed. It is shown that Ar{\i}kan's polarization
technique applied individually to each user transforms independent uses of a
-user binary input MAC into successive uses of extremal MACs. This
transformation has a number of desirable properties: (i) the `uniform sum rate'
of the original MAC is preserved, (ii) the extremal MACs have uniform rate
regions that are not only polymatroids but matroids and thus (iii) their
uniform sum rate can be reached by each user transmitting either uncoded or
fixed bits; in this sense they are easy to communicate over. A polar code can
then be constructed with an encoding and decoding complexity of
(where is the block length), a block error probability of o(\exp(- n^{1/2
- \e})), and capable of achieving the uniform sum rate of any binary input MAC
with arbitrary many users. An application of this polar code construction to
communicating on the AWGN channel is also discussed
On the optimization of bipartite secret sharing schemes
Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the tripartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.Peer ReviewedPostprint (author's final draft
Finding lower bounds on the complexity of secret sharing schemes by linear programming
Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants.
By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing schemePeer ReviewedPostprint (author's final draft
A Note on Extension Properties and Representations of Matroids
We discuss several extension properties of matroids and polymatroids and
their application as necessary conditions for the existence of different
matroid representations, namely linear, folded linear, algebraic, and entropic
representations. Iterations of those extension properties are checked for
matroids on eight and nine elements by means of computer-aided explorations,
finding in that way several new examples of non-linearly representable
matroids. A special emphasis is made on sparse paving matroids on nine points
containing the tic-tac-toe configuration. We present a clear description of
that family and we analyze extension properties on those matroids and their
duals
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