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 $m$-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 $m$-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 $O(n \log n)$ (where $n$ 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