On the fast computation of the weight enumerator polynomial and the tt value of digital nets over finite abelian groups

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$ dimensional unit cube $[0,1]^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further

Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition coincides the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures

Bounds on the size of codes

In this dissertation we determine new bounds and properties of codes in three different finite metric spaces, namely the ordered Hamming space, the binary Hamming space, and the Johnson space. The ordered Hamming space is a generalization of the Hamming space that arises in several different problems of coding theory and numerical integration. Structural properties of this space are well described in the framework of Delsarte's theory of association schemes. Relying on this theory, we perform a detailed study of polynomials related to the ordered Hamming space and derive new asymptotic bounds on the size of codes in this space which improve upon the estimates known earlier. A related project concerns linear codes in the ordered Hamming space. We define and analyze a class of near-optimal codes, called near-Maximum Distance Separable codes. We determine the weight distribution and provide constructions of such codes. Codes in the ordered Hamming space are dual to a certain type of point distributions in the unit cube used in numerical integration. We show that near-Maximum Distance Separable codes are equivalently represented as certain near-optimal point distributions. In the third part of our study we derive a new upper bound on the size of a family of subsets of a finite set with restricted pairwise intersections, which improves upon the well-known Frankl-Wilson upper bound. The new bound is obtained by analyzing a refinement of the association scheme of the Hamming space (the Terwilliger algebra) and intertwining functions of the symmetric group. Finally, in the fourth set of problems we determine new estimates on the size of codes in the Johnson space. We also suggest a new approach to the derivation of the well-known Johnson bound for codes in this space. Our estimates are often valid in the region where the Johnson bound is vacuous. We show that these methods are also applicable to the case of multiple packings in the Hamming space (list-decodable codes). In this context we recover the best known estimate on the size of list-decodable codes in a new way

Entanglement and quantum combinatorial designs

We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way than quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called $k$-uniform, i.e. multipartite pure states such that every reduction to $k$ parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite $k$-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome

Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

MacWilliams-type identities for fragment and sphere enumerators

AbstractLet P=n11⊕⋯⊕nt1 be the poset given by the ordinal sum of the antichains ni1 with ni elements. We derive MacWilliams-type identities for the fragment and sphere enumerators, relating enumerators for the dual C⊥ of the linear code C on P and those for C on the dual poset P̌. The linear changes of variables appearing in the identities are explicit. So we obtain, for example, the P-weight distribution of C⊥ as the P̌-weight distribution times an invertible matrix which is a generalization of the Krawtchouk matrix