38 research outputs found
Some combinatorial invariants determined by Betti numbers of Stanley-Reisner ideals
The papers in this thesis are not available in Munin:
Paper 1: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'Betti numbers associated to
the facet ideal of a matroid', available in Bulletin of the Brazilian Mathematical Society 45 no. 4, 727-744
Paper 2: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'A generalization of weight
polynomials to matroids', manuscript
Paper 3: Jan Roksvold, Hugues Verdure (2015): 'Betti numbers of skeletons', manuscriptThe thesis contains new results on the connection between the algebraic properties of certain ideals of a polynomial ring and properties of error-correcting linear codes, matroids and simplicial complexes. We demonstrate that the graded Betti numbers of the facet ideal of a matroid are determined by the Betti numbers of the blocks of the matroid. The extended weight enumerator of coding theory is generalized to matroids. We show that this generalization is equivalent to the Tutte-polynomial, and that the coefficients of this polynomial is determined by Betti numbers of the Stanley-Reisner ideal of the matroid and its elongations. The Betti numbers of the Stanley-Reisner ring of a skeleton of a simplicial complex is demonstrated to be an integral linear combination of the Betti numbers associated to the original complex
Results on zeta functions for codes
We give a new and short proof of the Mallows-Sloane upper bound for self-dual
codes. We formulate a version of Greene's theorem for normalized weight
enumerators. We relate normalized rank-generating polynomials to two-variable
zeta functions. And we show that a self-dual code has the Clifford property,
but that the same property does not hold in general for formally self-dual
codes.Comment: 12 page
MacWilliams Identities for -tuple Weight Enumerators
Since MacWilliams proved the original identity relating the Hamming weight
enumerator of a linear code to the weight enumerator of its dual code there
have been many different generalizations, leading to the development of
-tuple support enumerators. We prove a generalization of theorems of Britz
and of Ray-Chaudhuri and Siap, which build on earlier work of Kl{\o}ve,
Shiromoto, Wan, and others. We then give illustrations of these -tuple
weight enumerators.Comment: 17 pages. Accepted to SIAM Journal on Discrete Mathematic
Codes, arrangements, matroids, and their polynomial links
Codes, arrangements, matroids, and their polynomial links Many mathematical objects are closely related to each other. While studying certain aspects of a mathematical object, one tries to find a way to "view" the object in a way that is most suitable for a specific problem. Or, in other words, one tries to find the best way to model the problem. Many related fields of mathematics have evolved from one another this way. In practice, it is very useful to be able to transform a problem into other terminology: it gives a lot more available knowledge and that can be helpful to solve a problem. This thesis deals with various closely related fields in discrete mathematics, starting from linear error-correcting codes and their weight enumerator. We can generalize the weight enumerator in two ways, to the extended and generalized weight enumerators. The set of generalized weight enumerators is equivalent to the extended weight enumerator. Summarizing and extending known theory, we define the two-variable zeta polynomial of a code and its generalized zeta polynomial. These polynomials are equivalent to the extended and generalized weight enumerator of a code. We can determine the extended and generalized weight enumerator using projective systems. This calculation is explicitly done for codes coming from finite projective and affine spaces: these are the simplex code and the first order Reed-Muller code. As a result we do not only get the weight enumerator of these codes, but it also gives us information on their geometric structure. This is useful information in determining the dimension of geometric designs. To every linear code we can associate a matroid that is representable over a finite field. A famous and well-studied polynomial associated to matroids is the Tutte polynomial, or rank generating function. It is equivalent to the extended weight enumerator. This leads to a short proof of the MacWilliams relations for the extended weight enumerator. For every matroid, its flats form a geometric lattice. On the other hand, every geometric lattice induces a simple matroid. The Tutte polynomial of a matroid determines the coboundary polynomial of the associated geometric lattice. In the case of simple matroids, this becomes a two-way equivalence. Another polynomial associated to a geometric lattice (or, more general, to a poset) is the Möbius polynomial. It is not determined by the coboundary polynomial, neither the other way around. However, we can give conditions under which the Möbius polynomial of a simple matroid together with the Möbius polynomial of its dual matroid defines the coboundary polynomial. The proof of these relations involves the two-variable zeta polynomial, that can be generalized from codes to matroids. Both matroids and geometric lattices can be truncated to get an object of lower rank. The truncated matroid of a representable matroid is again representable. Truncation formulas exist for the coboundary and Möbius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the known truncation formula of the Tutte polynomial of a matroid. Several examples and counterexamples are given for all the theory. To conclude, we give an overview of all polynomial relations
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
Congruence conditions, parcels, and Tutte polynomials of graphs and matroids
Let be a matrix and be the matroid defined by linear dependence on
the set of column vectors of Roughly speaking, a parcel is a subset of
pairs of functions defined on to an Abelian group satisfying a
coboundary condition (that is a flow over relative to ) and a
congruence condition (that the size of the supports of and satisfy some
congruence condition modulo an integer). We prove several theorems of the form:
a linear combination of sizes of parcels, with coefficients roots of unity,
equals an evaluation of the Tutte polynomial of at a point
on the complex hyperbola $(\lambda - 1)(x-1) = |A|.
Applications of ordered weights in information transmission
This dissertation is devoted to a study of a class of linear codes related to a particular metric space that generalizes the Hamming space in that the metric function is defined by a partial order on the set of coordinates of the vector.
We begin with developing combinatorial and linear-algebraic aspects of linear ordered codes. In particular, we define multivariate rank enumerators for linear codes and show that they form a natural set of invariants in the study of the duality of linear codes. The rank enumerators are further shown to be connected to the shape distributions of linear codes, and enable us to give a simple proof of a MacWilliams-like theorem for the ordered case. We also pursue the connection between linear codes and matroids in the ordered case and show that the rank enumerator can be thought of as an instance of the classical matroid invariant called the Tutte polynomial. Finally, we consider the distributions of support weights of ordered codes and their expression via the rank enumerator. Altogether, these results generalize a group of well-known results for codes in the Hamming space to the ordered case.
Extending the research in the first part, we define simple probabilistic channel models that are in a certain sense matched to the ordered distance, and prove several results related to performance of linear codes on such channels. In particular, we define ordered wire-tap channels and establish several results related to the use of linear codes for reliable and secure transmission in such channel models.
In the third part of this dissertation we study polar coding schemes for channels with nonbinary input alphabets. We construct a family of linear codes that achieve the capacity of a nonbinary symmetric discrete memoryless channel with input alphabet of size q=2^r, r=2,3,.... A new feature of the coding scheme that arises in the nonbinary case is related to the emergence of several extremal configurations for the polarized data symbols. We establish monotonicity properties of the configurations and use them to show that total transmission rate approaches the symmetric capacity of the channel. We develop these results to include the case of ``controlled polarization'' under which the data symbols polarize to any predefined set of extremal configurations. We also outline an application of this construction to data encoding in video sequences of the MPEG-2 and H.264/MPEG-4 standards